![iv PREFACE
newclass of matrix-valued function of particular importance in construction
of an appropriate Liapunov function for the system under consideration.
It is knownthat the problem of stability is important not only for the
continuous systems modeledby ordinary differeatial equations. Therefore,
in this monographthe methods of qualitative analysis are presented for
discrete-time and impulsive systems. Further, in view of the importance
of the problem of estimating the domains of asymptotic stability, a new
methodfor its solution is set out in a separate chapter.
The monographcontains five chapter and is arranged as follows.
Thefirst chapter contains all necessary results associated with the me-
thod of matrix-valued Liapunovfunctions. It also provides general informa-
tion on scalar and vector functions including the cone-valued ones. General
theorems on various types of stability of the equilibrium state of the sys-
tems cited in this chapter are basic for establishing the sufficient stability
tests in subsequent chapters.
The second chapter deals with the construction of matrix-valued func-
tions and corresponding scalar auxiliary Liapunov functions. Here new
methodsof the initial system decomposition are discussed, including those
of hierarchical decomposition. Thecorresponding sufficient tests for var-
ious types of stability and illustrative examples are presented for every
case under consideration. Alongwith the classical notion of stability ma-
jor attention is paid to newtypes of motionstability, in particular, to the
exponential polystability of separable motions as well as the integral and
Lipschitz stability.
The third chapter addresses the methods of stability analysis of dis-
crete-time systems. Our attention is focussed mostly on the development
of the methodof matrix-valued functions in stability theory of discrete-time
systems.
In the fourth chapter the problems of dynamics of nonlinear systems
in the presence of impulsive perturbations are discussed. The method of
matrix-valued Liapunovfunctions is adapted here for the class of impulsive
systems that were studied before via the scalar Liapunov function. The
proposed development of the direct Liapunov method for the given class of
systems enables us to makean algorithm constructing the appropriate Liaptmov
functions and to increase efficiency of this method.
In the final chapter the problem of estimating the domains of
asymptotic stability is discussed in terms of the methodof matrix-valued
Liapunov functions. By meansof numerous examples considered earlier by
Abdullin,Anapolskii, et al. [1], Michel,Sarabudla, et al. [1], and ~iljak [1] it](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-15-320.jpg)
![2 1. PRELIMINARIES
large scale systems. Detailed discussion of possibilities of these approaches
mayprove to be useful for manybeginners in the field.
In the final Section 1.8, the formulations of theoremsof the direct Lia-
punov method are set out based on matrix-valued functions and intended
for application in stability investigation of large scale systems.
Generally, the results of this chapter are necessary to get a clear idea
of the results presented in Chapters 2-5. Throughout Chapters 2-5 refe-
rences to one or an other section of Chapter 1 are made.
1.2 Nonlinear Continuous Systems
1.2.1 General equations of nonlinear dynamics
Thesystems without nonintegrability differential constraints represent a
wide class of mechanicalsystems with a finite numberof degrees of freedom.
Let the state of such system in the phase space Rn, n = 2k, be determined
by the vectors
q = (ql,..., qk) w and ~ = (~1,..., ~k)
w.
It is knownthat the general motion equations of such a mechanical system
are
d (or’~ OT _ Us, s = 1,2,...,k.
(1.2.1)
Here T is the kinetic energy of the mechanical system and Us are the
generalized forces.
Thesystem of equations (1.2.1) is simplified, if for the forces affecting
the system a force function U = U(t,q~,... ,qk) exists such that
OU
us= , s=l,2,...,k.
Thesimplified system obtained so far,
d(O(T+V)~ O(T+U)_o, s = 1,2,...,k,
dt 00, ] Oqs](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-23-320.jpg)
![1.2 NONLINEAR
CONTINUOUS
SYSTEMS 3
can be presented in the canonical form
dqs OR dps OR
- s = 1,2,...,k,
dt Op~ dt Oqs
OT and R = T2-To-U. Here To is the totality of the
where p~ = 04--7
velocity-independent terms in the expression of the kinetic energy, and T2
is the totality of the secondorder terms with respect to velocities.
Thequalitative analysis of equations (1.2.1) and its particular cases
the principle point of the investigations in nonlinear dynamicsof continuous
systems.
1.2.2 Perturbed motion equations
Undercertain assumptions the equations (1.2.1) can be represented in the
scalar form
dy_~i = Y~(t, Yl,... ,Y2k), i =1,
2k,
or in the equivalent vector form
(1.2.2)
dy = Y(t, y),
dt
where* y = (Yl,Y2,... ,Y2k)T E 2k and Y= (Y1,Y~,... ,Y2k)T, Y:7-×
R
ek -+ R
~k. A motionof (1.2.2) is denotedby y(t; to, Y0), ~(to; to, Y0)
and the reference motionr/r(t; to, Yro). Fromthe physical point of view the
reference motion should be realizable by the system. Fromthe mathe-
matical point of view this meansthat the reference motion is a solution
of (1.2.2),
(1.2.3)
&?r
(t; t0, Yro)_=Y[t, ~/~(t; to, Y~0)].
dt
Let the Liapunov transformation of coordinates be used,
(1.2.4) x = y - Yr,
where yr(t) -- ~lr(t;to,Yro). Let f: T x R
~k -~ R
2k be defined by
(1.2.5) f(t, x) =Y[t, y~(t) +x] - Y[t, Yr].
! ! ~T
*InLiapunov’s
notationy ---- (ql,q2,... ,qk, ql,q2,’’’ ,qkJ](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-24-320.jpg)
![4 1. PRELIMINARIES
It is evident that
(1.2.6) f(t,O) =_
Now
(1.2.2)- (1.2.5) yield
dx
(1.2.7)
d~-=f(t, x).
In this way, the behavior of perturbed motions related to the reference
motion (in total coordinates) is represented by the behavior of the state
deviation x with respect to the zero state deviation. The reference motion
in the total coordinates Yi is represented by the zero deviation x = 0 in
state deviation coordinates xi. With this in mind, the following result em-
phasizes complete generality of both Liapunov’s second methodand results
represented by Liapunov [1] for the system (1.2.7). Let Q: R2k -r R
’~,
n = 2k is admissible but not required.
In the monograph
Grujid, et al. [1] the following assertion is proved.
Proposition 1.2.1. Stability of x = 0 of systena (1.2.7) with respect
to Q =x is necessary and sufficient for stability of the reference motion
of system (1.2.2) with respect to every vector function Qthat is continuous
in y.
This theorem reduced the problemof the stability of the reference motion
of (1.2.2) with respect to Q to the stability problem of x = 0 of (1.2.7)
with respect to x.
For the sake of clarity westate
Definition 1.2.1. State x* of the system (1.2.7) is its equilibrium state
over 7~ iff
(1.2.8) x(t;to,x*) = x*, for all t E To, and to
Theexpression "over 7~" is omitted iff 7~ = R.
Proposition 1.2.2. For x* ~ Rn to be an equilibrium state of the
system (1.2. 7) over Ti it is necessary and sufficient that both
(i) for every to q T/ there is the unique solution x(t; to, x*) of (1.2.7),
whichis defined for all to ~ To
and
(ii) f(t,x*) = 0, for a/l t e To, and to e 7~.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-25-320.jpg)
![1.3 DEFINITIONS
OFSTABILITY 5
The conditions for existence and uniqueness of the solutions of system
(1.2.7) can be found in manywell-knownbooks by Dieudonne[1], Hale [1],
Hirsch and Smale[1], Simmons
[1], Yoshizawa[1], etc.
Thenext result provides a set of sufficient conditions for the uniqueness
of solutions for initial value problem
(1.2.9)
d-¥ =f(t, x), X(to) =
Proposition 1.2.3. Let :D C Rn+l be an open and connected set.
Assume f ¯ C(:D, Rn) and for every compact K C ~), f satisfies the
Lipschitz condition
[]f(t,x) f( t,y)[[ <_L[[x - y[[
for all (t,x), (t,y) 6 K, where L is a constant dependingonly on
Then(1.2.9) has at most one solution on any interval [to, to + c), c > 0.
Definition 1.2.2. A solution x(t;to,Xo) of (1.2.7) defined on the inter-
val (a, b) is said to be bounded
if there exists /~ > 0 suchthat [[x(t; to, x0)[]
< fl for all t ¯ (a, b), where/~maydependon each solution.
For the system (1.2.7) the following result can be easily demonstrated.
Proposition 1.2.4. Assume f ¯ C(J x Rn,Rn), where J = (a,b)
a finite or infinite interval. Let every solution of (1.2.7) is bounded.Then
every solution of (1.2.7) can be continued on the entire interval (a, b).
1.3 Definitions of Stability
Consider the differential system (1.2.7), where f ¯ C(%n,Rn).Sup-
pose that the function f is smooth enoughto guarantee existence, unique-
ness and continuous dependenceof solutions x(t; to, x0) of (1.2.7). We
present various definitions of stability (see Grujid [1] and Grujid, et al. [1]).
Definition 1.3.1. The state x = 0 of the system (1.2.7) is:
(i) stable with respect to 7~ iff for every to ¯ T~and every e > 0 there
exists 5(to,e) > 0, such that [[Xo[[ < 5(to,e) implies
all t ¯ %;](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-26-320.jpg)
![1. PRELIMINARIES
(ii) uniformly stable with respect to Toiff both (i) holds and for every
¢ > 0 the corresponding maximal ~M obeying (i) satisfies
inf[t~M(t,~): t ¯ T/] >
(iii) stable in the wholewith respect to Ti iff both (i) holds and
5M(t,e)--~+oo as e-~+oo, for all t¯T/;
(iv) uniformly stable in the wholewith respect to T, iff both (ii) and (iii)
hold;
(v) unstable with respect to 7~ iff there are to ¯ T/, e ¯ (0, +oo) and
T ¯ To, V > tO, such that for every 5 ¯ (0,+oo) there is Xo,
Ilxoll<5, forwhich
IIx(T;
to,xo)ll
>~.
Theexpression "with respect to 7~" is omitted from (i)- (v) iff 7~
These stability properties hold as t -~ +oobut not for t = +oo.
Further the definitions on solution attraction are cited. The examples
by Hahn[2], Krasovskii [1], and Vinograd [1] showedthat the attraction
property does not ensure stability.
Definition 1.3.2. The state x = 0 of the system (1.2.7) is:
(i) attractive with respect to Ti ifffor every to ¯ 7~ there exists A(to)
0 and for every ~ > 0 there exists ~’(to;zo,~) ¯ [0,+oo) such
that Ilzoll < A(to) implies IIx(t;to,Xo)ll < ¢, for all t ¯ (to
r(to; xo, ¢), +oo);
(ii) Xo-uniformlyattractive with respect to 7~ iff both (i) is true and for
every to ¯ T/ there exists A(to) > 0 and for every ~ ¯ (0, +oc)
there exists r~,[to, A(to), ~] ¯ [0, +oo) such that
sup[T,~(t0; X0,¢): X0¯ T/] =T=(7~,X0,
(iii) to-uniformly attractive with respect to 7~ iff (i) is true, there is A>
0 and for every (x0, ~) ¯ Ba× (0, +o¢) there exists ru(Ti, Xo, ~)
[0, +o¢) such that
sup[rm(to); xo, (): to ¯ Ti] = ~’u(7~,x0,](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-27-320.jpg)
![1.3 DEFINITIONS
OF STABILITY 7
(iv) uniformlyattractive with respect to Ti iff both (ii) and (iii) hold,
is, that (i) is true, there exists A> 0 and for every ~ E (0, +c~)
there is T~(T/, A, ¢) E [0, +~) such that
sup [~-m(to;x0, ~): (to, x0) ~ T/xBa] = r(T/, A,
(v) Theproperties (i)- (iv) hold "in the whole" iff (i) true for every
A(t0) ~ (0, +oo) and every to ~
Theexpression "with respect to Ti" is omitted iff T/= R.
Definitions 1.3.1 and 1.3.2 enable us to define various types of asymptotic
stability as follows.
Definition 1.3.3. The state x =0 of the system (1.2.7) is:
(i) asymptotically stable with respect to Ti iff it is both stable with
respect to T/and attractive with respect to 7~;
(ii) equi-asymptoticallystable with respect to Ti iff it is both stable with
respect to
(iii) quasi-uniformlyasymptotically stable with respect to Ti iff it is both
uniformly stable with respect to 7~ and t0-uniformly attractive with
respect to 7~;
(iv) uniformly asymptotically stable with respect to 7~ iff it is both uni-
formly stable with respect to 7~ and uniformly attractive with re-
spect to
(v) the properties (i)- (iv) "in t he w
hole" iff b oth the c orrespond-
ing stability of x = 0 and the corresponding attraction of x = 0
hold in the whole;
(vi) exponentially stable with respect to Ti iff there are A> 0 and real
numbers c~ _> 1 and fl > 0 such that HXoll < A implies
IlX(t;to,xo)]] <_~llXoll exp[-fl(t- to)],
for all teTo, and for all to
This holds "in the whole" iff it is true for A= +oo.
Theexpression "with respect to 7~" is omitted iff 7~ = R.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-28-320.jpg)
![8 1. PRELIMINARIES
1.4 Scalar~ Vector and Matrix-Valued Liapunov Functions
In order that to avoid the problemof nonlinear equations nonintegrability in
their qualitative study, Liapunov[1] suggested to apply auxiliary functions
with the normproperties. The auxiliary function, being a function of one
variable (time) on the system trajectories, allows estimating the distance
from every point of the system integral curve to the origin (to the system
equilibrium state) whentime is changing from the fixed value to E "Yr.
1.4.1 Auxiliary scalar functions
Thesimplest type of auxiliary function for system (1.2.7) is the function
(1.4.1) v(t, x) e c(’ro×RR+),v(t, 0)
Further all functions (1.4.1) allowing the solution of the problemon stability
(instability) of the equilibrium state x = 0 of system(1.2.7) are called
Liapunov functions.
Theconstruction of the Liapunovfunctions still remains one of the cen-
tral problems of stability theory. These functions should satisfy special
requirements such as the property of having a fixed sign, decreasing, radial
unboundedness, etc. The Liapunov functions are often constructed as a
quadratic form of the phase variables whosecoefficients are constants or
time functions.
The following definitions are presented according to Gantmacher[1].
Definition 1.4.1. A matrix H = (hij) ~ nxn i s:
(i) positive (negative) semi-definite iff its quadratic form V(x) =xTHx
is positive (negative) semi-definite, respectively;
(ii) positive (negative) definite iff its quadratic form V(x) = xTHx is
positive (negative) definite, respectively.
Notice that a square matrix A with all real valued elements is (semi-)
definite iff its symmetricpart As=½(A +A
T) is (semi-) definite, and
square matrix Awith complexvalued elements is (semi-) definite iff its Her-
mitian part AH= ½(A+ A*) is (semi-) definite, where A*is the transpose
conjugate matrix of the matrix A.
Now,the fundamental theorem of the stability theory - the Liapunov
matrix theorem - can be stated in the form.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-29-320.jpg)
![1.4 SCALAR, VECTORANDMATRIX-VALUED
LIAPUNOVFUNCTIONS
9
Theorem1.4.1. In order that real parts of all eigenvalues of a matrix
A, A E R
’~×’*, be negative it is necessary andsufficient that for any positive
definite symmetric matrix G, G ~ Rn×n, there exists the unique solution
H, H ~ Rn×n, of the (Liapunov) matrix equation
(1.4.2) ATH + HA = -G,
which is also positive definite symmetric matrix.
If all the characteristic roots of Ahave negative real parts wecan solve
the matrix equations (1.4.2) in closed form (see Zubov[3], and Hahn[2])
H = / esATGesA ds.
For solving the Liapunov matrix equation (1.4.2), see also Aliev and
Larin [1], Barbashin[2], Barnett and Storey [1], etc.
1.4.2 Comparison functions
Comparisonfunctions are used as upper or lower estimates of the function V
and its total time derivative. Theyare usually denoted by ~, ~: R+-4 R+.
The main contributor to the investigation of properties of and use of the
comparison functions is Hahn[2]. Whatfollows is mainly based on his
definitions and results.
Definition 1.4.2. A function ~, qo: R+-4 R+, belongs to
(i) the class K[o,~), 0 < a < +~, iffboth it is defined, continuous and
strictly increasing on [0, a) and qa(0) =
(ii) the class K iff (i) holds for a = +~, K = K[0,+~);
(iii) the class KRiff both it belongs to the class K and ~(()
as ( -4
(iv) the class L[o,a) iff both it is defined, continuous and strictly de-
creasing on [0, a) and lim [~o(() : ( -4 +oc]
(v) the class L iff (iv) holds for a = +oo, L = L[0,+oo).
Let ~-1 denote the inverse function of ~, ~0-1[~(()] -- (.
The next result wasestablished by Hahn[2].](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-30-320.jpg)
![10 1. PRELIMINARIES
Proposition 1.4.1.
(1) If %0 6 K and ¢ ~ K then %0(¢) ~
(2)If %0 6 K anda 6 L then%0(a)
6
(3)If %06 K[o,a)
and%0(a)
= th
en%0
-I6 K[
o,e);
(4) If %06 K and lim [%0((): ( -+ +oo] then%0--1 is not def ined
on +oo];
(5) If%0 6 K[o,a), %b6 K[0,a) and %0(~) > ¢(() on [0,a) then %0-1(~)
¢-1(() on [0,~3], where f~ = ¢(a).
Definition 1.4.3. A ]unction %0, %0: R+x R+-~ R+, belongs to:
(i) the class KK[o;a,~) iff both %0(0, ~) 6 K[o,a) for every
and %0(~,0)6 K[o,~) for every ~ 6 [0,a);
(ii) the class KKiff (i) holds for a = ~3 = +oo;
(iii) the class KL[o;a,~) iff both %0(0, () 6 K[o,a) for
and %0((,0) 6 L[o,~) for every ~ 6 [0,a);
(iv) the class KLiff (iii) holds for a = f~ = +oo;
(v) the class CKiff %0(t,0) = 0, %0(t,u) 6 for ev ery t 6 R+
;
(vi) the class 14 iff %06 C(R+x R",R+), inf %0(t,x) = O, (t,x)
R+ x
(vii) the class 14o iff %0 ~ C(R+ x Rn,R+), inf%0(t,x) -- 0 for
each t ~ R+;
(viii) the class ~ iff %06 C(K,R+):%0(0) = 0, and %0(w)is increasing
with respect to cone K.
Definition 1.4.4. Two]unctions %01, %026 g (or %01, %026 KR) are
said to be of the sameorder of magnitudeif there exist positive constants
a, ~, such that
< %02(0
< Z%01(¢)
for all (or for all e [%00)).
