Numerical computation of an optimal control problem with homogenization in one-dimensional case

Transcription

1 Retrospective Theses and Dissertations Iowa State University Capstones, Theses and Dissertations 28 Numerical computation of an optimal control problem with homogenization in one-dimensional case Zhen Li Iowa State University Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Li, Zhen, "Numerical computation of an optimal control problem with homogenization in one-dimensional case" (28). Retrospective Theses and Dissertations This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact

2 Numerical computation of an optimal control problem with homogenization in one-dimensional case by Zhen Li A thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Applied Mathematics Program of Study Committee: L. Steven Hou, Major Professor Jue Yan Ananda Weerasinghe Iowa State University Ames, Iowa 28 Copyright c Zhen Li, 28. All rights reserved.

3

4 ii DEDICATION I would like to dedicate this thesis to my wife Yanfei without whose support I would not have been able to complete this work. I would also like to my friends and family for their loving guidance and financial assistance during the writing of this work.

5 iii TABLE OF CONTENTS List of Figures ACKNOWLEDGEMENTS ABSTRACT iv v vi CHAPTER. Overview Introduction One-dimensional case CHAPTER 2. Optimal Control Problem and Partial Differential Equation 4 CHAPTER 3. Compare the Results for a ɛ and a CHAPTER 4. Results of Optimal Control Problem BIBLIOGRAPHY

6 iv LIST OF FIGURES Figure 2. The graph of u ɛ and U(x) = sin(2πx), x, ɛ = Figure 2.2 The graph of v ɛ, x, ɛ = Figure 2.3 The graph of u ɛ and U(x) = sin(2πx), x, ɛ = Figure 2.4 The graph of v ɛ, x, ɛ = Figure 3. The graph of u ɛ and u, ɛ = Figure 3.2 The graph of error between u ɛ and u, ɛ = Figure 3.3 The graph of v ɛ and v, ɛ = Figure 3.4 The graph of error between v ɛ and v, ɛ = Figure 3.5 The graph of u ɛ and u, ɛ = Figure 3.6 The graph of error between u ɛ and u, ɛ = Figure 3.7 The graph of v ɛ and v, ɛ = Figure 3.8 The graph of error between v ɛ and v, ɛ =

7 v ACKNOWLEDGEMENTS I would like to take this opportunity to express my thanks to those who helped me with various aspects of conducting research and the writing of this thesis. First and foremost, Dr. Steven Hou for his guidance, patience and support throughout this research and the writing of this thesis. His insights and words of encouragement have often inspired me and renewed my hopes for completing my graduate education. I would also like to thank my committee members for their efforts and contributions to this work: Dr. Jue Yan and Dr. Ananda Weerasinghe.

8 vi ABSTRACT We consider an optimal control problem in which the state equation has rapidly oscillating coefficients(characterized by matrix A ɛ, where ɛ is a small parameter). Based on some important results from the paper by S. Kesavan and J. Saint Jean Paulin (997), we convert this optimal control problem to a partial differential equation problem. Therefore, solving optimal control problem is equivalent to solving this partial differential equation problem. By several numerical examples in one dimensional case, we also show that the limit satisfies a problem of the same type but with matrix A (the H-limit of A ɛ ).

9 CHAPTER. Overview This is the opening paragraph to my thesis which introduce the optimal control problem and the connection between this problem and partial differential equations. This thesis is mainly based on the results from the paper by S. Kesavan and J. Saint Jean Paulin (997).. Introduction We will discuss the homogenization of an optimal control problem in which the state equation (given by a second-order elliptic boundary value problem) has rapidly oscillating coefficients. We just consider the one-dimensional case in this thesis. Let f L 2 (Ω), A and B are matrices whose entries are functions on bounded domain Ω with smooth boundary. B is also symmetric and nonnegative. N > is a given constant. Let θ(x) be a control variable and the optimal control problem which can be found in the paper by S. Kesavan and J. Saint Jean Paulin (997) is defined as follows, div(a u) = f(x) + θ(x) in Ω, u = on Ω, and the state u = u(θ) is thus defined as the weak solution in H (Ω) of above problem. Then the cost function is given by J(θ) = 2 Ω (B u, u) + N 2 Ω θ 2 (x). Minimization of the above cost function is a standard minimization problem, a discussion of which can be found in the book by J. L. Lions (968) and we obtain a reduced form by

