![and sufficient condition on hpθq is that it is an element of L1
pSq, when the square of the
function is not involved, i.e. replacing u2
´u2
o with |u´uo|. Otherwise, one should naturally
consider L2
pSq functions, which is another benefit of considering the squares of the functions
instead. However, the space S is a compact boundariless manifold, and as such one has
Lp
pSq Ă L1
pSq for all p ľ 2.
2 A Brief Aside: Lesson of the Lambert W-Function
Here, it is defined briefly, and exposes a deep fact about, not only the Lambert W-function,
but in principle an entire class of nonlinear evolutions. The technical content is summarized
in some deep theorems found in the book by Haim Br´ezis, [B]. The theorems concern the
Cauchy problem, i.e. the time evolution of maximal monotone operators, in particular, in the
cases which lead to nonlinear equations, and their associated stationary states of stability.
The precise formulation is quite involved, and it will suffice to use some simple and clear
examples here below.
Consider the Lambert W-function defined as such:
fpxq “ x ex
ðñ x “ Wpxq eWpxq
whence f ´1
pxq “ Wpxq.(2.1)
The solution to the two following problems below are stated, and it is shown that they are
related.
Consider the linear first order differential delay equation given by:
dxpsq
ds
“ xps ´ τq
It has solution xpsq “ C e
W pτq
τ
s
Next consider the following nonlinear first order equation stemming from the dynamics
of the combustion of the fuel of a (toy or idealized) rocket. Again, the solution is quoted
without proof for brevity.
The differential equation is:
dyptq
dt
“ y2
p1 ´ yq
It has solution,
yptq “
1
1 ` WpDe´tq
,
4](https://image.slidesharecdn.com/fe07a882-0c5d-408c-bffd-337a49bed3a8-151114084514-lva1-app6891/85/PDE-Constrained-Optimization-and-the-Lambert-W-Function-4-320.jpg)
![where D is the constant 1´y0
y0
e
1´y0
y0 stemming from the initial conditions yp0q “
y0 ą 0.
Evidently, in the above two equations the transformation of dependent variables,
x “ Ce
1
y
´1
ðñ y “
1
log
`x
c
˘
` 1
,
and the transformation of independent varialbes,
s “ WpDe´t
q ðñ t “ ´ log
´ s
D
es
¯
,
changes a linear first order delay differential equation into a nonlinear first order differential
equation without delay. It is this interplay that the theory of maximal monotone operators
of formal Cauchy evolutions elucidates.
3 A Step Toward the Social Optimum
Consider the nonlinear first order partial differential equation below.
9D ´ A D D1
` B D1
´ A D “ 0,
where A and B are real positive constants. As an ansatz assume separability, i.e. let
Dpx, tq “ χpxqTptq. Then a simultaneous solution is obtained when:
a logpbT ` cq ´ logpTq “ dt ` g and χ ´ µ logpχq “ νx ` σ,
satisfy the proper connection conditions for the constants a, b, c, d, g, µ, ν and σ.
References
[B] Br´ezis, H., Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert: Mathe-
matics Studies 5, 1973, North-Holland, Amsterdam.
[FW] Fetter, A. and Walecka, J. D., Theoretical Mechanics of Particles and Continua, 2003, Dover, New York.
[GF] Gel’fand, I. and Fomin, S., Calculus of Variations, 1991, Dover, New York.
[HPUU] Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S., Optimization with PDE Constraints in Mathematical Modeling:
Theory and Applications, 23, 2009, Springer, Berlin.