In terms of the comparisonfunction’s existence, the special properties of
functions (1.4.1) or the function
(1.4.3) v(x) 6 C(R", R+), v(0) =
applied in the analysis of the autonomoussystem
dx
(1.4.4) d-~ = g(x), g(O)
where x 6 R
n, g 6 C(R
n, Rn), are specified in the following way.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-31-320.jpg)
![1.4 SCALAR,VECTOI~
ANDMATRIX-VALUED
LIAPUNOV
FUNCTIONS 11
Definition 1.4.5. A function v: Rn --4 R is
(i) positive semi-definite iff there is a time-invariant neighborhoodAf
of x = O, Af C Rn, such that
(a) v is continuous on Af: v E C(Af,
(b) v is non-negative on Af: v(x) >_for al l x e Af
;
(c) v vanishes at the origin: v(0)
(ii) positive semi-definite on a neighborhoodS of x = 0 iff (i) holds
for Af=S;
(iii) positive semi-definite in the wholeiff (i) holds for Af =Rn;
(iv) negative semi-definite (on a neighborhood S of x = 0 or in the
whole) iff (-v) is positive semi-definite (on the neighborhoodS
in the whole), respectively.
Remark1.4.1. It is to be noted that a function v defined by v(x) = for
all x ~ R
n is both positive and negative semi-definite. This ambiguity can
be avoided by introducing the notion of strictly positiveness (negativeness).
Definition 1.4.6. A function v: Rn -~ R is said to be strictly positive
(negative) semi-definite iff both it is positive (negative) semi-definite and
there is y ~ Af such that v(y) > 0 (v(y) < 0), respectively.
The H is strictly positive (negative) semi-definite iff v(x) = xWHx
is
strictly positive (negative) semi-definite, respectively.
Definition 1.4.7. A function v: R
n -~ R is:
(i) positive definite if there is a time-invariant neighborhoodAf, Af C_
R
n, or x = 0 such that both it is positive semi-definite on Af and
v(x) > 0 for all
(ii) positive definite on a neighborhood $ of x = 0 iff (i) holds
for Af = 3;
(iii) positive definite in the wholeiff (i) holds for Af =Rn;
(iv) negative definite (on a neighborhoodS of x = 0 or in the whole) iff
(-v) is positive definite (on the neighborhood~q or in the whole,
respectively).
Hahn[2] proved.
Proposition 1.4.2. Necessary and suftlcient for positive definiteness of
v on a neighborhoodAf of x = 0 is existence of comparisonfunctions
Kto,~), i = 1, 2, where a = sup{fix[f: z EAf}, such that both v(x) ~ C(Af)
a.d 1(11
11)< < 2(llxll) for x](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-32-320.jpg)
![12 1. PI:tELIMINARIES
Definition 1.4.8. A function v: R × R
n -~ R is:
(i) positive semi-definite on T~= [~’, +oo), r E R, iff there is a time-
invariant connected neighborhood Af of x = O, Af C_ R
n, such that
(a) v is continuous in (t,x) ~ 7-r × A/’: v(t,x) e C(Tr x A/’,R);
(b) v is non-negative on A/’: v(t,x) >_for al l (t ,x)
(c) v vanishes at the origin: v(t, O)=for al l t
(d) if[ the conditions (a)-(c) holds and for every t ~ T~
is y E A/" such that v(t, y) > 0, then v is strictly positive
semi-definite on T~;
(ii) positive semi-definite on 7"~×S iff (i) holds for Af=
(iii) positive semi-definite in the wholeon Tr iff (i) holds for Af =Rn;
(iv) negative semi-definite (in the whole) on T~ (on Tr × A/’) iff (-v)
positive semi-definite (in the whole)on Tr (on T~× A/’), respectively.
The expression "on 7"r " is omitted iff all corresponding requirements
hold for every r ~ R.
Definition 1.4.9. A function v: R × Rn -~ R is:
(i) positive definite on 7-~ , 7" E R, if[there is a time-invariant connected
neighborhoodA/" of x = 0, A/" c_ R
n, such that both it is positive
semi-definite on T~x Af and there exists a positive definite function
wonA/’, w: Rn --~ R, obeyingw(x) <_v(t,x) for all (t,x) T~
xA/
’;
(ii) positive definite on 7"? x S iff (i) holds for A/" =
(iii) positive definite in the wholeon 7-r iff (i) holds for Af =Rn;
(iv) negative definite (in the whole)on 7-r (on Tr xAf)iff (-v) is positive
definite (in the whole)on Tr (on Tr x iV’), respectively.
The expression "on 7"r " is omitted iff all corresponding requirements
hold for every r ~ R.
Thefollowing result is obtained directly from Proposition 1.4.2 and De-
finition 1.4.8.
Proposition 1.4.3. Necessary and sufficient for a function v : Rx R
n
R to be positive definite on 7-~ x Af whenA/" is a time-invariant neighbor-
hood of x =0 is that (a) and (c) of Det]nition 1.4.8 are fulfilled and there
is ~ ~ K[0,a], where a = sup(llxll: x E ~V},suchthat
v(t,z) =v÷(t,x)~(llxll) £or a/l T~
× ~
wherev+ ( t, x) is positive semi-definite](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-33-320.jpg)
![1.4 SCALAR,VECTOR
ANDMATRIX-VALUED
LIAPUNOV
FUNCTIONS
13
Definition 1.4.10. Set v~(t) is the largest connected neighborhood of
x = 0 at t E Rwhich can be associated with a function v, v: R x R
n -~ R,
so that x ~ v~(t) implies v(t,x) <
Definition 1.4.11. A function v: R x R
n --+ R is:
(i) decreasing on 7"~, r ~ R, iff there is a time-invariant neighborhood
Af of x = 0 and a positive definite function w on Af, w: Rn -+ R,
such that v(t, x) <_w(x) for all (t, x) e T~x Af;
(ii) decreasing on T~ x S iff (i) holds for Af =
(iii) decreasingin the wholeon 7-r iff (i) holds for AfR
n.
Theexpression "on T~" is omitted iff all corresponding conditions hold
for every ~- ~ R.
Definition 1.4.11 implies.
Proposition 1.4.4. Necessary and sufficient for v to be decreasing on
Tr x Af whenAf is a time-invariant neighborhood of x = 0 is existence of
a comparisonfunction ~ E K[o,a), where a = sup{[[x[[: x E .Af}, such that
v(t, z) =v_(t, z) ~(
llxll) for a~
T~× ~
wherev_ (t, x) is negative semi-definite on T~.
Barbashin and Krasovskii [1,2] discovered the concept of radially un-
boundedfunctions. Theyshowedthe necessity of it for asymptotic stability
in the whole.
Definition 1.4.12. A function v: R x R
n -~ R is:
(i) radially unboundedon T~, r ~ R, iff [[x[[ -~ +coimplies v(t, x)
+oofor all t ~ T~;
(ii) radially unboundediff [[x[[ -+ +ooimplies v(t, x) -~ +oofor all
t~Tr for all
Thenext can be easily verified (see Hahn[2]).
Proposition 1.4.5. Necessary and sufficient for a positive definite in
the whole (on ~ ) function v to be radially unboundedis that there exists
~ ~ KRobeying, respectively, v(t,x) >_~([[x[D/’or a11 x ~ R
n and for a11
t ~ R (for all t ~ ~).
For the details see Barbashin and Krasovskii [1,2], Gruji6, et al. [1],
Hahn[2], Martynyuk[9], etc.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-34-320.jpg)
![14 1. PRELIMINARIES
1.4.3 Vector Liapunov functions
Wereturn back to system (1.2.7) and assumethat for it the vector function
(1.4.5) = ,vm(t,x))
T
is constructed in some way, whose components va ¯ C(Tr × ~, R+), s
1, 2,..., m. For the function (1.4.5) to be suitable for stability analysis
the equilibrium state x = 0 of system(1.2.7) it is necessary for it to possess
the normtype properties (see Definitions 1.4.7-1.4.12). The presence
such properties of function (1.4.5) is established in terms of one of the
following functions (see Lakshmikantham,Matrosov, et al. [1])
(1.4.6) v(t,x) = max va(t,x);
8e[1
V(t,X, Ol) = o~TY(t,X), O~ m,
(1.4.7)
ai=const, i=l,2,...,m;
m
(1.4.8) v(t, x) = ~ vdt, x);
i=1
(1.4.9) v(t,x) = Q(Y(t,x)), Q(0)
Q ¯ C(R~, R+), the function Q(u) is nondecreasing in u. Since the func-
tions (1.4.6) - (1.4.9) are scalar and are constructed in terms of the vector
function (1.4.5), the special properties of the vector function (1.4.5)
established according to Definitions 1.4.7-1.4.12.
Remark1.4.2. Properties of positive definiteness, decrease and radial
unboundedness
of the function (1.4.5) follow from the algebraic inequalities,
provided that the componentsv8 (t, x) of the vector function (1.4.5) satisfy
the conditions
(1.4.10)
1 -1
ail¢~ (llxill) _< v~(t,x) <_a~2.C~(llx~ll), 1,2,...,m,
where ai~, aiu > 0 and ¢il and ¢i2 are of class K(KR), xi Rn’, n~+
T _1
¯ = IIxll =
Theconditions (1.4.10) are the broadest ones under which the algebraic
conditions of the property of having a fixed sign can be established for the
vector function (1.4.5).](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-35-320.jpg)
![1.4 SCALAR, VECTORANDMATRIX-VALUED
LIAPUNOVFUNCTIONS
15
The assumptions on the components vi(t, x) of the vector function are
knownbeing other than (1.4.10):
(a) ~li([[xi[[) _~ vi(t, xi) ~_ ~2i([[Xi[[), for all (t, xi) e ~ x n’, w
here
~u, ~o2i are of class K(KR), i =1, 2,..., rn;
(b) r~d[xd[ _<vi(t, xi) <_~i[[xi[[, for all (t, xi) e ~ x n’, w
here ~i a nd
hi are positive.constants, i : 1, 2,..., m,
(see MichelandMiller [1], ~iljak [1], etc.).
1.4.4 Matrix-valued metafunction
Assumethat for system (1.2.7) the two-indexes system of functions
Vll(t,X) ...
(1.4.11) II(t, x) = ... ".. ...
~Vtl(t,X) ... Vlk(t,X)
is constructed, where vii E C(T x n,R), i = 1,2,...,k; j = 1,2,...,l.
Definition 1.4.13. A function II: 7-r x R"~ -~ R
kxl is called the matrix-
valued meta~nction, if one of the Liapunov functions can be constructed
based on it, n~ely, a scMar, vector or simple matrix-valued one, which
solves the problem on stability of the equilibrium state x = 0 of sys-
tem (1.2.7).
Theproperties of having a fixed sign of metafunction (1.4.11) are estab-
lished by a general rule in terms of one of the functions
(1.4.12) ~(t,x) = m~ vij(t,x),
(1.4.13) vn(t, x,
where a ~ Rt, a = const ~ O, B~R
k, ~ = const ~ O;
k
0.4.14) ,.(t,
n(t,z)=¢(a(t,
where ~eC(Raxt,R+), ~(0) =0, ~(s) >0 for s>0, and lim ¢(s)
+~.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-36-320.jpg)
![16 1. PRELIMINARIES
Since the functions vri(t, .) determined by (1.4.12)- (1.4.15) are scalar,
the ordinary technique of the Liapunov functions method is used to check
their property of having a fixed sign, decreasing and radially unbounded-
hess.
Remark1.4.3. If k = I = min (1.4.11), then II(t,x) becomesan ordi-
nary matrix-valued function U(t, x)
(1.4.16) (
val(t,,) ... v~,~(t,x)
u(t,x) ... ".. ... ,
v,~(t,x) ... v,~,,(t,~)
where U: T~ × Rn --r R
"~×m
.
The property of having a fixed sign, decreasing and radial unbounded-
ness of the matrix-valued function (1.4.16) is established, provided that the
elements Vsk(t, x), s, k = 1, 2,..., m, satisfy the estimates
for all (t,x) EToxA/"
(for all (t,x) eT~xRn),
for all s = 1, 2,..., m, and (cf. Djordjevid [2])
_a,r¢81(llxsll)¢r1(llxrll
) <Vsr(t,x)
<~,r¢,2(llzsII)¢r2(llx~ll)
for all (t,x)~ToxAf (for all (t,x)~T~xR’~),
whenall s ¢ r.
Weshall formulate the assertions on the property of having a fixed sign
of the matrix metafunction similar to howit has been done for the ordinary
matrix-valued function (see Martynyuk[5- 7, 20]).
Proposition 1.4.6. A metafunction H: Tr x Rn -~ R~x~ is positive
definite on 7"r, r ~ R iff there exists a ~ Rt, ~ ~ R
k, and a ~ K, and it
can be written as
via(t,z,~,~)=~wII+(t,
x)~
a(
llxll),
whereII+(t, x) is positive semi-definite on "!’~.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-37-320.jpg)
![1.5 COMPARISON
PRINCIPLE 17
Proposition 1.4.7. A metafunction II: 7-r × R’~ -~ RTM is decreasing
on 7"~, T E R if[ there exists a 6 R~, /9 6 R
k, and b 6 K, and it can be
written as
= Tri_(t,
x)/9+b(llxll),
wherelI_ (t, x) is negative semi-definite on 7-~.
Proposition 1.4.8. A metafunction l-I: To x Rn -~ R
TM is radially
unboundedin the whole (on ~ ) if[ it can be written
Vll(t,x,o~,/9) = o~TII+(t,X)/9+ c(llxll),
where II+(t,x) is positive semi-definite in the whole (on T~), 6 R~,
B 6 Rk, and c 6 KR.
Remark 1.4.4. If k = l = m, the vectors a and/9 are replaced by one
vector y 6 Rmand Propositions 1.4.6-1.4.8 become the knownones (see
Martynyuk [20]).
1.5 Comparison Principle
In this section weformulate the basic comparisonresults in terms of Liapu-
nov-like functions and the theory of differential inequalities that are neces-
sary for our later discussion (see also Yoshizawa
[1], Szarski [1], etc.).
For system(1.2.7) weshall consider a continuous function v(t, x) defined
on an open set in T~x Af. Weassumethat v(t, x) satisfies locally Lipschitz
condition with respect to x that is, for each point in Tr x Af there are a
neighborhood T~ x ,9 and a positive number L > 0 such that
Iv(t,x)
vC
t, y)
l <LI
Iz -Yl
l
for any (t, x) ~ Tr x S, (t, y) e Tr
Definition 1.5.1. Let v be a continuous (either scalar, vector or matrix-
valued) function, v: T~xR’~ ~ R
sx’, v(t,x) ~ C(T~×Af), and let solutions
X of the system (1.2.7) exist and be defined on T~ x Af. Then, for all
(t, e
(i) D+v(t,x)= limsup (v[t+t~,~;(t+o;t,z)]-v(t,z)
0 :8-+0 + is the upper
right Dini derivative of v along the motion Xat (t, x);](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-38-320.jpg)
![18 1. PRELIMINARIES
(ii) D+v(t,x) li minfr ~ ~[t+e,x(t+~t,x)]-~(t,x): 0 +}
is the lower
right Dini derivative of v along the motionX at (t, x);
(iii) D-v(t,x) li msup {’[t+°’x(~+~’x)]-~{t’~) : 0 ~ 0-) i s upper
left Dini derivative of v along the motionXat (t, x);
(iv) D_v(t,x) li minf- {~[~+e,x(~+~t,~)]-~(~,~): 0 ~ 0-~ is lower
left Dini derivative of v along the motionX at (t, x).
(v) The function v has Eulerian derivative ~, ~)(t,x) = ~ v(t,x), at
(t, x) along the motionXiff
D+v(t, x) = D+v(t, x) = D-v(t, x) = D_v(t, x) =
and then i~(t, x) =Dr(t,
If v is a scalar function and differentiable at (t, x) then (see Liapunov
[1])
Ov
O(t, x) = ~- + (grad v)Wf(t, x),
where
gradv = ~-~ ’ Ox2’" " Ox,~ ] "
Effective application of D+vin the framework of the second Liapunov
methodis based on the next result by Yoshizawa[1], which enables calcu-
lation of D+vwithout utilizing system motion themselves.
Theorem1.5.1. Let v be continuous and locally Lipshitzian in x over
T~ x 8 and ~q be an open set. Then,
D+v(t’~)[(1.2.7) lim su
p { v[t +0,x +
Of
(t,O x)] - v(t, ~)
holds along solutions X of ehe system (1.2. 7) at (t, x) ~ T~
D*v will mean that both D+vand D+vcan be used.
The system of equations (1.2.7) is considered with the matrix-valued
function U(t, x).
Definition 1.5.2. All scalar function of the type
(1.5.1) v(t, X, Or) = otTU(t, x)a, a ~ rn,
where U ~ C(T~x Af, R’~×’~), are attributed to the class SL.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-39-320.jpg)
![1.5 COMPARISON
PRINCIPLE 19
The vector a can be determined in several ways(see Martynyuk[12]) and
its choice can effect the property of having a fixed sign of function (1.5.1).
ByDefinition 1.5.1 for function (1.5.1) whenall (t, x) ¯ Tr xAfthe total
derivative is calculated by virtue of system(1.2.7)
(1.5.2) D+v(t, x, a)[(1.2.~) = aTD+U(t,
where D+U(t,x) is calculated element-wise.
Let us consider the following scalar differential equation
du
(1.5.3) d-~ = g(t,u), u(to) = Uo>_ O, to ¯ R (to ¯
where g ¯ C(RxR, R) (or g ¯ C(Tr×R,R)) and g(t,O) = for al l t ¯ To
.
Definition 1.5.3. Let ~,(t) be a solution of (1.5.3) existing on
interval J = [to, to + a), 0 < a _< +~, to ¯ R (to ¯ Tr). Then~,(t) is
to be the maximal
solution of (1.5.3) if for every solution u(t) -~ u(t; t0,x0)
of (1.5.3) existing on J, the following inequalities hold
(1.5.4) u(t) <_~/(t), t ¯ g, to ¯ R
A minimalsolution is defined similarly by reversing the inequality (1.5.4).
Proposition 1.5.1. Let U: ~ × Af ~ Rm×’~, U(t,x) be locally Lip-
schitzian in x. Assumethat
(1) function g ¯ C(TrxRnxR+,R), g(t,O,O) = 0 existsforall t
such that
D+v(t,x,a)[(1.2.~ ) <_g(t,x,v(t,x,a)) for all (t,x,a) xJ~fx Rm;
(2) solution x(t) =x(t;to,Xo) of system (1.2.7) is definite and
nuousfor a11(t; to, x0) ¯ To× Tr × Af;
(3) maximalsolution of the comparison equation
du
d--~ = g(t,x,u), u(to) = uo, x(to)
exist for all t ¯ T~.
Then the estimate
v(t,x(t),a) <_ r(t;to,xo,uo) for all
holds wheneverv(to, xo, a) <_uo.
For the proof see monographsby Lakshmikantham,Leela, et al. [1].](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-40-320.jpg)
![20 1. PRELIMINARIES
Proposition 1.5.2. Let U: ~ x Af -~ R"~×’~, U(t,x) be locally Lip-
schitzian in x. Assumethat
(i) function g 6 C(Tr x n xR+
, R) exi sts suc h tha
D+v(t,x,a)[(1.2.~ ) >_g(t,x,v(t,x,a)) for all (t,x,a) e T~ x Af
(ii) solution of system(1.2. 7) is definite andcontinuousfor (t; to, xo)
Tox ~x.,V’;
(iii) minimalsolution r- (t; to, xo, wo)of the comparisonequation
d-~ =g(t, x, w), w(to) = wo
exists for all t ~ 7"~.
Theninequality v(t, Xo, a) >_woyields the estimate
~(t,x(t),~)>_
~-(t;to,xo,~,o)
for all t ~ ~.
Propositions 1.5.1 and 1.5.2 are a scalar version of the principle of com-
parison with the matrix-valued function.
In the monographby Zubov[4] the following assertions are proved.
Corollary
1.5.1.
Let
(i) function (1.5.1) obey the bilateral inequality
~(t)~
~(t)<~(t, ~,~)<_~2(t)~
~(t
),
where~oi(t) >0 for all 6 To
and
p(t) = (xT
(t to, Xo)x(t; to, Xo))
(ii) function g(t, x, v) satisfy the estimates
--¢1(t)p2(t)
where ¢i(t) _>0 for all t 6 To and functions ¢i(t)/qoi(t), i =
are integrable.
Then for the solutions of system (1.2. 7) the estimates
1 1 ¢~(~)
<: Po~2~ (t0)~- ½ (t) exp - ~ dT
to
are valid for all t 6 Toand to 6](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-41-320.jpg)
![1.5 COMPARISON PRINCIPLE 21
Corollary 1.5.2. Let
(i) function (1.5.1) obey the bilateral inequality
qo~
(t)ph(t) _<v(t, x, a) <_qo~
(t)p
~=(t),
where ~oi(t) are piece-wise continuous positive functions given for
all t E A = [to,to +T], 11 >_12 are positive numbers;
(ii) function g(t, x, v) satisfy the estimates
-¢1(t)p
~’(t) <g(t,x,v)<-¢~.(t)p
~2
(t),
where ¢i(t), i = 1, 2 are positive piece-wise continuous functions
for all t ~ A, kl ~ k2.
Thenfor the solutions of system (1.2. 7) the estimates
1 1
to
are valid for all t ~ A, v0 = v(to,xo,a), Ai = ~, ,~i > 1 for i = 1,2.
Corollary 1.g.3. Let both conditions (i) and (ii) of Corollary 1.5.2
satisfied ~mclAi >1, i = 1, 2. Thenfor !1 =12 =l the solutions of system
(1.2. 7) satisfy the estimate
~o;
l(t)vo 1+(~1- 1)v~
~-1I~(’~la~ <_
o(t)
~ ~r~(t)Vo 1 + (A2 - 1)v~’-~ f~(~)d~
~o
for all t ~ A, where
{¢l(t)~i
-~’(t)
¢1(t)~
~’(t)
t"
¢2()~2(t)
¢~ ~1 (~)
if ~)l(t) ~
if ¢1(t)<
if ¢2(t)>o,
if ~)2 (t) <](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-42-320.jpg)
![1.6 LIAPUNOV-LIKE
THEOREMS 23
Thenthe inequaJity L( to, Xo, b) <_~0 implies the estimate
(1.5.7) L(t, x(t), <w
+(t; to, x0, w0)
for all t¯ ~.
Besides, estimate (1.5.7) holds component-wise.
Theproof of Proposition 1.5.3 is similar to that of Theorem
3.1.2. from
Lakshmikantham,
Leela, et al. [1].
1.6 Liapunov-Like Theorems
There are several directions in stability theory to search for newconditions
which weakenone of other suppositions of the original Liapunov’s theorems.
Werecall the classification of these directions:
(1) search of minimal weakassumptions on the properties of auxiliary
functions (semi-definite functions, integral positive functions, etc.);
(2) modification of assumptions on the properties of total derivative
scalar function along solutions of perturbed motion equations;
(3) construction and application of multicomponentauxiliary functions
(vector, matrix-valued, metafunctions).
It is natural to expect the developmentof both the first and the second
directions within the frameworkof the third one.
Further on this section basic theorems of the direct Liapunov method
are set out in terms of the matrix-valued functions. Also, maindefinitions
of the class of matrix-valued functions are presented, here.
1.6.1 Matrix-valued function and its properties
Together with the system (1.2.7) weshall consider a two-indices system
functions
(1.6.1) V(t,x) [v~(~,~)], i, j = 1, 2,...,,~,
where v~ ¯ C(Tr x R’*,R+), and v~j ¯ C(T~ x n,R) f or a ll i ¢ j.
Moreover the next conditions are making
(i) vlj (t, x) are locally Lipschitzian in
(ii) v~j(t,O) for all t ¯ R(t ¯ T~), i, j = 1,2,... ,m;
(iii) v~j (t, x) = v~.~ (t, x) in any open connectedneighborhood
Afof
x = O for all t e R+](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-44-320.jpg)
![24 1. PRELIMINARIES
Let y E /~rn, y ~ 0, be given. By means of the vector y and matrix-
valued function (1.6.1) weintroduce the following function
v(t, x, y) =yWU(t,x)y.
The following definitions will be used throughout the book, which are
based on the corresponding results by Djordjevid [3], Grujid [2], Hahn[2],
Krasovskii [1], Liapunov [1], and Martynyuk[3-7].
is:
Definition 1.6.1. The matrix-valued function U: Tr × Rn -~ R
mxm
(i) positive semi-definite on Tr = [r, +~), r E R iff there are time-
invariant connected neighborhood Af of x = O, Af C Rn, and vector
yERm, y#0 such that
(a) v(t,x,y) is continuous in (t,x) ~ 7"~ x Af × Rm;
(b) v(t, x, y) is non-negativeon A;, v(t, x, y) >_for al l (t , x,
O) e T~ x Af x m and
(c) vanishes at the origin: v(t, 0, y) = 0 for all (t, y) E T~Rm
;
(d) iff the conditions (a)- (c) hold and for every t e T~, there
w E Af such that v(t,w,y) > 0, then v is strictly positive
semi-definite on
(ii) positive semi-definite on 7-r x S iff (i) holds for Af = ~q;
(iii) positive semi-definite in the wholeon 7-r iff (i) holds for Af
(iv) negative semi-definite (in the whole) on 7"r (on Tr x Af) iff (-v)
positive semi-definite (in the whole)on Tr (on T~x Af), respectively.
The expression "on 7"~" is omitted if[ all corresponding requirements
hold for every ~ E R.
Definition 1.6.2. The matrix-valued function U: ~ x Rn -~ R
"~×
’~
is:
(i) positive definite on ~, ~ ~ R, iff there are a time-invariant con-
nected neighborhood iV" of x = O, Af _c R
n and a vector y ~ R
m,
y ¢ 0 such that it is both positive semi-definite on Tr x Af and
there exists a positive definite function w on A/’, w: R
n -~ R+,
obeyingw(x) <_v(t, x, forall (t, x, y) ~ Tr x Af x R’~
(ii) positive definite on T~x S iff (i) holds for Af =
(iii) positive definite in the wholeon T~iff (i) holds for fl/"
(iv) negative definite (in the whole) on 7"r (on 7-~ x Af m)
iff (-v)
positive definite (in the whole)on Tr (on Tr x.h/’x Rm),respectively;](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-45-320.jpg)
![1.6 L1APUNOV-LIKE THEOREMS 25
(v) weakly decreasing on ~ if there exists a Aa > 0 and a function
a e CKsuch that v(t,x,y) < a(t, Ilxll) as soon as Hxll < A1and
(t, y)e T~
x
(vi) asymptotically decreasing on "]-r if there exists a A2 > 0 and a
function b ¯ KLsuch that v(t,x,y) < b(t, llxll ) as soon as I]xll <
As and (t, y) ¯ T~ R
m
.
The expression "on T~" is omitted iff all corresponding requirements
hold for every ~- ¯ R.
Proposition 1.6.1. The matrix-valued function U: R x R
n -~ R
rnx m
is positive definite on Tr, ~- ¯ R iff it can be written as
yTu(t, x)y = yTU+(t,z)y a(llxll),
whereU+(t, x) is positive se mi-definite matrix-valued fu nction and a ¯ K.
Definition 1.6.3. (cf. Gruji6, et al. [1]). Set v¢(t) is the largest con-
nected neighborhood of x = 0 at t 6 R which can be associated with a
function U: R x Rn -~ Rmxmso that x ¯ v¢(t) implies v(t,x,y) <
y¯R
m.
Definition 1.6.4. The matrix-valued function U: R x Rn --+ Rs×s is:
(i) decreasing on T~, ~" ¯ R, if[ there is a time-invariant neighborhood
Afof x = 0 and a positive definite function won A/’, ~: R
n --r R,
such that yWU(t,x)y <_~(x), for all (t, x) ¯ T~x A/’;
(ii) decreasing on 7"r x S iff (i) holds for Af=
(iii) decreasingin the wholeon ~ iff (i) holds for A/" = ~.
Theexpression "on "/-r " is omitted if[ all correspondingconditions still
hold for every T ¯ R.
Proposition 1.6.2. The matrix-valued function U: R x Rn -+ R
mxm
is decreasing on ~, ~- ¯ R, iff it can be written as
yTu(t, z)y = yTu_ (t, x)y b(llxll), (y # 0) m,
where U_(t,x) is a negative semi-definite matrix-valued
and b¯ K.
is:
function
Definition 1.6.5. The matrix-valued function U: R x R
~ ~ R
"~x’~
(i) radially unbounded
on Tr, r ¯ R, iff Ilxll-+~ implies yTU(t, x)y
+cx~, for all t ¯ Tr, y ¯ Rm;
(ii) radially unbounded,
iff Ilxll --r ~x~implies yWU(t,x)y -+q-oo, for all
t ¯ "Fr, and for all 7- E R, y ¯ R
"~, y ~ 0.](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-46-320.jpg)
![26 1.PRELIMINARIES
Proposition 1.6.3. The matrix-valued function U: 7-r x Rn -+ R
m×m
is radially unbounded
in the whole (on T~ ) iff it can be written
yTu(t,X)y : yTU.b($,x)y-[-a([[x[[ ) for all x e a
n,
where U+(t,x) is a positive semi-definite matrix-valued function in the
whole (on Tr) and a E K
For the proof of Proposition 1.6.1-1.6.3 see Martynyuk[9, 20].
1.6.2 A version of the original theorems of Liapunov
Thefollowing results are useful in the subsequent sections.
Proposition 1.6.4. Suppose re(t) is continuous on (a, b). Then re(t)
is nondecreasing(nonincreasing) on (a, b)
D+m(t) >_0 (<_0) ever y t E (a ,b ),
where
D+m(t)li msup {[m(t + 8) - m(t )]8-1 : 8 -+
FollowingLiapunov[1], Persidskii [1], and Yoshizawa[1] the next result
follows (see Martynyuk[20]).
Theorem1.6.1. Let the vector function ] in system (1.2.7) be conti-
nuous on R x Af (on Tr x fir). If there exist
(1) an open connected time-invariant neighborhood ~q C_ Af of
point x = O;
(2) positive de finite on Af (onTr xAf)ma
trix -valued functi on U ( t,
and vector y ~ Rm such that function v(t,x,y) = yWU(t,x)y
locally Lipschitzian in x and D+v(t, x, y) <_
Then
(a)
(b)
the state x = 0 of system (1.2. 7) is stable (on Tr), providedU(t,
is weakly decreasing on Af (on Tr x Af);
the state x =0 of system (1.2. 7) is uniformly stable (on Tr),
vided U(t, x) is decreasing on Af (on Tr x Af).](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-47-320.jpg)
![1.6 LIAPUNOV-LIKE
THEOREMS 27
Corollary 1.6.1. Assumethat the functions vii(t, x) in (1.6.1) are con-
tinuously differentiable for all (t, x) E T~x Af, and
m m
(1.6.2) Dv(t,x,y)[(1.2.7
) =~, ~y,yjDv,~(t,x)[(1.2.7
i=1
j----1
If there exist functions Oij(t,x,y), #)ij(t,O,y) =0 for y # 0 for all i,
i,2,...,
m,such that
(1.6.3) yiyjVvij (t, x)[(1.2.7) -~ ~ij (t,
for all (t, x) ~ Tr x Af, then (cf. Djordjevid[3])
(1.6.4) DvM(t,x,y)[(1.2.7) ~’~¢ij(t,x,y) = eWO
(t,x,y)e,
i=1 j=l
where e = (1, 1,..., 1)wE R~.
Further we denote
1[¢(~,
~,v)+cT(t,
~,V)]
(1.6.5) B(t, x, y) =
and assume that there exist comparison functions Wl([[x[D,...
,
of class K and matrix/~(t, x, y) such that
(1.6.6) eTB(t,
=,y)e
<_
~T(llxll)h(~,
=,Y)~(ll=ll)
for all (t, x, y) E T~x A/" x m.
Compile the equation
(1.6.7) det [~(t, x, y) - AE]=
where E is an mx midentity matrix. Designate the roots of this equation by
Ai = Ai(t, x, y), i = 1, 2,..., m. It is easy to verify that DvM(t,x, y) <_
in domain Tr x Af x R
m if
(1.6.8) A~(t,x,y) <_for al l i = 1,2 .. .. ,m,
and all (t, x, y) e Tr x A/" R
ra.
Conditions (1.6.8) together with the other conditions of Theorem1.6.1
is a sufficient test for stability and uniformstability of the state x = 0 of
system(1.2.7).](https://image.slidesharecdn.com/87816-250506111431-f5d11507/85/Qualitative-methods-in-nonlinear-dynamics-novel-approaches-to-Liapunov-s-matrix-functions-1st-Edition-A-A-Martynyuk-48-320.jpg)
Qualitative methods in nonlinear dynamics novel approaches to Liapunov s matrix functions 1st Edition A.A. Martynyuk
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- 6. PURE AND APPLIED MATHEMATICS A Programof Monographs,Textbooks, and Lecture Notes EXECUTIVE EDITORS EarlJ. Taft Rutgers Univers#y NewBrunswick, NewJersey Zuhair Nashed Universityof Delaware Newark, Delaware EDITORIAL BOARD M. S. Baouendi Universityof California, San Diego Jane Cronin RutgersUniversity Jack K. Hale Georgia Institute of Technology Anil Nerode CornellUniversity Donald Passman Universityof Wisconsin, Madison FredS. Roberts RutgersUniversity S. Kobayashi UniversityofCalifornia, Berkeley DavidL. Russell VirginiaPolytechnic Institute andState University Marvin Marcus Universityof California, Santa Barbara W.S. Massey YaleUniversity Walter Schempp UniversitiitSiegen MarkTeply Universityof Wisconsin, Milwaukee
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- 12. QUALITATIVE METHODS IN NONLINEAR DYNAMICS Novel Approaches to Liapunov’sMatrix Functions A. A. Martynyuk Institute of Mechanics National Academyof Sciences of Ukraine Kiev, Ukraine MARCEL DEKKER MARCEL DEKKER, INC. NEWYO~K- BASEL
- 13. ISBN:0-8247-0735-4 This bookis printed onacid-free paper. Headquarters MarcelDekker,Inc. 270 Madison Avenue, NewYork, NY10016 tel: 212-696-9000;fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse4, Posffach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482;fax: 41-61-261-8896 World Wide Web http://www.dekker.com Thepublisher offers discountson this bookwhenorderedin bulk quantities. For moreinfor- mation,write to Special Sales/Professional Marketing at the headquartersaddressabove. Copyright©2002 by MarcelDekker,Inc. All Rights Reserved. Neither this booknor any part maybe reproduced or transmitted in any form or by any means,electronic or mechanical,including photocopying,microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 1098765432 1 PRINTED IN THE UNITED STATES OF AMERICA
- 14. PREFACE An important place among modern qualitative methods in nonlinear dynamics of systems is occupied by those associated with the development of Poincar~’s and Liapunov’s ideas for investigating nonlinear systems of differential equations. Liapunovdivides into two categories all methodsfor the solution of the problemof stability of motion. Heincludes in the first category those meth- ods that reduce the consideration of the disturbed motion to the determi- nation of the general or particular solution of the equation of perturbed motion. It is usually necessary to search for these solutions in a variety of forms, of which the simplest are those that reduce to the usual method of successive approximations. Liapunovcalls the totality of all methodsof this first category the "first method". It is possible, however,to indicate other methodsof solution of the prob- lem of stability which do not necessitate the calculation of a particular or the general solution of the equations of perturbed motion, but which re- duce to the search for certain functions possessing special properties. Lia- punovcalls the totality of all methodsof this second category the "second method". During the post-Liapunov period both the first and second Liapunov’s methods have been developed considerably. The second method, or the direct Liapunovmethod, based first on scalar auxiliary function, w~sre- plenished with newideas and newclasses of auxiliary functions. This al- lowed one to apply this fruitful technique in the solution of manyapplied problems. The ideas of the direct Liapunov methodare the source of new moderntechniques of qualitative analysis in nonlinear systems dynamics. A considerable numberof publications appearing annually in this direction provide a moderntool for qualitative analysis of processes and phenomena in the real world. Theaim of this monograph is to introduce the reader to a newdirection in nonlinear dynamicsof systems. This direction is closely connected with a iii
- 15. iv PREFACE newclass of matrix-valued function of particular importance in construction of an appropriate Liapunov function for the system under consideration. It is knownthat the problem of stability is important not only for the continuous systems modeledby ordinary differeatial equations. Therefore, in this monographthe methods of qualitative analysis are presented for discrete-time and impulsive systems. Further, in view of the importance of the problem of estimating the domains of asymptotic stability, a new methodfor its solution is set out in a separate chapter. The monographcontains five chapter and is arranged as follows. Thefirst chapter contains all necessary results associated with the me- thod of matrix-valued Liapunovfunctions. It also provides general informa- tion on scalar and vector functions including the cone-valued ones. General theorems on various types of stability of the equilibrium state of the sys- tems cited in this chapter are basic for establishing the sufficient stability tests in subsequent chapters. The second chapter deals with the construction of matrix-valued func- tions and corresponding scalar auxiliary Liapunov functions. Here new methodsof the initial system decomposition are discussed, including those of hierarchical decomposition. Thecorresponding sufficient tests for var- ious types of stability and illustrative examples are presented for every case under consideration. Alongwith the classical notion of stability ma- jor attention is paid to newtypes of motionstability, in particular, to the exponential polystability of separable motions as well as the integral and Lipschitz stability. The third chapter addresses the methods of stability analysis of dis- crete-time systems. Our attention is focussed mostly on the development of the methodof matrix-valued functions in stability theory of discrete-time systems. In the fourth chapter the problems of dynamics of nonlinear systems in the presence of impulsive perturbations are discussed. The method of matrix-valued Liapunovfunctions is adapted here for the class of impulsive systems that were studied before via the scalar Liapunov function. The proposed development of the direct Liapunov method for the given class of systems enables us to makean algorithm constructing the appropriate Liaptmov functions and to increase efficiency of this method. In the final chapter the problem of estimating the domains of asymptotic stability is discussed in terms of the methodof matrix-valued Liapunov functions. By meansof numerous examples considered earlier by Abdullin,Anapolskii, et al. [1], Michel,Sarabudla, et al. [1], and ~iljak [1] it
- 16. PREFACE v is shown that the application of matrix-valued functions involves an essential extension of the domainsof asymptotic stability constructed previously. I wish to acknowledgethe essential technical assistance provided by my colleagues in the Stability of Processes Departmentof S.P.TimoshenkoIn- stitute of Mechanics, National Academy of Sciences of Ukraine. The bibliographical information used in the monograph was checked by CD-ROM Compact MATH,which was kindly provided by Professor, Dr. Bernd Wegnerand Mrs. Barbara Strazzabosco from the Zentralblatt MATH. I express mysincere gratitude to all persons mentioned above. I am also grateful to the staff of MarcelDekker,Inc., for their initiative and kind assistance. A. A. Martynyuk