5](https://image.slidesharecdn.com/fe07a882-0c5d-408c-bffd-337a49bed3a8-151114084514-lva1-app6891/85/PDE-Constrained-Optimization-and-the-Lambert-W-Function-5-320.jpg)
PDE Constrained Optimization and the Lambert W-Function
- 1. PDE Constrained Optimization and the Lambert W-Function Michael Maroun June 22, 2010 1 A Simple Example of Robin-Type Optimization Consider the unit disk D “ tx2 ` y2 ĺ 1u Ă R2 comprised of an uniform metal with the boundary thermal conductivity transfer function k : BD “ S Ñ R` “ p0, 8q : x ÞÑ kpxq given as a property of the metal used to construct the disk. The disk is heated and used to ignite solid rocket fuel as a propellant for a rocket. The throttle or controllable ignition is a heating element surrounding the disk. The heating element is a bounded function h : S Ñ R taking values between h´ ĺ h ĺ h`, where h´ is the minimum value of the throttle heating element and h` is the maximum value. The ideal or optimal temperature distribution of the disk is the known temperature function uo : D Ñ R` : x ÞÑ uopxq with x P D, which gives the maximum efficiency for fuel combustion. However, uo : D Ñ R` is not necessarily attainable given the set up. One seeks to find the realizable temperature function upxq ą 0, which is closest to the function uo. That is given the functional J : R` Ñ R` , which measures the difference between the temperature distribution functions u and uo, one wishes to minimize J. One refers to J as a functional because if, for example, u were known to be a smooth function, i.e. u P C8 pRq, then the map J : C8 pRq Ñ R` obviously takes in an element of a vector space of real-valued functions and returns a subset of the ground field associated to the vector space. Additionally. the function u is determined by Laplace’s equation, which governs thermal conduction. Lastly, the boundary conditions are such that changes in the temperature function u are caused by the interface of the disk’s boundary with the heating control and 1
- 2. the thermal conductivity transfer function. More precisely, one has: min uPR` tJrusu :“min uPR` $ & % ż D ru2 pxq ´ u2 opxqsd2 x , . - (1.1) ´ 2 upxq “0 @ x P D(1.2) ´ ˇ ˇ r“1 u ¨ ˆr “kpθq rup1, θq ´ hpθqs P R @ θ P S(1.3) h´pθq ĺ hpθq ĺ h`pθq on S.(1.4) Thus given the thermal conductivity transfer function kpθq and the ideal temperature distribution function uopxq, one is asked to find the temperature distribution function upxq, and the setting for the throttle heating element function hpθq, which brings upxq as close to uopxq as possible. Proceed as follows. The constraint 1.2 is one of great importance and its general solution being a family of solutions in the absence of precise boundary conditions is one of the more important features of the problem. The fundamental solution of 1.2 is the singular1 Poisson kernel. It admits a spectral decomposition on the unit disk which gives rise to the expression for upxq with the extra requirement that upxq be finite at the origin, r “ 0. Hence, one has: (1.5) upxq “ a0 ` ÿ nPN ranrn cospnθq ` bnrn sinpnθqs. Here the coefficients an and bn are to be determined from the given data just as they would be in the usual boundary value problem. Now equation 1.3 is equivalent to: (1.6) ÿ nPN rn an cospnθq ` n bn sinpnθqs “ kpθq « hpθq ´ a0 ´ ÿ nPN ran cospnθq ` bn sinpnθqs ff The above expression now contains explicitly the two unknowns namely hpθq and the coef- ficients an and bn. From the minimization condition 1.1,2 one sees immediately that if upxq 1 One calls the Poisson kernel on the unit circle singular, since for every θ “ 2πn with n P Z, the Poisson kernel diverges as r approaches unity. 2 Incidentally, the appearance of the squares of the temperature functions is necessary to avoid the trivial solution of 0 “ 0. It suffices to minimize the square of the difference of the functions u and uo, in order to minimize the linear difference. This is because the temperature is measured in Kelvin, a strictly non-negative (in day to day macroscopic phenomena) temperature scale. Slight changes in the functional, that exploit physical properties as inequalities with mathematically equivalent extrema, appear quite frequently in the calculus of variation. 2
- 3. were to equal uopxq this is the obvious minimum for Jrus and if in turn this led to a throttle heating element function hpθq that fell into the inequality condition 1.4, the problem would be solved trivially. Hardly should one be this lucky. Instead, one typically must employ Lagrange-Newton- like methods for smoothly stepping away from uo as little as possible while closing into the inequality 1.4. To do this, construct a functional over the boundary of the disk with a parameter λ, which is used to step away from uopxq. Given the functional, (1.7) λ Krhs :“ 1 2 λ ż S h2 pθqdθ, one can now add this expression to the functional in 1.1 to control the departure from uo. In principle this solves the problem for the combustion of rocket fuel on a heated metal disk with controllable throttle heating element. Convergence can be checked by examining the functional’s values at the bounds Krh´s and