- 18. CONTENTS Preface 1 Preliminaries 1.1 Introduction 1.2 Nonlinear Continuous Systems 1.2.1 General equations of nonlinear dynamics 1.2.2 Perturbed motion equations 1.3 Definitions of Stability 1.4 Scalar, Vector and Matrix-Valued Liapunov Functions 1.4.1 Auxiliary scalar functions 1.4.2 Comparison functions 1.4.3 Vector Liapunov functions 1.4.4 Matrix-valued metafunction 1.5 Comparison Principle 1.6 Liapunov-Like Theorems 1.6.1 Matrix-valued function and its properties 1.6.2 Aversion of the original theorems of Liapunov 1.7 Advantages of Cone-Valued Liapunov Functions 1.7.1 Stability with respect to two measures 1.7.2 Stability analysis of large scale systems 1.8 Liapunov’s Theoremsfor Large Scale Systems in General 1.8.1 Whyare matrix-valued Liapunov functions needed? 1.8.2 Stability and instability of large scale systems 1.9 Notes 2 Qualitative Analysis of Continuous Systems 2.1 Introduction 111 1 1 14 23 23 ~4 41 41 42 47 49 vii
- 19. viii CONTENTS 2.2 Nonlinear Systems with Mixed Hierarchy of Subsystems 50 2.2.1 Mixedhierarchical structures 50 2.2.2 Hierarchical matrix function structure 52 2.2.3 Structure of hierarchical matrix function derivative 56 2.2.4 Stability and instability conditions 59 2.2.5 Linear autonomous system, 60 2.2.6 Examples of third order systems 63 2.3 Dynamicsof the Systems with Regular Hierarchy Subsystems 68 2.3.1 Ikeda-~iljak hierarchical decomposition 68 2.3.2 Hierarchical Liapunov’s matrix-valued functions 69 2.3.3 Stability and instability conditions 74 2.3.4 Linear nonautonomous systems 79 2.4 Stability Analysis of Large Scale Systems 90 2.4.1 A class of large scale systems 90 2.4.2 Construction of nondiagonal elements of matrix-valued function 91 2.4.3 Test for stability analysis 94 2.4.4 Linear large scale system 94 2.4.5 Discussion and numerical example 97 2.5 Overlapping Decomposition and Matrix-Valued Function Construction 100 2.5.1 Dynamical system extension 100 2.5.2 Liapunov matrix-valued function construction 105 2.5.3 Test for stability of system(2.5.1) 105 2.5.4 Numerical example 106 2.6 Exponential Polystability Analysis of Separable Motions 108 2.6.1 Statement of the Problem 108 2.6.2 A method for the solution of the problem 110 2.6.3 Autonomous system 118 2.6.4 Polystability by the first order approximations 122 2.7 Integral and Lipschitz Stability 127 127 2.8 Definitions Sufficient conditions for integral and asymptotic integral stability 128 2.7.3 UniformLipschitz stability 133 Notes 135
- 20. CONTENTS 3 Qualitative Analysis of Discrete-Time Systems 3.1 Introduction 3.2 Systems Described by Difference Equations 3.3 Matrix-Valued Liapunov Functions Method 3.3.1 Auxiliary results 3.3.2 Comparisonprinciple application 3.3.3 General theorems on stability 3.4 Large Scale System Decomposition 3.5 Stability and Instability of Large Scale Systems 3.5.1 Auxiliary estimates 3.5.2 Stability and instability conditions 3.6 AutonomousLarge Scale Systems 3.7 Hierarchical Analysis of Stability 3.7.1 Hierarchical decomposition and stability conditions 3.7.2 Noveltests for connective stability 3.8 Controlled Systems 3.9 Notes 4 Nonlinear Dynamics of Impulsive Systems 4.1 Introduction 4.2 Large Scale Impulsive Systems in General 4.2.1 Notations and definitions 4.2.2 Auxiliary results 4.2.3 Sufficient stability conditions 4.2.4 Instability conditions 4.3 Hierarchical Impulsive Systems 4.4 Analytical Construction of Liapunov Function 4.4.1 Structure of hierarchical matrix-valued Liapunov function 4.4.2 Structure of the total derivative of hierarchical matrix-valued function 4.5 Uniqueness and Continuability of Solutions 4.6 On Boundedness of the Solutions 4.7 Novel Methodologyfor Stability 4.7.1 Stability conditions 4.8 Notes ix 139 139 140 143 143 144 147 149 151 151 157 159 166 166 172 179 181 183 183 184 184 186 195 197 201 204 204 207 215 222 228 228 238
- 21. x CONTENTS 5 Applications 239 Introduction 239 Estimations of Asymptotic Stability Domainsin General 239 5.2.1 A fundamental Zubov’s result 239 5.2.2 Someestimates for quadratic matrix-valued functions 241 5.2.3 Algorithm of constructing a point network covering boundary of domain E 245 5.2.4 Numerical realization and discussion of the algorithm 250 5.2.5 Illustrative examples 254 Construction of Estimate for the DomainE of Power System 263 Oscillations and Stability of SomeMechanical Systems 267 5.4.1 Three-mass systems 267 5.4.2 Nonautonomousoscillator 269 Absolute Stability of Discrete Systems 270 Notes 274 295 5.1 5.2 5.3 5.4 5.5 5.6 References Subject Index
- 22. 1 PRELIMINARIES 1.1 Introduction This chapter contains an extensive overview of the qualitative methodsin nonlinear dynamicsand is arranged as follows. Section 1.2 is short and gives information about continuous nonlinear sys- tems that is important for applications in investigation of the mechanical, electrical and electromechanical systems. Also discussed are the equations of perturbed motion of nonlinear systems which are the object of investi- gation in this monograph. For the reader’s convenience, in Section 1.3 the definitions weuse of motion stability of various types are formulated. These formulations re- sult from an adequate description of stability properties of nonlinear and nonautonomous systems. Section 1.4 deals with three classes of Liapunovfunctions: scalar, vec- tor and matrix-valued ones, as well as the possibilities of their application in motion stability theory. Along with the well-known results, some new notions are introduced, for example, the notion of the "Liapunov metafunc- tion". Basic theorems of the comparisonprinciple for SL-class and VL-class of the Liapunov functions are set out in Section 1.5. Also, some important corollaries of the comparisonprinciple related to the results of Zubovare presented here. Section 1.6 deals with generalization of the main Liapunov and Barba- shin-Krasovskii theorems established by the author in terms of matrix- valued functions. Somecorollaries of general theorems contain newsuf- ficient stability (instability) tests for the equilibrium state of the system under consideration. In Section 1.7 the vector and cone-valued functions are applied in the problemof stability with respect to two measures and in stability theory of
- 23. 2 1. PRELIMINARIES large scale systems. Detailed discussion of possibilities of these approaches mayprove to be useful for manybeginners in the field. In the final Section 1.8, the formulations of theoremsof the direct Lia- punov method are set out based on matrix-valued functions and intended for application in stability investigation of large scale systems. Generally, the results of this chapter are necessary to get a clear idea of the results presented in Chapters 2-5. Throughout Chapters 2-5 refe- rences to one or an other section of Chapter 1 are made. 1.2 Nonlinear Continuous Systems 1.2.1 General equations of nonlinear dynamics Thesystems without nonintegrability differential constraints represent a wide class of mechanicalsystems with a finite numberof degrees of freedom. Let the state of such system in the phase space Rn, n = 2k, be determined by the vectors q = (ql,..., qk) w and ~ = (~1,..., ~k) w. It is knownthat the general motion equations of such a mechanical system are d (or’~ OT _ Us, s = 1,2,...,k. (1.2.1) Here T is the kinetic energy of the mechanical system and Us are the generalized forces. Thesystem of equations (1.2.1) is simplified, if for the forces affecting the system a force function U = U(t,q~,... ,qk) exists such that OU us= , s=l,2,...,k. Thesimplified system obtained so far, d(O(T+V)~ O(T+U)_o, s = 1,2,...,k, dt 00, ] Oqs
- 24. 1.2 NONLINEAR CONTINUOUS SYSTEMS 3 can be presented in the canonical form dqs OR dps OR - s = 1,2,...,k, dt Op~ dt Oqs OT and R = T2-To-U. Here To is the totality of the where p~ = 04--7 velocity-independent terms in the expression of the kinetic energy, and T2 is the totality of the secondorder terms with respect to velocities. Thequalitative analysis of equations (1.2.1) and its particular cases the principle point of the investigations in nonlinear dynamicsof continuous systems. 1.2.2 Perturbed motion equations Undercertain assumptions the equations (1.2.1) can be represented in the scalar form dy_~i = Y~(t, Yl,... ,Y2k), i =1, 2k, or in the equivalent vector form (1.2.2) dy = Y(t, y), dt where* y = (Yl,Y2,... ,Y2k)T E 2k and Y= (Y1,Y~,... ,Y2k)T, Y:7-× R ek -+ R ~k. A motionof (1.2.2) is denotedby y(t; to, Y0), ~(to; to, Y0) and the reference motionr/r(t; to, Yro). Fromthe physical point of view the reference motion should be realizable by the system. Fromthe mathe- matical point of view this meansthat the reference motion is a solution of (1.2.2), (1.2.3) &?r (t; t0, Yro)_=Y[t, ~/~(t; to, Y~0)]. dt Let the Liapunov transformation of coordinates be used, (1.2.4) x = y - Yr, where yr(t) -- ~lr(t;to,Yro). Let f: T x R ~k -~ R 2k be defined by (1.2.5) f(t, x) =Y[t, y~(t) +x] - Y[t, Yr]. ! ! ~T *InLiapunov’s notationy ---- (ql,q2,... ,qk, ql,q2,’’’ ,qkJ
- 25. 4 1. PRELIMINARIES It is evident that (1.2.6) f(t,O) =_ Now (1.2.2)- (1.2.5) yield dx (1.2.7) d~-=f(t, x). In this way, the behavior of perturbed motions related to the reference motion (in total coordinates) is represented by the behavior of the state deviation x with respect to the zero state deviation. The reference motion in the total coordinates Yi is represented by the zero deviation x = 0 in state deviation coordinates xi. With this in mind, the following result em- phasizes complete generality of both Liapunov’s second methodand results represented by Liapunov [1] for the system (1.2.7). Let Q: R2k -r R ’~, n = 2k is admissible but not required. In the monograph Grujid, et al. [1] the following assertion is proved. Proposition 1.2.1. Stability of x = 0 of systena (1.2.7) with respect to Q =x is necessary and sufficient for stability of the reference motion of system (1.2.2) with respect to every vector function Qthat is continuous in y. This theorem reduced the problemof the stability of the reference motion of (1.2.2) with respect to Q to the stability problem of x = 0 of (1.2.7) with respect to x. For the sake of clarity westate Definition 1.2.1. State x* of the system (1.2.7) is its equilibrium state over 7~ iff (1.2.8) x(t;to,x*) = x*, for all t E To, and to Theexpression "over 7~" is omitted iff 7~ = R. Proposition 1.2.2. For x* ~ Rn to be an equilibrium state of the system (1.2. 7) over Ti it is necessary and sufficient that both (i) for every to q T/ there is the unique solution x(t; to, x*) of (1.2.7), whichis defined for all to ~ To and (ii) f(t,x*) = 0, for a/l t e To, and to e 7~.
- 26. 1.3 DEFINITIONS OFSTABILITY 5 The conditions for existence and uniqueness of the solutions of system (1.2.7) can be found in manywell-knownbooks by Dieudonne[1], Hale [1], Hirsch and Smale[1], Simmons [1], Yoshizawa[1], etc. Thenext result provides a set of sufficient conditions for the uniqueness of solutions for initial value problem (1.2.9) d-¥ =f(t, x), X(to) = Proposition 1.2.3. Let :D C Rn+l be an open and connected set. Assume f ¯ C(:D, Rn) and for every compact K C ~), f satisfies the Lipschitz condition []f(t,x) f( t,y)[[ <_L[[x - y[[ for all (t,x), (t,y) 6 K, where L is a constant dependingonly on Then(1.2.9) has at most one solution on any interval [to, to + c), c > 0. Definition 1.2.2. A solution x(t;to,Xo) of (1.2.7) defined on the inter- val (a, b) is said to be bounded if there exists /~ > 0 suchthat [[x(t; to, x0)[] < fl for all t ¯ (a, b), where/~maydependon each solution. For the system (1.2.7) the following result can be easily demonstrated. Proposition 1.2.4. Assume f ¯ C(J x Rn,Rn), where J = (a,b) a finite or infinite interval. Let every solution of (1.2.7) is bounded.Then every solution of (1.2.7) can be continued on the entire interval (a, b). 1.3 Definitions of Stability Consider the differential system (1.2.7), where f ¯ C(%n,Rn).Sup- pose that the function f is smooth enoughto guarantee existence, unique- ness and continuous dependenceof solutions x(t; to, x0) of (1.2.7). We present various definitions of stability (see Grujid [1] and Grujid, et al. [1]). Definition 1.3.1. The state x = 0 of the system (1.2.7) is: (i) stable with respect to 7~ iff for every to ¯ T~and every e > 0 there exists 5(to,e) > 0, such that [[Xo[[ < 5(to,e) implies all t ¯ %;
- 27. 1. PRELIMINARIES (ii) uniformly stable with respect to Toiff both (i) holds and for every ¢ > 0 the corresponding maximal ~M obeying (i) satisfies inf[t~M(t,~): t ¯ T/] > (iii) stable in the wholewith respect to Ti iff both (i) holds and 5M(t,e)--~+oo as e-~+oo, for all t¯T/; (iv) uniformly stable in the wholewith respect to T, iff both (ii) and (iii) hold; (v) unstable with respect to 7~ iff there are to ¯ T/, e ¯ (0, +oo) and T ¯ To, V > tO, such that for every 5 ¯ (0,+oo) there is Xo, Ilxoll<5, forwhich IIx(T; to,xo)ll >~. Theexpression "with respect to 7~" is omitted from (i)- (v) iff 7~ These stability properties hold as t -~ +oobut not for t = +oo. Further the definitions on solution attraction are cited. The examples by Hahn[2], Krasovskii [1], and Vinograd [1] showedthat the attraction property does not ensure stability. Definition 1.3.2. The state x = 0 of the system (1.2.7) is: (i) attractive with respect to Ti ifffor every to ¯ 7~ there exists A(to) 0 and for every ~ > 0 there exists ~’(to;zo,~) ¯ [0,+oo) such that Ilzoll < A(to) implies IIx(t;to,Xo)ll < ¢, for all t ¯ (to r(to; xo, ¢), +oo); (ii) Xo-uniformlyattractive with respect to 7~ iff both (i) is true and for every to ¯ T/ there exists A(to) > 0 and for every ~ ¯ (0, +oc) there exists r~,[to, A(to), ~] ¯ [0, +oo) such that sup[T,~(t0; X0,¢): X0¯ T/] =T=(7~,X0, (iii) to-uniformly attractive with respect to 7~ iff (i) is true, there is A> 0 and for every (x0, ~) ¯ Ba× (0, +o¢) there exists ru(Ti, Xo, ~) [0, +o¢) such that sup[rm(to); xo, (): to ¯ Ti] = ~’u(7~,x0,
- 28. 1.3 DEFINITIONS OF STABILITY 7 (iv) uniformlyattractive with respect to Ti iff both (ii) and (iii) hold, is, that (i) is true, there exists A> 0 and for every ~ E (0, +c~) there is T~(T/, A, ¢) E [0, +~) such that sup [~-m(to;x0, ~): (to, x0) ~ T/xBa] = r(T/, A, (v) Theproperties (i)- (iv) hold "in the whole" iff (i) true for every A(t0) ~ (0, +oo) and every to ~ Theexpression "with respect to Ti" is omitted iff T/= R. Definitions 1.3.1 and 1.3.2 enable us to define various types of asymptotic stability as follows. Definition 1.3.3. The state x =0 of the system (1.2.7) is: (i) asymptotically stable with respect to Ti iff it is both stable with respect to T/and attractive with respect to 7~; (ii) equi-asymptoticallystable with respect to Ti iff it is both stable with respect to (iii) quasi-uniformlyasymptotically stable with respect to Ti iff it is both uniformly stable with respect to 7~ and t0-uniformly attractive with respect to 7~; (iv) uniformly asymptotically stable with respect to 7~ iff it is both uni- formly stable with respect to 7~ and uniformly attractive with re- spect to (v) the properties (i)- (iv) "in t he w hole" iff b oth the c orrespond- ing stability of x = 0 and the corresponding attraction of x = 0 hold in the whole; (vi) exponentially stable with respect to Ti iff there are A> 0 and real numbers c~ _> 1 and fl > 0 such that HXoll < A implies IlX(t;to,xo)]] <_~llXoll exp[-fl(t- to)], for all teTo, and for all to This holds "in the whole" iff it is true for A= +oo. Theexpression "with respect to 7~" is omitted iff 7~ = R.
- 29. 8 1. PRELIMINARIES 1.4 Scalar~ Vector and Matrix-Valued Liapunov Functions In order that to avoid the problemof nonlinear equations nonintegrability in their qualitative study, Liapunov[1] suggested to apply auxiliary functions with the normproperties. The auxiliary function, being a function of one variable (time) on the system trajectories, allows estimating the distance from every point of the system integral curve to the origin (to the system equilibrium state) whentime is changing from the fixed value to E "Yr. 1.4.1 Auxiliary scalar functions Thesimplest type of auxiliary function for system (1.2.7) is the function (1.4.1) v(t, x) e c(’ro×RR+),v(t, 0) Further all functions (1.4.1) allowing the solution of the problemon stability (instability) of the equilibrium state x = 0 of system(1.2.7) are called Liapunov functions. Theconstruction of the Liapunovfunctions still remains one of the cen- tral problems of stability theory. These functions should satisfy special requirements such as the property of having a fixed sign, decreasing, radial unboundedness, etc. The Liapunov functions are often constructed as a quadratic form of the phase variables whosecoefficients are constants or time functions. The following definitions are presented according to Gantmacher[1]. Definition 1.4.1. A matrix H = (hij) ~ nxn i s: (i) positive (negative) semi-definite iff its quadratic form V(x) =xTHx is positive (negative) semi-definite, respectively; (ii) positive (negative) definite iff its quadratic form V(x) = xTHx is positive (negative) definite, respectively. Notice that a square matrix A with all real valued elements is (semi-) definite iff its symmetricpart As=½(A +A T) is (semi-) definite, and square matrix Awith complexvalued elements is (semi-) definite iff its Her- mitian part AH= ½(A+ A*) is (semi-) definite, where A*is the transpose conjugate matrix of the matrix A. Now,the fundamental theorem of the stability theory - the Liapunov matrix theorem - can be stated in the form.
- 30. 1.4 SCALAR, VECTORANDMATRIX-VALUED LIAPUNOVFUNCTIONS 9 Theorem1.4.1. In order that real parts of all eigenvalues of a matrix A, A E R ’~×’*, be negative it is necessary andsufficient that for any positive definite symmetric matrix G, G ~ Rn×n, there exists the unique solution H, H ~ Rn×n, of the (Liapunov) matrix equation (1.4.2) ATH + HA = -G, which is also positive definite symmetric matrix. If all the characteristic roots of Ahave negative real parts wecan solve the matrix equations (1.4.2) in closed form (see Zubov[3], and Hahn[2]) H = / esATGesA ds. For solving the Liapunov matrix equation (1.4.2), see also Aliev and Larin [1], Barbashin[2], Barnett and Storey [1], etc. 1.4.2 Comparison functions Comparisonfunctions are used as upper or lower estimates of the function V and its total time derivative. Theyare usually denoted by ~, ~: R+-4 R+. The main contributor to the investigation of properties of and use of the comparison functions is Hahn[2]. Whatfollows is mainly based on his definitions and results. Definition 1.4.2. A function ~, qo: R+-4 R+, belongs to (i) the class K[o,~), 0 < a < +~, iffboth it is defined, continuous and strictly increasing on [0, a) and qa(0) = (ii) the class K iff (i) holds for a = +~, K = K[0,+~); (iii) the class KRiff both it belongs to the class K and ~(() as ( -4 (iv) the class L[o,a) iff both it is defined, continuous and strictly de- creasing on [0, a) and lim [~o(() : ( -4 +oc] (v) the class L iff (iv) holds for a = +oo, L = L[0,+oo). Let ~-1 denote the inverse function of ~, ~0-1[~(()] -- (. The next result wasestablished by Hahn[2].
- 31. 10 1. PRELIMINARIES Proposition 1.4.1. (1) If %0 6 K and ¢ ~ K then %0(¢) ~ (2)If %0 6 K anda 6 L then%0(a) 6 (3)If %06 K[o,a) and%0(a) = th en%0 -I6 K[ o,e); (4) If %06 K and lim [%0((): ( -+ +oo] then%0--1 is not def ined on +oo]; (5) If%0 6 K[o,a), %b6 K[0,a) and %0(~) > ¢(() on [0,a) then %0-1(~) ¢-1(() on [0,~3], where f~ = ¢(a). Definition 1.4.3. A ]unction %0, %0: R+x R+-~ R+, belongs to: (i) the class KK[o;a,~) iff both %0(0, ~) 6 K[o,a) for every and %0(~,0)6 K[o,~) for every ~ 6 [0,a); (ii) the class KKiff (i) holds for a = ~3 = +oo; (iii) the class KL[o;a,~) iff both %0(0, () 6 K[o,a) for and %0((,0) 6 L[o,~) for every ~ 6 [0,a); (iv) the class KLiff (iii) holds for a = f~ = +oo; (v) the class CKiff %0(t,0) = 0, %0(t,u) 6 for ev ery t 6 R+ ; (vi) the class 14 iff %06 C(R+x R",R+), inf %0(t,x) = O, (t,x) R+ x (vii) the class 14o iff %0 ~ C(R+ x Rn,R+), inf%0(t,x) -- 0 for each t ~ R+; (viii) the class ~ iff %06 C(K,R+):%0(0) = 0, and %0(w)is increasing with respect to cone K. Definition 1.4.4. Two]unctions %01, %026 g (or %01, %026 KR) are said to be of the sameorder of magnitudeif there exist positive constants a, ~, such that < %02(0 < Z%01(¢) for all (or for all e [%00)). In terms of the comparisonfunction’s existence, the special properties of functions (1.4.1) or the function (1.4.3) v(x) 6 C(R", R+), v(0) = applied in the analysis of the autonomoussystem dx (1.4.4) d-~ = g(x), g(O) where x 6 R n, g 6 C(R n, Rn), are specified in the following way.
- 32. 1.4 SCALAR,VECTOI~ ANDMATRIX-VALUED LIAPUNOV FUNCTIONS 11 Definition 1.4.5. A function v: Rn --4 R is (i) positive semi-definite iff there is a time-invariant neighborhoodAf of x = O, Af C Rn, such that (a) v is continuous on Af: v E C(Af, (b) v is non-negative on Af: v(x) >_for al l x e Af ; (c) v vanishes at the origin: v(0) (ii) positive semi-definite on a neighborhoodS of x = 0 iff (i) holds for Af=S; (iii) positive semi-definite in the wholeiff (i) holds for Af =Rn; (iv) negative semi-definite (on a neighborhood S of x = 0 or in the whole) iff (-v) is positive semi-definite (on the neighborhoodS in the whole), respectively. Remark1.4.1. It is to be noted that a function v defined by v(x) = for all x ~ R n is both positive and negative semi-definite. This ambiguity can be avoided by introducing the notion of strictly positiveness (negativeness). Definition 1.4.6. A function v: Rn -~ R is said to be strictly positive (negative) semi-definite iff both it is positive (negative) semi-definite and there is y ~ Af such that v(y) > 0 (v(y) < 0), respectively. The H is strictly positive (negative) semi-definite iff v(x) = xWHx is strictly positive (negative) semi-definite, respectively. Definition 1.4.7. A function v: R n -~ R is: (i) positive definite if there is a time-invariant neighborhoodAf, Af C_ R n, or x = 0 such that both it is positive semi-definite on Af and v(x) > 0 for all (ii) positive definite on a neighborhood $ of x = 0 iff (i) holds for Af = 3; (iii) positive definite in the wholeiff (i) holds for Af =Rn; (iv) negative definite (on a neighborhoodS of x = 0 or in the whole) iff (-v) is positive definite (on the neighborhood~q or in the whole, respectively). Hahn[2] proved. Proposition 1.4.2. Necessary and suftlcient for positive definiteness of v on a neighborhoodAf of x = 0 is existence of comparisonfunctions Kto,~), i = 1, 2, where a = sup{fix[f: z EAf}, such that both v(x) ~ C(Af) a.d 1(11 11)< < 2(llxll) for x
- 33. 12 1. PI:tELIMINARIES Definition 1.4.8. A function v: R × R n -~ R is: (i) positive semi-definite on T~= [~’, +oo), r E R, iff there is a time- invariant connected neighborhood Af of x = O, Af C_ R n, such that (a) v is continuous in (t,x) ~ 7-r × A/’: v(t,x) e C(Tr x A/’,R); (b) v is non-negative on A/’: v(t,x) >_for al l (t ,x) (c) v vanishes at the origin: v(t, O)=for al l t (d) if[ the conditions (a)-(c) holds and for every t ~ T~ is y E A/" such that v(t, y) > 0, then v is strictly positive semi-definite on T~; (ii) positive semi-definite on 7"~×S iff (i) holds for Af= (iii) positive semi-definite in the wholeon Tr iff (i) holds for Af =Rn; (iv) negative semi-definite (in the whole) on T~ (on Tr × A/’) iff (-v) positive semi-definite (in the whole)on Tr (on T~× A/’), respectively. The expression "on 7"r " is omitted iff all corresponding requirements hold for every r ~ R. Definition 1.4.9. A function v: R × Rn -~ R is: (i) positive definite on 7-~ , 7" E R, if[there is a time-invariant connected neighborhoodA/" of x = 0, A/" c_ R n, such that both it is positive semi-definite on T~x Af and there exists a positive definite function wonA/’, w: Rn --~ R, obeyingw(x) <_v(t,x) for all (t,x) T~ xA/ ’; (ii) positive definite on 7"? x S iff (i) holds for A/" = (iii) positive definite in the wholeon 7-r iff (i) holds for Af =Rn; (iv) negative definite (in the whole)on 7-r (on Tr xAf)iff (-v) is positive definite (in the whole)on Tr (on Tr x iV’), respectively. The expression "on 7"r " is omitted iff all corresponding requirements hold for every r ~ R. Thefollowing result is obtained directly from Proposition 1.4.2 and De- finition 1.4.8. Proposition 1.4.3. Necessary and sufficient for a function v : Rx R n R to be positive definite on 7-~ x Af whenA/" is a time-invariant neighbor- hood of x =0 is that (a) and (c) of Det]nition 1.4.8 are fulfilled and there is ~ ~ K[0,a], where a = sup(llxll: x E ~V},suchthat v(t,z) =v÷(t,x)~(llxll) £or a/l T~ × ~ wherev+ ( t, x) is positive semi-definite
- 34. 1.4 SCALAR,VECTOR ANDMATRIX-VALUED LIAPUNOV FUNCTIONS 13 Definition 1.4.10. Set v~(t) is the largest connected neighborhood of x = 0 at t E Rwhich can be associated with a function v, v: R x R n -~ R, so that x ~ v~(t) implies v(t,x) < Definition 1.4.11. A function v: R x R n --+ R is: (i) decreasing on 7"~, r ~ R, iff there is a time-invariant neighborhood Af of x = 0 and a positive definite function w on Af, w: Rn -+ R, such that v(t, x) <_w(x) for all (t, x) e T~x Af; (ii) decreasing on T~ x S iff (i) holds for Af = (iii) decreasingin the wholeon 7-r iff (i) holds for AfR n. Theexpression "on T~" is omitted iff all corresponding conditions hold for every ~- ~ R. Definition 1.4.11 implies. Proposition 1.4.4. Necessary and sufficient for v to be decreasing on Tr x Af whenAf is a time-invariant neighborhood of x = 0 is existence of a comparisonfunction ~ E K[o,a), where a = sup{[[x[[: x E .Af}, such that v(t, z) =v_(t, z) ~( llxll) for a~ T~× ~ wherev_ (t, x) is negative semi-definite on T~. Barbashin and Krasovskii [1,2] discovered the concept of radially un- boundedfunctions. Theyshowedthe necessity of it for asymptotic stability in the whole. Definition 1.4.12. A function v: R x R n -~ R is: (i) radially unboundedon T~, r ~ R, iff [[x[[ -~ +coimplies v(t, x) +oofor all t ~ T~; (ii) radially unboundediff [[x[[ -+ +ooimplies v(t, x) -~ +oofor all t~Tr for all Thenext can be easily verified (see Hahn[2]). Proposition 1.4.5. Necessary and sufficient for a positive definite in the whole (on ~ ) function v to be radially unboundedis that there exists ~ ~ KRobeying, respectively, v(t,x) >_~([[x[D/’or a11 x ~ R n and for a11 t ~ R (for all t ~ ~). For the details see Barbashin and Krasovskii [1,2], Gruji6, et al. [1], Hahn[2], Martynyuk[9], etc.
- 35. 14 1. PRELIMINARIES 1.4.3 Vector Liapunov functions Wereturn back to system (1.2.7) and assumethat for it the vector function (1.4.5) = ,vm(t,x)) T is constructed in some way, whose components va ¯ C(Tr × ~, R+), s 1, 2,..., m. For the function (1.4.5) to be suitable for stability analysis the equilibrium state x = 0 of system(1.2.7) it is necessary for it to possess the normtype properties (see Definitions 1.4.7-1.4.12). The presence such properties of function (1.4.5) is established in terms of one of the following functions (see Lakshmikantham,Matrosov, et al. [1]) (1.4.6) v(t,x) = max va(t,x); 8e[1 V(t,X, Ol) = o~TY(t,X), O~ m, (1.4.7) ai=const, i=l,2,...,m; m (1.4.8) v(t, x) = ~ vdt, x); i=1 (1.4.9) v(t,x) = Q(Y(t,x)), Q(0) Q ¯ C(R~, R+), the function Q(u) is nondecreasing in u. Since the func- tions (1.4.6) - (1.4.9) are scalar and are constructed in terms of the vector function (1.4.5), the special properties of the vector function (1.4.5) established according to Definitions 1.4.7-1.4.12. Remark1.4.2. Properties of positive definiteness, decrease and radial unboundedness of the function (1.4.5) follow from the algebraic inequalities, provided that the componentsv8 (t, x) of the vector function (1.4.5) satisfy the conditions (1.4.10) 1 -1 ail¢~ (llxill) _< v~(t,x) <_a~2.C~(llx~ll), 1,2,...,m, where ai~, aiu > 0 and ¢il and ¢i2 are of class K(KR), xi Rn’, n~+ T _1 ¯ = IIxll = Theconditions (1.4.10) are the broadest ones under which the algebraic conditions of the property of having a fixed sign can be established for the vector function (1.4.5).
- 36. 1.4 SCALAR, VECTORANDMATRIX-VALUED LIAPUNOVFUNCTIONS 15 The assumptions on the components vi(t, x) of the vector function are knownbeing other than (1.4.10): (a) ~li([[xi[[) _~ vi(t, xi) ~_ ~2i([[Xi[[), for all (t, xi) e ~ x n’, w here ~u, ~o2i are of class K(KR), i =1, 2,..., rn; (b) r~d[xd[ _<vi(t, xi) <_~i[[xi[[, for all (t, xi) e ~ x n’, w here ~i a nd hi are positive.constants, i : 1, 2,..., m, (see MichelandMiller [1], ~iljak [1], etc.). 1.4.4 Matrix-valued metafunction Assumethat for system (1.2.7) the two-indexes system of functions Vll(t,X) ... (1.4.11) II(t, x) = ... ".. ... ~Vtl(t,X) ... Vlk(t,X) is constructed, where vii E C(T x n,R), i = 1,2,...,k; j = 1,2,...,l. Definition 1.4.13. A function II: 7-r x R"~ -~ R kxl is called the matrix- valued meta~nction, if one of the Liapunov functions can be constructed based on it, n~ely, a scMar, vector or simple matrix-valued one, which solves the problem on stability of the equilibrium state x = 0 of sys- tem (1.2.7). Theproperties of having a fixed sign of metafunction (1.4.11) are estab- lished by a general rule in terms of one of the functions (1.4.12) ~(t,x) = m~ vij(t,x), (1.4.13) vn(t, x, where a ~ Rt, a = const ~ O, B~R k, ~ = const ~ O; k 0.4.14) ,.(t, n(t,z)=¢(a(t, where ~eC(Raxt,R+), ~(0) =0, ~(s) >0 for s>0, and lim ¢(s) +~.
- 37. 16 1. PRELIMINARIES Since the functions vri(t, .) determined by (1.4.12)- (1.4.15) are scalar, the ordinary technique of the Liapunov functions method is used to check their property of having a fixed sign, decreasing and radially unbounded- hess. Remark1.4.3. If k = I = min (1.4.11), then II(t,x) becomesan ordi- nary matrix-valued function U(t, x) (1.4.16) ( val(t,,) ... v~,~(t,x) u(t,x) ... ".. ... , v,~(t,x) ... v,~,,(t,~) where U: T~ × Rn --r R "~×m . The property of having a fixed sign, decreasing and radial unbounded- ness of the matrix-valued function (1.4.16) is established, provided that the elements Vsk(t, x), s, k = 1, 2,..., m, satisfy the estimates for all (t,x) EToxA/" (for all (t,x) eT~xRn), for all s = 1, 2,..., m, and (cf. Djordjevid [2]) _a,r¢81(llxsll)¢r1(llxrll ) <Vsr(t,x) <~,r¢,2(llzsII)¢r2(llx~ll) for all (t,x)~ToxAf (for all (t,x)~T~xR’~), whenall s ¢ r. Weshall formulate the assertions on the property of having a fixed sign of the matrix metafunction similar to howit has been done for the ordinary matrix-valued function (see Martynyuk[5- 7, 20]). Proposition 1.4.6. A metafunction H: Tr x Rn -~ R~x~ is positive definite on 7"r, r ~ R iff there exists a ~ Rt, ~ ~ R k, and a ~ K, and it can be written as via(t,z,~,~)=~wII+(t, x)~ a( llxll), whereII+(t, x) is positive semi-definite on "!’~.
- 38. 1.5 COMPARISON PRINCIPLE 17 Proposition 1.4.7. A metafunction II: 7-r × R’~ -~ RTM is decreasing on 7"~, T E R if[ there exists a 6 R~, /9 6 R k, and b 6 K, and it can be written as = Tri_(t, x)/9+b(llxll), wherelI_ (t, x) is negative semi-definite on 7-~. Proposition 1.4.8. A metafunction l-I: To x Rn -~ R TM is radially unboundedin the whole (on ~ ) if[ it can be written Vll(t,x,o~,/9) = o~TII+(t,X)/9+ c(llxll), where II+(t,x) is positive semi-definite in the whole (on T~), 6 R~, B 6 Rk, and c 6 KR. Remark 1.4.4. If k = l = m, the vectors a and/9 are replaced by one vector y 6 Rmand Propositions 1.4.6-1.4.8 become the knownones (see Martynyuk [20]). 1.5 Comparison Principle In this section weformulate the basic comparisonresults in terms of Liapu- nov-like functions and the theory of differential inequalities that are neces- sary for our later discussion (see also Yoshizawa [1], Szarski [1], etc.). For system(1.2.7) weshall consider a continuous function v(t, x) defined on an open set in T~x Af. Weassumethat v(t, x) satisfies locally Lipschitz condition with respect to x that is, for each point in Tr x Af there are a neighborhood T~ x ,9 and a positive number L > 0 such that Iv(t,x) vC t, y) l <LI Iz -Yl l for any (t, x) ~ Tr x S, (t, y) e Tr Definition 1.5.1. Let v be a continuous (either scalar, vector or matrix- valued) function, v: T~xR’~ ~ R sx’, v(t,x) ~ C(T~×Af), and let solutions X of the system (1.2.7) exist and be defined on T~ x Af. Then, for all (t, e (i) D+v(t,x)= limsup (v[t+t~,~;(t+o;t,z)]-v(t,z) 0 :8-+0 + is the upper right Dini derivative of v along the motion Xat (t, x);
- 39. 18 1. PRELIMINARIES (ii) D+v(t,x) li minfr ~ ~[t+e,x(t+~t,x)]-~(t,x): 0 +} is the lower right Dini derivative of v along the motionX at (t, x); (iii) D-v(t,x) li msup {’[t+°’x(~+~’x)]-~{t’~) : 0 ~ 0-) i s upper left Dini derivative of v along the motionXat (t, x); (iv) D_v(t,x) li minf- {~[~+e,x(~+~t,~)]-~(~,~): 0 ~ 0-~ is lower left Dini derivative of v along the motionX at (t, x). (v) The function v has Eulerian derivative ~, ~)(t,x) = ~ v(t,x), at (t, x) along the motionXiff D+v(t, x) = D+v(t, x) = D-v(t, x) = D_v(t, x) = and then i~(t, x) =Dr(t, If v is a scalar function and differentiable at (t, x) then (see Liapunov [1]) Ov O(t, x) = ~- + (grad v)Wf(t, x), where gradv = ~-~ ’ Ox2’" " Ox,~ ] " Effective application of D+vin the framework of the second Liapunov methodis based on the next result by Yoshizawa[1], which enables calcu- lation of D+vwithout utilizing system motion themselves. Theorem1.5.1. Let v be continuous and locally Lipshitzian in x over T~ x 8 and ~q be an open set. Then, D+v(t’~)[(1.2.7) lim su p { v[t +0,x + Of (t,O x)] - v(t, ~) holds along solutions X of ehe system (1.2. 7) at (t, x) ~ T~ D*v will mean that both D+vand D+vcan be used. The system of equations (1.2.7) is considered with the matrix-valued function U(t, x). Definition 1.5.2. All scalar function of the type (1.5.1) v(t, X, Or) = otTU(t, x)a, a ~ rn, where U ~ C(T~x Af, R’~×’~), are attributed to the class SL.