Krh`s. The solution is in hand because notice that 1.5 is necessarily a harmonic function, i.e. any upr, θq given by the expansion 1.5 solves the Laplace equation 1.2. Lastly, recall that equation 1.6 gives an expression for the unknown function hpθq in terms of the known function kpθq. Of course, if uopxq itself was already harmonic, and obeyed the inequalities given, then determination of the unknown hpθq would be a standard problem of finding the coefficients an and bn. However, the extra consideration is due to the fact that uopxq need not be harmonic, hence the need for the functionals J and K. Now suppose uopxq vanishes at the boundary and takes a particular value at the origin. This boundary data can now be used to compute the unknown function hpθq. However, this does not necessarily minimize J. In particular, due to the lack of uniqueness of boundary value problems even for ordinary differential equations, such a condition, is neither necessary nor sufficient. One should instead integrate over the disk’s radial variable, r, and have remaining an expression in terms of an integral over θ, known functions of θ, and the unknown function hpθq. Applying integration be parts appropriately creates a functional K, which will depend upon θ, hpθq, and h1 pθq. At this point, one can employ the Euler-Lagrange minimization of the functional. The solution to this resulting ordinary differential equation will be the final answer since, again, determining hpθ in turn determines upr, θq via the coefficients an and bn. There are several instances where the expression for the coefficients an and bn through a self-consistency relationship with hpθq leads to an ordinary delay differential equation (ODDE). In these cases, the need for the Lambert-W function becomes obvious. For instance, one may be forced to find inequalities on unknown coefficients that arise in solving the ODDE that keep hpθq a bounded function. Finally, it will be stated without proof that a necessary 3
- 4. and sufficient condition on hpθq is that it is an element of L1 pSq, when the square of the function is not involved, i.e. replacing u2 ´u2 o with |u´uo|. Otherwise, one should naturally consider L2 pSq functions, which is another benefit of considering the squares of the functions instead. However, the space S is a compact boundariless manifold, and as such one has Lp pSq Ă L1 pSq for all p ľ 2. 2 A Brief Aside: Lesson of the Lambert W-Function Here, it is defined briefly, and exposes a deep fact about, not only the Lambert W-function, but in principle an entire class of nonlinear evolutions. The technical content is summarized in some deep theorems found in the book by Haim Br´ezis, [B]. The theorems concern the Cauchy problem, i.e. the time evolution of maximal monotone operators, in particular, in the cases which lead to nonlinear equations, and their associated stationary states of stability. The precise formulation is quite involved, and it will suffice to use some simple and clear examples here below. Consider the Lambert W-function defined as such: fpxq “ x ex ðñ x “ Wpxq eWpxq whence f ´1 pxq “ Wpxq.(2.1) The solution to the two following problems below are stated, and it is shown that they are related. Consider the linear first order differential delay equation given by: dxpsq ds “ xps ´ τq It has solution xpsq “ C e W pτq τ s Next consider the following nonlinear first order equation stemming from the dynamics of the combustion of the fuel of a (toy or idealized) rocket. Again, the solution is quoted without proof for brevity. The differential equation is: dyptq dt “ y2 p1 ´ yq It has solution, yptq “ 1 1 ` WpDe´tq , 4
- 5. where D is the constant 1´y0 y0 e 1´y0 y0 stemming from the initial conditions yp0q “ y0 ą 0. Evidently, in the above two equations the transformation of dependent variables, x “ Ce 1 y ´1 ðñ y “ 1 log `x c ˘ ` 1 , and the transformation of independent varialbes, s “ WpDe´t q ðñ t “ ´ log ´ s D es ¯ , changes a linear first order delay differential equation into a nonlinear first order differential equation without delay. It is this interplay that the theory of maximal monotone operators of formal Cauchy evolutions elucidates. 3 A Step Toward the Social Optimum Consider the nonlinear first order partial differential equation below. 9D ´ A D D1 ` B D1 ´ A D “ 0, where A and B are real positive constants. As an ansatz assume separability, i.e. let Dpx, tq “ χpxqTptq. Then a simultaneous solution is obtained when: a logpbT ` cq ´ logpTq “ dt ` g and χ ´ µ logpχq “ νx ` σ, satisfy the proper connection conditions for the constants a, b, c, d, g, µ, ν and σ. References [B] Br´ezis, H., Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert: Mathe- matics Studies 5, 1973, North-Holland, Amsterdam. [FW] Fetter, A. and Walecka, J. D., Theoretical Mechanics of Particles and Continua, 2003, Dover, New York. [GF] Gel’fand, I. and Fomin, S., Calculus of Variations, 1991, Dover, New York. [HPUU] Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S., Optimization with PDE Constraints in Mathematical Modeling: Theory and Applications, 23, 2009, Springer, Berlin. 5