- 40. 1.5 COMPARISON PRINCIPLE 19 The vector a can be determined in several ways(see Martynyuk[12]) and its choice can effect the property of having a fixed sign of function (1.5.1). ByDefinition 1.5.1 for function (1.5.1) whenall (t, x) ¯ Tr xAfthe total derivative is calculated by virtue of system(1.2.7) (1.5.2) D+v(t, x, a)[(1.2.~) = aTD+U(t, where D+U(t,x) is calculated element-wise. Let us consider the following scalar differential equation du (1.5.3) d-~ = g(t,u), u(to) = Uo>_ O, to ¯ R (to ¯ where g ¯ C(RxR, R) (or g ¯ C(Tr×R,R)) and g(t,O) = for al l t ¯ To . Definition 1.5.3. Let ~,(t) be a solution of (1.5.3) existing on interval J = [to, to + a), 0 < a _< +~, to ¯ R (to ¯ Tr). Then~,(t) is to be the maximal solution of (1.5.3) if for every solution u(t) -~ u(t; t0,x0) of (1.5.3) existing on J, the following inequalities hold (1.5.4) u(t) <_~/(t), t ¯ g, to ¯ R A minimalsolution is defined similarly by reversing the inequality (1.5.4). Proposition 1.5.1. Let U: ~ × Af ~ Rm×’~, U(t,x) be locally Lip- schitzian in x. Assumethat (1) function g ¯ C(TrxRnxR+,R), g(t,O,O) = 0 existsforall t such that D+v(t,x,a)[(1.2.~ ) <_g(t,x,v(t,x,a)) for all (t,x,a) xJ~fx Rm; (2) solution x(t) =x(t;to,Xo) of system (1.2.7) is definite and nuousfor a11(t; to, x0) ¯ To× Tr × Af; (3) maximalsolution of the comparison equation du d--~ = g(t,x,u), u(to) = uo, x(to) exist for all t ¯ T~. Then the estimate v(t,x(t),a) <_ r(t;to,xo,uo) for all holds wheneverv(to, xo, a) <_uo. For the proof see monographsby Lakshmikantham,Leela, et al. [1].
- 41. 20 1. PRELIMINARIES Proposition 1.5.2. Let U: ~ x Af -~ R"~×’~, U(t,x) be locally Lip- schitzian in x. Assumethat (i) function g 6 C(Tr x n xR+ , R) exi sts suc h tha D+v(t,x,a)[(1.2.~ ) >_g(t,x,v(t,x,a)) for all (t,x,a) e T~ x Af (ii) solution of system(1.2. 7) is definite andcontinuousfor (t; to, xo) Tox ~x.,V’; (iii) minimalsolution r- (t; to, xo, wo)of the comparisonequation d-~ =g(t, x, w), w(to) = wo exists for all t ~ 7"~. Theninequality v(t, Xo, a) >_woyields the estimate ~(t,x(t),~)>_ ~-(t;to,xo,~,o) for all t ~ ~. Propositions 1.5.1 and 1.5.2 are a scalar version of the principle of com- parison with the matrix-valued function. In the monographby Zubov[4] the following assertions are proved. Corollary 1.5.1. Let (i) function (1.5.1) obey the bilateral inequality ~(t)~ ~(t)<~(t, ~,~)<_~2(t)~ ~(t ), where~oi(t) >0 for all 6 To and p(t) = (xT (t to, Xo)x(t; to, Xo)) (ii) function g(t, x, v) satisfy the estimates --¢1(t)p2(t) where ¢i(t) _>0 for all t 6 To and functions ¢i(t)/qoi(t), i = are integrable. Then for the solutions of system (1.2. 7) the estimates 1 1 ¢~(~) <: Po~2~ (t0)~- ½ (t) exp - ~ dT to are valid for all t 6 Toand to 6
- 42. 1.5 COMPARISON PRINCIPLE 21 Corollary 1.5.2. Let (i) function (1.5.1) obey the bilateral inequality qo~ (t)ph(t) _<v(t, x, a) <_qo~ (t)p ~=(t), where ~oi(t) are piece-wise continuous positive functions given for all t E A = [to,to +T], 11 >_12 are positive numbers; (ii) function g(t, x, v) satisfy the estimates -¢1(t)p ~’(t) <g(t,x,v)<-¢~.(t)p ~2 (t), where ¢i(t), i = 1, 2 are positive piece-wise continuous functions for all t ~ A, kl ~ k2. Thenfor the solutions of system (1.2. 7) the estimates 1 1 to are valid for all t ~ A, v0 = v(to,xo,a), Ai = ~, ,~i > 1 for i = 1,2. Corollary 1.g.3. Let both conditions (i) and (ii) of Corollary 1.5.2 satisfied ~mclAi >1, i = 1, 2. Thenfor !1 =12 =l the solutions of system (1.2. 7) satisfy the estimate ~o; l(t)vo 1+(~1- 1)v~ ~-1I~(’~la~ <_ o(t) ~ ~r~(t)Vo 1 + (A2 - 1)v~’-~ f~(~)d~ ~o for all t ~ A, where {¢l(t)~i -~’(t) ¢1(t)~ ~’(t) t" ¢2()~2(t) ¢~ ~1 (~) if ~)l(t) ~ if ¢1(t)< if ¢2(t)>o, if ~)2 (t) <
- 43. 22 1. PRELIMINARIES Corollary1.5.4. Let both conditions (i) - (ii) of Corollary 1.5.2 be tisfied and A1 = A2= 1. Then for ¢i _> O, i = 1,2 every solution ofsystem (1.2. 7) satisfies the estimates qo~l(t)vo exp ¢l (s)~o-~l(s) as <_p(t) x tO for all t ¯ A. < ~?l(t)vo exp - to Definition 1.5.4. All vector functions of the type (1.5.5) L(t, x, b) =AU(t, x)b, are attributed to the class VL. Here U ¯ C(T~ x Af, R’~×m), A is constant m x m-matrix, and b m-vector. For the vector function (1.5.5) wecalculate (1.5.6) D+L(t, x, b)[(1.9..7) AD +U(t, z) for all (t, x, b) ¯ T~x Af x R~. Proposition 1.5.3. Let U: Tr x Af --r Rre×m, U(t,x) be locally Lip- schitzian in x. Assumethat (1) constant m x m-matrix A, a v ector b ¯ Rr~,a vec to r y ¯ R m and a function a ¯ K exist such that yWL(t,x, b) >_a(llxll) for all (t, x, b) ¯ 7"r x Af x Rr~; (2) vector function G ¯ C(7-~ x n xR m , R m ) is such tha t G(t , x, u) is quasimonotone nondecreasing in u for any t ¯ Tr and D+L(t,x, b)1(1.2.7) < G(t, x, L(t, x, b)); (3) solution x(t) = x(t;to,xo) of system (1.2.7) is definite and tinuous for (t;to,xo) ¯ Tr x Ti x Af and the maximal solution w +(t; to, Wo)of the comparisonsystem d--[ = G(t,x,w), w(to) exists for all t E T~.
- 44. 1.6 LIAPUNOV-LIKE THEOREMS 23 Thenthe inequaJity L( to, Xo, b) <_~0 implies the estimate (1.5.7) L(t, x(t), <w +(t; to, x0, w0) for all t¯ ~. Besides, estimate (1.5.7) holds component-wise. Theproof of Proposition 1.5.3 is similar to that of Theorem 3.1.2. from Lakshmikantham, Leela, et al. [1]. 1.6 Liapunov-Like Theorems There are several directions in stability theory to search for newconditions which weakenone of other suppositions of the original Liapunov’s theorems. Werecall the classification of these directions: (1) search of minimal weakassumptions on the properties of auxiliary functions (semi-definite functions, integral positive functions, etc.); (2) modification of assumptions on the properties of total derivative scalar function along solutions of perturbed motion equations; (3) construction and application of multicomponentauxiliary functions (vector, matrix-valued, metafunctions). It is natural to expect the developmentof both the first and the second directions within the frameworkof the third one. Further on this section basic theorems of the direct Liapunov method are set out in terms of the matrix-valued functions. Also, maindefinitions of the class of matrix-valued functions are presented, here. 1.6.1 Matrix-valued function and its properties Together with the system (1.2.7) weshall consider a two-indices system functions (1.6.1) V(t,x) [v~(~,~)], i, j = 1, 2,...,,~, where v~ ¯ C(Tr x R’*,R+), and v~j ¯ C(T~ x n,R) f or a ll i ¢ j. Moreover the next conditions are making (i) vlj (t, x) are locally Lipschitzian in (ii) v~j(t,O) for all t ¯ R(t ¯ T~), i, j = 1,2,... ,m; (iii) v~j (t, x) = v~.~ (t, x) in any open connectedneighborhood Afof x = O for all t e R+
- 45. 24 1. PRELIMINARIES Let y E /~rn, y ~ 0, be given. By means of the vector y and matrix- valued function (1.6.1) weintroduce the following function v(t, x, y) =yWU(t,x)y. The following definitions will be used throughout the book, which are based on the corresponding results by Djordjevid [3], Grujid [2], Hahn[2], Krasovskii [1], Liapunov [1], and Martynyuk[3-7]. is: Definition 1.6.1. The matrix-valued function U: Tr × Rn -~ R mxm (i) positive semi-definite on Tr = [r, +~), r E R iff there are time- invariant connected neighborhood Af of x = O, Af C Rn, and vector yERm, y#0 such that (a) v(t,x,y) is continuous in (t,x) ~ 7"~ x Af × Rm; (b) v(t, x, y) is non-negativeon A;, v(t, x, y) >_for al l (t , x, O) e T~ x Af x m and (c) vanishes at the origin: v(t, 0, y) = 0 for all (t, y) E T~Rm ; (d) iff the conditions (a)- (c) hold and for every t e T~, there w E Af such that v(t,w,y) > 0, then v is strictly positive semi-definite on (ii) positive semi-definite on 7-r x S iff (i) holds for Af = ~q; (iii) positive semi-definite in the wholeon 7-r iff (i) holds for Af (iv) negative semi-definite (in the whole) on 7"r (on Tr x Af) iff (-v) positive semi-definite (in the whole)on Tr (on T~x Af), respectively. The expression "on 7"~" is omitted if[ all corresponding requirements hold for every ~ E R. Definition 1.6.2. The matrix-valued function U: ~ x Rn -~ R "~× ’~ is: (i) positive definite on ~, ~ ~ R, iff there are a time-invariant con- nected neighborhood iV" of x = O, Af _c R n and a vector y ~ R m, y ¢ 0 such that it is both positive semi-definite on Tr x Af and there exists a positive definite function w on A/’, w: R n -~ R+, obeyingw(x) <_v(t, x, forall (t, x, y) ~ Tr x Af x R’~ (ii) positive definite on T~x S iff (i) holds for Af = (iii) positive definite in the wholeon T~iff (i) holds for fl/" (iv) negative definite (in the whole) on 7"r (on 7-~ x Af m) iff (-v) positive definite (in the whole)on Tr (on Tr x.h/’x Rm),respectively;
- 46. 1.6 L1APUNOV-LIKE THEOREMS 25 (v) weakly decreasing on ~ if there exists a Aa > 0 and a function a e CKsuch that v(t,x,y) < a(t, Ilxll) as soon as Hxll < A1and (t, y)e T~ x (vi) asymptotically decreasing on "]-r if there exists a A2 > 0 and a function b ¯ KLsuch that v(t,x,y) < b(t, llxll ) as soon as I]xll < As and (t, y) ¯ T~ R m . The expression "on T~" is omitted iff all corresponding requirements hold for every ~- ¯ R. Proposition 1.6.1. The matrix-valued function U: R x R n -~ R rnx m is positive definite on Tr, ~- ¯ R iff it can be written as yTu(t, x)y = yTU+(t,z)y a(llxll), whereU+(t, x) is positive se mi-definite matrix-valued fu nction and a ¯ K. Definition 1.6.3. (cf. Gruji6, et al. [1]). Set v¢(t) is the largest con- nected neighborhood of x = 0 at t 6 R which can be associated with a function U: R x Rn -~ Rmxmso that x ¯ v¢(t) implies v(t,x,y) < y¯R m. Definition 1.6.4. The matrix-valued function U: R x Rn --+ Rs×s is: (i) decreasing on T~, ~" ¯ R, if[ there is a time-invariant neighborhood Afof x = 0 and a positive definite function won A/’, ~: R n --r R, such that yWU(t,x)y <_~(x), for all (t, x) ¯ T~x A/’; (ii) decreasing on 7"r x S iff (i) holds for Af= (iii) decreasingin the wholeon ~ iff (i) holds for A/" = ~. Theexpression "on "/-r " is omitted if[ all correspondingconditions still hold for every T ¯ R. Proposition 1.6.2. The matrix-valued function U: R x Rn -+ R mxm is decreasing on ~, ~- ¯ R, iff it can be written as yTu(t, z)y = yTu_ (t, x)y b(llxll), (y # 0) m, where U_(t,x) is a negative semi-definite matrix-valued and b¯ K. is: function Definition 1.6.5. The matrix-valued function U: R x R ~ ~ R "~x’~ (i) radially unbounded on Tr, r ¯ R, iff Ilxll-+~ implies yTU(t, x)y +cx~, for all t ¯ Tr, y ¯ Rm; (ii) radially unbounded, iff Ilxll --r ~x~implies yWU(t,x)y -+q-oo, for all t ¯ "Fr, and for all 7- E R, y ¯ R "~, y ~ 0.
- 47. 26 1.PRELIMINARIES Proposition 1.6.3. The matrix-valued function U: 7-r x Rn -+ R m×m is radially unbounded in the whole (on T~ ) iff it can be written yTu(t,X)y : yTU.b($,x)y-[-a([[x[[ ) for all x e a n, where U+(t,x) is a positive semi-definite matrix-valued function in the whole (on Tr) and a E K For the proof of Proposition 1.6.1-1.6.3 see Martynyuk[9, 20]. 1.6.2 A version of the original theorems of Liapunov Thefollowing results are useful in the subsequent sections. Proposition 1.6.4. Suppose re(t) is continuous on (a, b). Then re(t) is nondecreasing(nonincreasing) on (a, b) D+m(t) >_0 (<_0) ever y t E (a ,b ), where D+m(t)li msup {[m(t + 8) - m(t )]8-1 : 8 -+ FollowingLiapunov[1], Persidskii [1], and Yoshizawa[1] the next result follows (see Martynyuk[20]). Theorem1.6.1. Let the vector function ] in system (1.2.7) be conti- nuous on R x Af (on Tr x fir). If there exist (1) an open connected time-invariant neighborhood ~q C_ Af of point x = O; (2) positive de finite on Af (onTr xAf)ma trix -valued functi on U ( t, and vector y ~ Rm such that function v(t,x,y) = yWU(t,x)y locally Lipschitzian in x and D+v(t, x, y) <_ Then (a) (b) the state x = 0 of system (1.2. 7) is stable (on Tr), providedU(t, is weakly decreasing on Af (on Tr x Af); the state x =0 of system (1.2. 7) is uniformly stable (on Tr), vided U(t, x) is decreasing on Af (on Tr x Af).
- 48. 1.6 LIAPUNOV-LIKE THEOREMS 27 Corollary 1.6.1. Assumethat the functions vii(t, x) in (1.6.1) are con- tinuously differentiable for all (t, x) E T~x Af, and m m (1.6.2) Dv(t,x,y)[(1.2.7 ) =~, ~y,yjDv,~(t,x)[(1.2.7 i=1 j----1 If there exist functions Oij(t,x,y), #)ij(t,O,y) =0 for y # 0 for all i, i,2,..., m,such that (1.6.3) yiyjVvij (t, x)[(1.2.7) -~ ~ij (t, for all (t, x) ~ Tr x Af, then (cf. Djordjevid[3]) (1.6.4) DvM(t,x,y)[(1.2.7) ~’~¢ij(t,x,y) = eWO (t,x,y)e, i=1 j=l where e = (1, 1,..., 1)wE R~. Further we denote 1[¢(~, ~,v)+cT(t, ~,V)] (1.6.5) B(t, x, y) = and assume that there exist comparison functions Wl([[x[D,... , of class K and matrix/~(t, x, y) such that (1.6.6) eTB(t, =,y)e <_ ~T(llxll)h(~, =,Y)~(ll=ll) for all (t, x, y) E T~x A/" x m. Compile the equation (1.6.7) det [~(t, x, y) - AE]= where E is an mx midentity matrix. Designate the roots of this equation by Ai = Ai(t, x, y), i = 1, 2,..., m. It is easy to verify that DvM(t,x, y) <_ in domain Tr x Af x R m if (1.6.8) A~(t,x,y) <_for al l i = 1,2 .. .. ,m, and all (t, x, y) e Tr x A/" R ra. Conditions (1.6.8) together with the other conditions of Theorem1.6.1 is a sufficient test for stability and uniformstability of the state x = 0 of system(1.2.7).
- 49. Another Random Scribd Document with Unrelated Content
- 50. that of the foe).⁴⁰³ And the steeds of my car were worn out by carrying it; and they were battered, and helpless, and perspired like kine in a shower. And bad omens fast sprang up before us. And on these occurring, I perceived that things would go against us. O thou endowed with exceeding might, a charioteer should be conversant with season and place, with omens, and the expressions of emotions; as also with depression of spirits, exhilaration, and grief. And he should have a knowledge of low, level and uneven grounds, and the time for conflict, and he should be able to perceive the shortcomings of the enemy. And a charioteer mounted on a car, should know when to draw near an enemy, when to turn away from him; when to stay; and when to turn round from before the foe—all these (he should know). What I, for bringing respite unto thee as well as the horses of the car, have done by way of removing the terrific exhaustion, is proper. I did not, O hero, turn away the car of my own sheer will. What I have done, O lord, had been dictated by my affection for thee. Command me. What thou sayest, O destroyer of foes; I will do every way, O hero, with my whole soul". Thereat, well-pleased with the speech of the charioteer, Rāvana, eager for encounter, after praising him in various ways, said,—"O charioteer, do thou swiftly take the car towards Rāghava. Without slaying his foe in fight; Rāvana turneth not away (from the field)". Speaking thus, Rāvana—lord of Rākshasas—gave the charioteer on the car an excellent ornament for the hand. Hearing Rāvana’s words, the charioteer drove the car. And urged on by the speech of Rāvana, the charioteer drove on the steeds,—and in a moment the mighty chariot of the Rākshasa chief appeared before Rāma in the field of battle. ⁴⁰² On the ascension of the Moon.
- 51. ⁴⁰³ Two negatives in this verse amounting to an affirmitive. This is the only instance of double negatives in Vālmiki.—T. SECTION CVI. Then the revered Agastya, who, desirous of witnessing the fight, had along with the deities come there,—seeing Rāvana spent with the toil of conflict, staying in the field plunged in thought, and stationed before Rāma for engaging in encounter,—addressed Rāma, drawing near to him, saying,—"Rāma, Rāma, O mighty-armed one, hearken to the eternal secret, whereby, my child, thou wilt conquer all foes in fight,—_Aditya-hridaya,_⁴⁰⁴ sacred, capable of destroying all foes, bringing victory—the recitation, enduring and indestructible,—and supremely good; fraught with all welfare, removing every sin,— chasing away anxiety and grief, bringing length of days; and excellent. Do thou worship Vivaçwata’s offspring—the Sun—lord of the world, furnished with rays,—who maketh people engage in work, and who is bowed down to by deities and Asuras. This effulgent one, producing rays is instinct with the spirits of all the deities; and he with his rays ruleth all creatures—and the hosts of celestials and Asuras. This Sun is Brahmā and Vishnu and Siva and Skanda⁴⁰⁵ and Prajapati,⁴⁰⁶ and Mahendra and Dhanada⁴⁰⁷ and the Destroyer— Yama—and Soma⁴⁰⁸ and the Lord of waters; and the Pitris,⁴⁰⁹ and the Vasus, and the Sadhyas⁴¹⁰ and the two Açwinis⁴¹¹ and the Maruts and Manu,⁴¹² and the Wind-god and the God of fire and the creatures and the Creator of life and the seasons. And he is Aditya⁴¹³ and Savitā⁴¹⁴ and Suryya⁴¹⁵ and Khaga⁴¹⁶ and Pusha⁴¹⁷ and Gavastimān⁴¹⁸ and the Golden-looking and Bhānu⁴¹⁹ and
- 52. Hiranyaretā⁴²⁰ and Divākara.⁴²¹ And he is Haridaçwa⁴²² and Saltasrārchi⁴²³ and Saptasapti⁴²⁴ and Marichimān.⁴²⁵ And he subdueth darkness, and he is Sambhu⁴²⁶ and Tashta⁴²⁷ and Mārtandaka⁴²⁸ and Ançumān.⁴²⁹ And he is Hiranyagarbha,⁴³⁰ Sicira,⁴³¹ and Tapana,⁴³² and Ahaskara⁴³³ and Ravi,⁴³⁴ and Agnigarbha,⁴³⁵ and Aditi’s son,⁴³⁶ and Sankha,⁴³⁷ and Siciranāçana,⁴³⁸ Byomanātha⁴³⁹ and Tamabheda,⁴⁴⁰ the one proficient in Rik, Yajus and Sāma; and Ghanavrishti,⁴⁴¹ and the friend of the Apas,⁴⁴² and he that swiftly courseth in the Vindhya way. And he is Ātapi⁴⁴³ and Mandali⁴⁴⁴ and Mrityu.⁴⁴⁵ And he is Pingala,⁴⁴⁶ and the destroyer of everything, and the Omniscient, and he having the universe for his form,⁴⁴⁷ and the exceedingly energetic one, and the beloved of all, and that one lording it over all kinds of actions. And he is the lord⁴⁴⁸ of stars and planets and constallations, and the origin of everything, and the one powerful pre-eminently of powerful things⁴⁴⁹—and the one having twelve forms.⁴⁵⁰ I bow unto thee (having these forms and functions). Salutation unto the Eastern mount and the mount of the West. Salutation unto the lord of the stellar bodies and salutation also unto the lord of day. Salutation and salutation unto him that bringeth victory, and the joy that springeth up from victory; and unto him of yellow steeds. Salutation, salutation, O thousand-rayed one; Salutation and salutation unto Āditya. Salutation unto him that keepeth his senses under subjection; Salutation and salutation unto the Hero,⁴⁵¹ and unto Sāranga⁴⁵² and unto him that awakenest the Lotus.⁴⁵³ And (salutation) unto thee, O fierce one. Salutation unto the Lord himself of Brahmā, Içāna⁴⁵⁴ and Achchyuta,⁴⁵⁵ and unto Sura⁴⁵⁶ and unto him that constitutes the knowledge of Āditya, and unto him that unfoldeth me and not-me; and unto the devourer of all, and unto the form of the destroyer of the darkness of ignorance,
- 53. Salutation unto the destroyer of darkness, and unto the destroyer of enemies, and unto him of immeasurable Soul, and unto the destroyer of the ingrate, and unto the deity, and unto the lord of all stellar bodies. And salutation unto him that boasteth of the splendour of burning gold, unto the destroyer of all mental obscurity, —and unto the maker of the universe. Salutation unto the remover of darkness; unto the illuminator of the Soul; unto the all-beholding one of all the worlds. The lord createth everything and verily destroyeth it. And with his rays he sucketh up, and destroyeth and createth (everything). When all are asleep, this one waketh, and he is resident in the hearts of all creatures. This one is both Agnihotra as well as the fruit reaped by the sacrifices thereof. And he constitutes the gods and the sacrifices and the fruit also thereof; and he is the lord of all acts that are performed by creatures. If a person recites this (hymn), he, O Rāghava, doth not come by misfortune, when he is in peril of his life, or is ill, or in a lonely place, or in fear. Do thou, with concentration, worship this god of gods, this lord of the universe. By reciting (this hymn) instinct with the three virtues, thou wilt obtain victory in battle. This very instant, O mighty-armed one, thou wilt conquer Rāvana". Having said this, Agastya went whither from he had come. Hearing this, that exceedingly energetic one had his grief gone, Then, well pleased, Rāghava, exerting himself, contemplated (the hymn). And reciting this, he beholding the Sun, attained excess of joy. And sipping water again and again, and becoming purified, that powerful one, taking up his bow, and viewing Rāvana, advanced with a delighted heart, to obtain victory. And he became intent on his death with his dearest energies. Then exceedingly delighted, and filled with rejoicings, the Sun, in the midst of the celestial hosts knowing that the destruction of the
- 54. Sovereign of the night-rangers was at hand,—spoke unto Rāma "Bestir thyself". ⁴⁰⁴ Lit—The heart of the Sun.—the designation of a Vedic Hymn. ⁴⁰⁵ The celestial generallissimo. The commentator gives a spiritual interpretation. ’He that by means of his rays openeth up the five organs of perception.’ ⁴⁰⁶ The lord of all creatures, by virtue of his bringing forth all beings through his energy. ⁴⁰⁷ Dispenser of riches, a name of Kuvera. ⁴⁰⁸ Furnished with splendour, a name of the Moon. According to some ’endowed with energy.’ ⁴⁰⁹ Lit. the ancestral manes. Here the generator of everything. ⁴¹⁰ An order of semi-divine beings. Here, ’He who is adored by the spiritual.’ ⁴¹¹ In virtue of his omnipresence and his being the healer of all ailments. ⁴¹² All-knowing and being the primaeval sovereign. ⁴¹³ ’He from whom all derive sustenance.’ ⁴¹⁴ ’The producer of heart and the spiritual faculties by heat, and corn, etc. by showers.’
- 55. ⁴¹⁵ ’Coursing alone, according to the commentator. It may also means—’He that sets people to work.’ ⁴¹⁶ ’Coursing the highest heavens,’ or says the commentator ’the heavens of the heart.’ ⁴¹⁷ ’The maintainer.’ ⁴¹⁸ Gavastimān—’Ray-furnished, or having the all-permeating Spirit of Auspiciousness.’ ⁴¹⁹ Bhānu—’having brightness.’ ⁴²⁰ Hiranyaretā—’instinct with the cosmic energy.’ ⁴²¹ Divākara—’maker of day’ ⁴²² Haridaçwa—’pervading all sides’ or ’having black steeds.’ ⁴²³ Sahasrārchi—’thousand-rayed. The commentator explains spiritually, He whose cognition points in infinite directions.’ ⁴²⁴ ’He from whom proceed the seven organs of sense of people.’ Or ’he who has seven steeds.’ ⁴²⁵ ’Having rays.’ ⁴²⁶ ’He from whom proceed the several sorts of happiness.’ ⁴²⁷ ’He who removes the misfortunes of his votaries.’ ⁴²⁸ ’He that infuses life into the lifeless mundane egg.’
- 56. ⁴²⁹ ’Having rays.’ ⁴³⁰ The cause of the creation, preservation and destruction of the Universe. ⁴³¹ ’Good-natured.’ ⁴³² Tapa—means ’wealth.’ Tapana—the possessor of all riches. ⁴³³ Ahas—day and Kara—maker. ⁴³⁴ Rauti—teacheth—Rāvi—he that teacheth. ⁴³⁵ Lit. fire-wombed. He that carries the fire of doom within himself. ⁴³⁶ Aditi—’without destruction’—means ’Brahma knowledge.’ ⁴³⁷ Supreme happiness. ⁴³⁸ The remover of intellectual stupor or evil-mindedness. ⁴³⁹ Lord of the welkin. ⁴⁴⁰ Dispeller of darkness. ⁴⁴¹ He from whom floweth the fruit of acts: or he from whom come downpours. The latter epithet is justified on grounds of Physical Geography, rain being ultimately dependant on solar heat. ⁴⁴² Apas may mean either ’the good,’ or ’water.’ Vindhya way means either the way known as Brahmanari or the orbit of the San.
- 57. ⁴⁴³ He that is intent on creating the cosmos. ⁴⁴⁴ Ray-crowned or adorned with gems. ⁴⁴⁵ The bringer of death. ⁴⁴⁶ The motive force of the blood-tube called Pingalā. ⁴⁴⁷ Or the ornament of the Universe. ⁴⁴⁸ i.e. the controller of them, remarks Rāmānuja. ⁴⁴⁹ Such as, observes the commentator, as fire. ⁴⁵⁰ i.e. the months of the year. ⁴⁵¹ Him that leadeth the senses, and that is endowed with the prowess of slaying Tripura etc. ⁴⁵² Him that deservest the pranaba, the holiest formula in all Hindu Scripture. ⁴⁵³ That awakenest the external lotus as well as the lotus of the heart. ⁴⁵⁴ Siva. ⁴⁵⁵ Vishnu. ⁴⁵⁶ The sun.
- 58. SECTION CVII. Then the charioteer fully drove with speed Rāvana’s car, capable of bringing down the hosts of foes, resembling in form a city of the Gandharvas, having elevated streamers, yoked with surpassingly superb steeds, engarlanded in gold; stocked with war-like implements; furnished with ensigns and standards; appearing to devour the welkin; making the earth herself resound; destructive to hostile hosts; and filling its own party with delight. And as it speedily descended, the monarch of men beheld that Rākshasa-king’s resounding car, having huge standards, yoked with black chargers, and endowed with fierce splendour; as if flaming in the firmament; having the resplendance of the Sun himself; with thronging lightning pennons; displaying the glow of Indra’s weapon;⁴⁵⁷ showering arms; and resembling rain-charged clouds. Seeing the enemy’s car resembling a mass of clouds having a chatter resembling the sounds sent by a cleaving mountain rived by the thunder, Rāma, vehemently drawing his bow curved like the infant moon, addressed Mātali— charioteer unto the thousand-eyed (deity),⁴⁵⁸ saying,—"O Mātali, behold the enraged chariot of my foe as it courseth on. From the furious speed with which he is again wheeling at my right, it appeareth that he hath set his heart on slaying me in encounter. Do thou therefore heedfully drive the car right against the vehicle of my foe. I wish to destroy this one even as the wind scattereth clouds that have appeared. Do thou with all thy wits about thee, without trepidation, and holding thy heart as well as thy eye in calmness, swiftly drive the chariot ruled by the reins. Worthy of Purandara’s car, thou ought not to be taught by me. Desirous of encounter and my whole soul bent on fight, I simply remind thee—not teach thee". Pleased with these words of Rāma, the excellent celestial charioteer
- 59. —Mātali—drove the car. Then leaving Rāvana’s mighty car on the right, he enveloped Rāvana with the dust raised by the wheels. Thereat the Ten-necked one, enraged, with his eyes coppery and dilated (in passion), covered with arrows Rāma staying in front of his car. Enraged at the smiting, Rāma, with his ire aroused, but summoning up patience, took up in the encounter the bow of Indra endowed with exceeding vehemence; as well as highly impetuous shafts having the resplendance of the solar rays. And then there began a furious encounter between those (two) eager for slaying each other; confronting each other like unto flaming lions. And then desirous of destruction of Rāvana,—celestials with Gandharvas, and Siddhas and supersaints assembled to go to behold the encounter taking place between the two cars. And for the destruction of Rāvana and the success of Rāghava, there occurred round about the cars terrible bodements capable of making people’s down stand on end. The god poured down showers of blood on the car of Rāvana; and a violent tornado eddied on his right. And a mighty swarm of vultures, wheeling in the heavens, pursued the car wherever it moved. And Lankā was enveloped with evening resembling the red javā flowers and even in day appeared ablaze. Lightnings and firebrands accompanied by a terrible sound began to fall down on all sides. And beholding these omens inauspicious unto Rāvana all the Rākshasas were greatly sorry. And wherever Rāvana moved the earth shook and the hands of all the Rākshasas fighting were as if paralysed. The copper coloured, the yellow, the red, and the white rays of the sun falling before Rāvana appeared like melted metals of a mountain. And the jackals followed by vultures, vomitting forth fire and casting their looks at him, began to emit inauspicious cries. And in that battle-field the unfavourable wind began to blow raising dust and obstructing the vision of the king of Rākshasas. On the
- 60. Rākshasa host on all sides dreadful lightnings were showered without the sound of the clouds. All the quarters were enshrouded with darkness and the welkin became invisible being covered with darkness. And setting up a dreadful quarrel hundreds of terrible _Sharikas_⁴⁵⁹ began to fall down on his chariot. The horses emitted forth sparks of fire from their hips and tears from their eyes. These and various other dreadful omens arose there announcing the destruction of Rāvana. And there appeared on all sides many an auspicious and good sign intimating the approach of Rāma’s victory. And beholding all those auspicious marks announcing Rāma’s success, Lakshmana was greatly delighted and considered Rāvana as slain. Thereupon beholding all those auspicious signs, Rāghava, well qualified to decipher them attained an excess of delight and became anxious to display a greater prowess. ⁴⁵⁷ The rain-bow. ⁴⁵⁸ Indra. ⁴⁵⁹ A kind of bird (Turdus Salica, Buch). SECTION CVIII. Thereupon there ensued a mighty and dreadful encounter of two cars between Rāma and Rāvana, creating terror unto all people. And the army of Rākshasas and the mighty host of the monkeys, although they had weapons in their hands, became stupified (for the time being). And beholding them (Rāma and Rāvana) fight, all the Rākshasas and monkeys, having their minds agitated, were greatly
- 61. surprised. With various weapons and hands uplifted for fight, they, greatly wondered, stood there beholding them and did not address themselves to fight with each other. The Rākshasas beholding Rāvana, and the monkeys beholding Rāma with wonder-stricken eyes, the whole army appeared like a picture. And espieing all omens Rāghava and Rāvana began to fight, undaunted, firm, resolute and unagitated by anger. And determining that Kākutstha would win victory and Rāvana would die, they began to display their own prowess. Thereupon the highly powerful Rāvana, setting his arrows in anger, discharged them at the pennon stationed on Rāghava’s car. Those arrows reaching the flag staff of the Purandara chariot and perceiving its might fell down on the earth. Thereupon the highly powerful Rāma, wroth, stretching his bow, made up his mind to return the blow. And aiming at Rāvana’s flag staff he discharged a sharpened shaft flaming unbearably by its own lustre like a huge serpent. And the effulgent Rāma discharged a shaft aiming at (Rāvana’s) banner which, piercing the Ten-necked demon’s flag fell, down on the earth. And beholding his flag staff thus broken down the highly powerful Rāvana became ablaze as if burning down every tiling with his unbearable ire And being possessed by wrath he began to make a downpour of shafts. Rāvana then, with flaming arrows, pierced Rāma’s steeds. The celestial horses were not bewildered thereby nor their course was slackened. And they remained thoroughly unagitated as if they were stricken with lotus stalks. Beholding the steeds thus unmoved Rāvana was again exercised with wrath and began to discharge afresh his various weapons—gadās, parighas, chakras and musalas, mountain tops, trees, darts and parashus and thousands of other shafts by virtue of his illusive powers. And unmoved was his energy. And that downpour of various weapons became huge and terrible in the
- 62. conflict creating terror and making a dreadful noise. Thereupon leaving aside Rāghava’s car he began to assail the monkey-host and enveloped the sky with a continual discharge of arrows. The Ten- necked demon let loose many a weapon even at the risk of his own life. And beholding Rāvana in the encounter thus actively engaged in the discharge of arrows, Kākuthstha, smiling, set up pointed shafts, and discharged them by hundreds and thousands. Beholding them Rāvana again filled the welkin with arrows—and thus with shafts discharged by them both another flaming sky was created. None (of the arrows) missed the aim, none of them failed to pierce another and none of them was fruitless. And the arrows discharged by Rāma and Rāvana stricking each other fell down on the earth. And they standing on their right and left began to make a continued downpour of arrows and enveloped the sky entirely. And they opposing each other, Rāvana slew Rāma’s steeds and Rāma in his turn slew Rāvanan’s. They, thus enraged fought with each other and for sometime there ensued a terrible encounter capable of making ones down stand on end. And the highly powerful Rāvana and Rāma⁴⁶⁰ righting with each other in the conflict by means of sharpened arrows, the lord of Rākshasas beholding his flag staff broken down became enraged with the foremost of Raghus. ⁴⁶⁰ The epithet in the text is Lakshmana’s elder brother. SECTION CIX. Rāma and Rāvana thus opposing each other in battle, all the animals beheld them, stricken with astonishment. And those two great
- 63. heroes, highly angered, began to dash towards and assail each other; and being determined to slay each other they looked greatly dreadful. And their charioteers drove the cars on, displaying their skill by moving in circles, in rows and diverse other ways. And those two excellent heroes, discharging their shafts and influenced by illusions, assailed each other proceeding and receding, Rāma attacking Rāvana and Rāvana withstanding him. And these two cars coursed the earth for sometime like clouds accompanied by showers. And displaying many a movement in the conflict they again stood facing each other, the forepart of one car touching that of the other and the heads of the steeds touching each other; and the pennons, stationed on one touched those of the other. Rāma, with four sharpened arrows, shot off his bow, removed the flaming horses of Rāvana to some distance. And finding his steeds thus removed he was exercised with wrath. And the Ten-necked one discharged sharpened arrows at Rāghava. And he was pierced by those arrows coming from the powerful Ten-necked demon. He was neither overwhelmed nor pained therewith and he again discharged arrows resembling the thunder-bolts. And the Ten-necked demon again discharged arrows at the charioteer, which fell with great vehemence on the person of Mātali. Mātali was not the least pained or overwhelmed in that encounter. And beholding his charioteer thus assailed Rāma was excited with wrath and overwhelmed his foe with a net of arrows. And the heroic Rāghava showered on his enemy’s chariot shafts by twentys, thirtys, sixtys, hundreds and thousands. And the lord of Rākshasas, Rāvana, who was stationed on the car, wroth, attacked Rāma in the conflict with maces and Musalas. And there again ensued a terrible conflict capable of making one’s down stand on end. And the seven oceans were overwhelmed with the sound of maces, musalas, Parighas and gold feathered arrows. And
- 64. those inhabiting the regions under the agitated oceans, all the Dānavas and thousands of Pannagas were greatly pained. And greatly shook the earth with her mountains, forests and gardens. The Sun was shorn of its resplendance and the wind blew very rough. Thereupon the celestials, with Gandharbas, Siddhas, great saints, Kinnaras and serpents were all worked up with anxiety. And beholding the dreadful encounter between Rāma and Rāvana capable of making people’s down stand on end, the celestials with ascetics began to pray,—"May good betide the Brahmins and cows, may people live in peace and may Rāghava defeat Rāvana, the lord of Rākshasas, in the conflict". And the crowd of Gandharbas and Apsaras beholding that wonderful battle between Rāma and Rāvana, said,—"The ocean resembleth the sky and the sky resembleth the ocean—forsooth this encounter between Rāma and Rāvana befits them only". Thereupon Rāma of long-arms, the enhancer of the glory of Raghu’s race, enraged, set his arrow, resembling a serpent, on his bow, and cut assunder Rāvana’s head wearing shinning Kundalas. And that head in the presence of the inhabitants of the three regions fell down on the earth. Instantly there arose another head resembling the former; and it was speedily cut off by the light- handed Rāma. As soon as the second head was chopped off in the encounter by means of shafts another appeared again. And that was again severed by Rāma’s shafts resembling thunder-bolts. And thus were severed hundred heads all equal in brilliance. But the end of Rāvana’s life was not seen by him. Thereupon the heroic Rāghava, conversant with the use of all weapons, the enhancer of Kauçalyi’s joy, began to reason within himself in various Ways,—"Verily these are the arrows by which Māricha was killed, and Khara with Dushana was slain—Viradha was destroyed in the forest of Krauncha—the headless demon in the forest of Dandaka—Salas and mountains
- 65. were broken—the ocean was agitated—and Vāli was killed;—I do not perceive the reason, why they are becoming fruitless when discharged at Rāvana". Thinking thus Rāghava made himself ready In the conflict and began to shower arrows on Rāvana’s breast. Thereupon Rāvana too, the lord of Rākshasas, seated in a car and highly enraged, assailed Rāma in the conflict with a downpour of maces and Musalas. That dreadful and huge conflict, capable of making hairs stand on end, continued for seven nights before the eyes of the celestials, Dānavas, Yakshas, Pisāchas, Uragas and serpents stationed in the sky, on the earth or on the mountain-tops. Neither for the night nor for the day, nor for a moment did the fight between Rāma and Rāvana cease. And beholding the conflict between Daçaratha’s son and the lord of Rākshasas, and Rāghava’s victory, the high-souled charioteer of the lord of the celestials spake speedily unto Rāma engaged in the conflict. SECTION CX. Thereupon Mātali, reminding him, spake unto Rāghava—"Why dost thou, O hero, as if not knowing, fear him? Do thou, O lord, discharge at him the weapon obtained from the great Patriarch. The time for (his) destruction, as described by the celestials, hath arrived". Being reminded by those words of Mātali, Rāma took up the flaming shaft, breathing as if like a serpent. The great Rishi Agastya first conferred this upon him. This is a huge and dreadful shaft given by Brahmā, and highly useful in battle. It was made by Brahmā of undecaying prowess for Indra and conferred by him upon the Lord of celestials desirous of acquiring victory. In its wings there is wind, in its head
- 66. there is fire and the Sun, in its body there is the sky and in its weight there are the (hill) Meru and Mandara. It is resplendent by its own lustre, well feathered and adorned with gold—made of the essence of all objects and bright as the rays of the Sun. It is like the fire of dissolution enveloped in smoke—like the flaming serpent, capable of piercing men, serpents and horses and was swift- coursing. (It can) rend the gateways, Parighas and hills—is soaked in blood, dipped in marrow, and extremely dreadful. It is hard as the lightning—producing a dreadful sound, assailing various (divisions of the) army, creating terror unto all, dreadful and (as if) breathing like a serpent. It is terrible as the Death in the conflict and provides food always for the herons, vultures, cranes, jackals and the Rākshasas. It is the enhancer of the monkey-leaders’ joy and the repressor of the Rākshasas and is feathered like unto a bird with many a picturesque wing. And the highly powerful Rāma, consecrating in accordance with the mantras laid down in the Vedas, that huge shaft —the foremost of all in the world, removing the fear of the Ikshwāku race, destroying the fame of the enemies and conducing to the joy of its own party, set it on his bow. And that excellent arrow being mounted on his bow by Rāghava all the animals were stricken with fear and the earth shook. And (Rāma) highly enraged, and greatly wary, suppressing (his breath) discharged that shaft at Rāvana— piercing to the vitals. (That Brahmā weapon) irrepressible as the thunder, dreadful as the Death and discharged by Rāma, fell down on Rāvana’s breast. And that shaft, capable of bringing about death and gifted with velocity, when discharged, cleft the breast of the vicious-souled Rāvana. And that body-ending arrow, bathed in blood, stealing away the life of Rāvana, entered the earth. That shaft, slaying Rāvana, soaked in blood and successful, again entered the quiver⁴⁶¹ humbly. And from his hand, who was deprived of his life,
- 67. fell down instantly on earth his shafts and bow. And fell down on the earth from the chariot, the highly effulgent Rāvana, gifted with dreadful velocity and shorn of his life. And beholding him thus fallen down, the remaining night-rangers, deprived of their lord and stricken with terror fled away to various quarters. And beholding the destruction of the Ten-necked (demon) and the victory of Rāghava, the monkeys, fighting with trees, pursued them on all sides. And being assailed by the monkeys and having their countenances full of tears in consequence of their lord being slain they fled away to Lankā in fear. Thereupon the monkeys being greatly delighted roared out the victory of Rāma. The celestial bugle was sounded in the sky and there blew the excellent air carrying the celestial fragrance. Flowers were showered upon Rāma’s car which was covered therewith. The celestials in the sky began to chaunt the glory of Rāma and praise him. And Rāvana, the dread of all people, being slain, the celestials with the Charanas were greatly delighted. And slaying that foremost of the Rākshasas, Rāma satisfied the desire of Sugriva, Angada and Bibhishana. Thereupon the celestials attained their peace, the quarters were delighted, the atmosphere was clear, calm air began to prevail all over the earth, and the Sun appeared in its full rays. Thereupon Sugriva, Bibhishana and Lakshmana, welcomed Rāma, of unmitigated prowess, singing his glory. And there appeared beautiful at the battle-field Rāma of firm promise, slaying his enemy and encircled by his army and friends, like unto the Lord of the celestials surrounded by the gods. ⁴⁶¹ In some texts there is "napunarabishat" i.e. did not enter the quiver.—T.
- 68. SECTION CXI. Beholding his brother defeated, slain and lying down on the battle- field, Bibhishana, overpowered with the weight of his grief, began to lament—"O hero, well-known for thy prowess, wise and conversant with polity, thou wert used to excellent beds, why dost thou lie down on the earth, spreading (on the earth) thy long and actionless arms, always adorned with Angadas and being shorn of thy helmet having the resplendance of the Sun? O hero, thou hast come by what I had anticipated and what did not please thee who wert possessed by delusions. Prahasta, Indrajit, Kumbhakarna, Atikāya, Atiratha, Narāntaka, yourself and others—none of you paid heed, out of haughtines, to what I had said which hath now been brought about. Oh! the bridge of the pious hath been broken, the figure of the virtue hath been spoiled, the refuge of the strong and powerful hath disappeared and thou hast attained to the state of the heroes! The sun hath fallen down on the earth, the moon hath been shorn of its lustre, the fire hath been extinguished and virtue hath desisted from its action, this hero, the foremost of those using weapons, falling down on the earth. O thou the foremost of the Rākshasas lying down in the dust on the battle field like one asleep, whom else have these remaining (Rākshasas) deprived of their power and energy, got? The huge tree, of the lord of Rākshasas, having patience for its leaves, velocity for its flowers, the power of asceticism and heroism for its firm roots, hath been uprooted by the Rāghava wind. Mad- elephant-like Rāvana, having prowess for its tusk, family rank for its back bone, anger for its legs, and delightedness for its trunk, hath been laid low on the ground by the lion of the Ikshwāku race. The powerful Rākshasa-fire, having prowess and energy for its rays, angry breath for its smoke, own strength for its power of burning,
- 69. hath been extinguished in battle by Rāma-*like cloud. The Rākshasa bull ever defeating others and powerful as the wind, having Rākshasas for its tail, hump and horns, and fickleness for its ears and eyes, hath been slain to-day by *Rāma-tiger". Hearing these words, pregnant with sound reasonings from Bibhishana and beholding him overwhelmed with grief Rāma said,—"(This lord of Rākshasas) of dreadful prowess hath not been slain in battle disabled. He is gifted with great prowess and energy and devoid of the fear of death.⁴⁶² The heroes abiding by the virtues of the Kshatriyas, who fall at the battle field for enhancing their glory, when dead, should not be mourned for. This is not the time to mourn for him although possessed by death, by whom gifted with intellect, Indra with the three worlds was terrified in conflict. Besides success in battle is not perpetual; either one slays his enemy or meets with his destruction at his hands in the conflict This procedure of the Kshatryas was laid down by the ancient preceptors that a Kshatrya, when slain in battle should not be mourned for. Beholding this to be certain and attaining calmness, do thou be freed from thy sorrow and think what should be done now. Thereupon Bibhishana stricken with grief addressed the powerful son of the king speaking thus with words tending to his brother’s well-being. "Thou hast, like the ocean breaking down its banks, broken him down, who had not been ere this even defeated by Bāsava and the celestials. By him were conferred many a gift on those who wanted them, were enjoyed many a luxury, were maintained many a servant, distributed wealth unto friends and slain the enemies. He propitiated fire, performed great austerities, was conversant with the Vedas and the great performer of sacrifices. I desire to perform, by thy instructions, his becoming obsequies". Being thus addressed by Bibhishana with piteous accents, the high-souled son of the lord of men, gifted with
- 70. great energy, ordered him to perform his obsequies and said. "With death our enmity hath terminated and our object hath been accomplished: he is as dear unto me as unto thee: perform (therefore) his funeral rites". ⁴⁶² i.e. he has accidentally met with death. SECTION CXII. Beholding Rāvana slain by the high-souled Rāghava Rākshasees, stricken with grief, issued out of the inner appartments. Stricken with grief and with dishevelled hairs they rolled in the dust albeit prevented again and again like unto cows separated from their calves. And coming out by the northern gate along with the Rākshasas, entering the dreadful arena of battle and searching their slain lord the she-demons cried piteously—"O lord, O husband, O our all" and moved along the battle field soaked in blood and filled with headless corpses. With eyes full of tears and overwhelmed with the grief of their husband they began to move about like she-elephants without the lord of their herd. Thereupon they beheld there on the earth the huge-bodied and the highly powerful and effulgent Rāvana slain like red collyrium. And beholding their lord lying down on the battle-field they all fell on his body like creepers torn assunder. Some wept embracing him respectfully—some holding his feet and some placing themselves around his neck. And some taking up his hand rolled on the ground and some were beside themselves (with grief) beholding the slain (Rāvana’s) countenance. And some placing her head on his lap, and beholding his face, wept, bathing it with tears
- 71. like a lotus enveloped with snow. Seeing their husband Rāvana thus slain on the earth, they stricken with grief, bewailing again and again in sorrow, wept profusely. He by whom the king Vaisrabana was deprived of his flower car, who terrified the high-souled Gandharbas, ascetics and the celestials in battle field, who did not know of any fear from the Asuras, celestials and the Pannagas, hath now been overpowered by a man. He, whom the celestials, the Dānavas and the Rakshas could not slay, hath been slain in conflict by a man walking on foot. He, who was incapable of being killed by the celestials, Yakshas and Asuras, hath met with death like one devoid of prowess at the hands of a mortal". Speaking in this wise, the she- demons, over-powered with sorrow, wept and bewailed again and again (saying),—"Not hearing (the counsels) of thy friends, always pointing out thy welfare, thou didst bring Sitā for thy destruction as well as that of the Rākshasas. Although thy brother Bibhishana addressed thee with words pregnant with thy welfare—thou, for thy own destruction, out of thy misgivings, didst excite his anger and hast (now) seen (the result thereof). Hadst thou returned Sitā the daughter of the king of Mithilā to Rāma, this mighty and dreadful disaster, destroying the very root, would not have befallen us. Rāma’s desire would have been encompassed—his friends would have been successful (through Bibhishana)—we would not been widowed and our enemies would not have got their desires fulfilled. By thee, Sitā was kept by force in captivity in a ruthless manner, and the Rākshasas, ourselves and thyself—all three equally have been slain. O foremost of the Rākshasas, forsooth this is not thy own folly —it is Accident that uniteth all things and it is Accident again that bringeth about destruction. O thou of huge arms—the destruction of the monkeys and the Rākshasas as well as that of thyself hath been brought about by Accident. When the course of accident is about to
- 72. bring about result—wealth, desire, prowess or command—nothing is capable of with-standing it". Thus wept piteously the wives of the lord of Rākshasas like unto so many she-elephants—rendered poorly, stricken with grief and with tears in their eyes. SECTION CXIII. The foremost of the wives of the Rākshasa (king) bewailing piteously cast her looks poorly towards her husband. And beholding her Ten- necked husband slain by Rāma of inconcievable actions Mandodari bewailed there piteously,—"O thou of huge arms! O younger brother of Baishravana! Even Purandara feared to stand before thee when enraged. The great Rishis—the far-famed Gandharbas and the Chāranas, fled away to different quarters in thy fear. And then (how) hast thou been overpowered in battle by Rāma who is a mere man? Why is it that thou art not ashamed of it, O king, O lord of the Rākshasas? Conquering the three worlds with thy prowess thou didst attain thy glory; and it is unbearable, that a man, ranging in the forest, hath slain thee. Thou, who art capable of assuming shapes at will, hast been slain in conflict by Rāma, in the city of Lankā unapproachable by men. I do not believe that thy destruction,—who hadst always been crowned with success, before the forces, is work of Rāma. (Methinks) Death (himself) came there in the shape of Rāma and spread illusions unconsciously for thy destruction, O thou of great strength. Or thou hast been slain by Vāsava—(no) what power has he got to face thee in the conflict gifted with great strength, prowess and energy and an enemy of the celestials as thou art? It is evident that the great ascetic Vishnu, having truth for his
- 73. prowess—the soul of all beings, ever existing, without beginning, middle or end, greater than the great, the Preserver of the Nature, holding conch, discus and club, having _Sribatsa_⁴⁶³ on his breast, always beautiful, incapable of being conquered, without destruction, devoid of end, and the lord of all men, assuming this shape of a man and encircled by the celestials, assuming monkey shapes, hath for the behoof of mankind, slain (thee) the dreadful enemy of the gods with all (thy) family and Rākshasas. Subduing all thy passions, thou didst conquer the three worlds—remembering their grudge they have now over-powered thee. Rāma is not a man since he slew at Janasthāna thy brother Khara encircled by many a Rākshasa. We were sore-distressed when Hanumān, by his own prowess, entered the city of Lankā, incapable of being approached even by the celestials. And I prevented thee from creating enmity with Rāghava, but thou didst not pay heed to my words and this is the result (thereof). O foremost of Rākshasas—thou of a vicious understanding, for the destruction of thy wealth, thyself and thy relatives, thou didst suddenly cherish amour for Sitā, greater than (even) Arundhuti and Rohini. Forsooth thou didst perpetrate an unbecoming deed by distressing the adorable Sitā ever devoted unto her lord, an earth⁴⁶⁴ even unto Earth herself and a Sree⁴⁶⁵ even unto Sree herself. And having brought in a false guise from the solitary forest the sorrowful and chaste Sitā, having a blameless person, thou didst bring on the destruction of thy family. Thou didst fail to encompass thy desire for the company of Sitā. But now, forsooth, O my lord, thou hast been burnt down by her devotion who is devoted unto her husband.⁴⁶⁶ Thou wert not burnt down when thou didst captivate that middle-statured (damsel) whom fear all the celestials headed by Indra and Agni.⁴⁶⁷ O husband, (proper) time appearing the perpetrator gets the result of his vicious deeds:
- 74. there is not the least doubt in this. The performer of the good receives good (result)—the perpetrator of the vicious (deeds) meets with bad (result). Bibhishana hath attained to happiness and thou hast met with this thy end. There are damsels in your seraglio far more beautiful than she but thou possessed by cupid couldst not perceive this. Maithilee is not my equal nor superior either in birth, beauty or accomplishments, but thou didst not perceive this by thy misgivings. Death doth not always visit all men without any cause— and Maithilee is the cause of thy death. And death, in consequence of Maithilee, hath taken thee far away. And she, shorn of all sorrow, shall enjoy in the company of Rāma, and I, of limited piety, am now sunk in the the ocean of grief. Roaming at large with thee in a car unequalled in beauty on the hill Kailāça, Mandara, Maru, in the garden of Chaitraratha and ail other celestial gardens, beholding many a country, wearing variegated clothes and garlands, I have been deprived of all pleasures and enjoyments, O hero, by thy death. And I am a widow now. O fie on fickle fortune! O king, in lustre of countenance thou art like the Sun, in grace like the Moon, in beauty like the lotus; thou art graceful, O thou having excellent brows; thou hast got excellent skin, high nose and thy countenance is graced with a brilliant crown and Kundalas. Oh how beautiful, thou didst look with various garlands when thy eye whirled with intoxication on the drinking ground and how beautiful were thy smiles, O lord. Thy countenance doth not shine now—being severed with Rāma’s shafts, bathed in a pool of blood, having thy back-bone and brain deranged and covered with the dust raised by the wheels of the chariots. Alas! by my ill-luck I have attained to that after state making me a widow which I did never think of. My father is the king of Dānavas, my husband the lord of Rākshasas, and my son the subduer of Sakra. I was greatly proud of this and always confident
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