10 2 introducing a new adjoint state p, div(a u) = f(x) + θ(x) in Ω div(a t p B u) = in Ω, where u, p H (Ω), and the optimal control θ can be characterized by such inequality (p + Nθ )(θ θ ) θ S, Ω where S is a subset of L 2 (Ω). What we are interested in is that given a parameter ɛ > which tends to zero, the matrices A and B above depend on ɛ. And we also have the same assumptions on A ɛ and B ɛ. In Kesavan s paper, there are also following conclusions. Suppose A ɛ is matrix depending on ɛ, then θɛ exists and is bounded in L 2 (Ω). Thus, we have θ ɛ θ weakly in L 2 (Ω), where θ is also an optimal control defined by a problem of the same type with matrices A and B. That paper also gives the following theorem. The solution (u ɛ, p ɛ ) of system div(a ɛ u ɛ ) = f(x) + θ(x) in Ω div(a t ɛ p ɛ B u ɛ ) = in Ω, u ɛ = p ɛ = on Ω, is bounded and also have the following weak convergence result in (H (Ω))2, u ɛ u, as ɛ p ɛ p, as ɛ where u, p satisfy the following system of equations, div(a u ) = f(x) + θ(x) in Ω div(a t p B u ) = in Ω.

11 3.2 One-dimensional case For the one-dimensional case, d d ( ) du aɛ ɛ = f(x) + θ(x) in (, ), ) ( dp a ɛ ɛ b ɛ duɛ = in (, ), we also have the similar results. Suppose (, ) R, if a ɛ a weakly in L (, ) and b = a2 g where g = b ɛ a 2 ɛ g weakly in L (, ). Then we have the following weak convergence in H (, ), u ɛ u, as ɛ p ɛ p, as ɛ where u, p satisfy the equations ( d d a du ( dp a b du ) = f(x) + θ(x) x (, ), ) = x (, ), u = p = x =,. In this thesis, we rewrite this optimal control problem and consider the following forms, d (a ɛ(x) duɛ ) = f(x) + c(x) x (, ), And the cost function is u ɛ = x =,. J(c) = β 2 u ɛ U c 2 (x), where β and U(x) is a given function. a ɛ (x) is a function defined on [, ]. Thus, the optimal control c is the function in [, ] which minimizes J(c) for c(x) L 2 (, ).

12 4 CHAPTER 2. Optimal Control Problem and Partial Differential Equation It is difficult to solve this optimal problem directly. From the numerical analysis viewpoint, it is advantageous to convert this problem to an equivalent PDE problem. Then we are able to analyze it by finite element or finite difference methods on numerical analysis. Let s consider the following optimal control problem d (a ɛ(x) duɛ ) = f(x) + c(x) x (, ), u ɛ = x =,, min β 2 u ɛ U c2 (x). Let v ɛ (x) L 2 (, ) and v ɛ =, if x =,, then L(u ɛ, c) = β 2 = β 2 u ɛ U u ɛ U c 2 (x) c 2 (x) By the integration by parts and v ɛ =, u ɛ = if x =,, we find that d v ɛ (a ɛ(x) du ɛ ) = v ɛa ɛ (x) du ɛ = v ɛ ( d (a ɛ(x) du ɛ ) f(x) c(x)). a ɛ (x) du ɛ dv ɛ = u ɛ a ɛ (x) dv ɛ + = u ɛ d (a ɛ(x) dv ɛ ). a ɛ (x) du ɛ dv ɛ u ɛ d (a ɛ(x) dv ɛ )

13 5 Therefore, L(u ɛ, c) = β 2 + = β 2 + u ɛ U d v ɛ (a ɛ(x) du ɛ ) + u ɛ U u ɛ d (a ɛ(x) dv ɛ ) + Then for any t(x), w(x) L 2 (, ), we should have L, w = β u ɛ = L c, t = = (u ɛ U)w + ( β(u ɛ U) + d c(x)t(x) + c 2 (x) v ɛ f(x) + c 2 (x) v ɛ f(x) + w d (a ɛ(x) dv ) ( a ɛ (x) dv ɛ v ɛ (x)t(x) (c(x) + v ɛ (x)) t(x) =,. v ɛ c(x) v ɛ c(x). )) w =, Therefore, β(u ɛ U) + d ( a ɛ (x) dv ) ɛ =, c(x) + v ɛ (x) =. i.e. d ( a ɛ (x) dv ) ɛ βu ɛ = βu(x), v ɛ (x) = c(x). Hence, v(x) = c(x) and the optimal problem is equivalent to the following partial differential equation problem, d (a ɛ(x) duɛ ) + v ɛ(x) = f(x) x (, ), d ( ) aɛ (x) dvɛ βuɛ (x) = βu(x) x (, ) u ɛ = x =,, v ɛ = x =,.

14 6 Now let s look several numerical examples. We solve this partial differential equation with finite difference. The finite difference for this problem is as follows, a ɛ,i+ u i+ (a 2 ɛ,i+ 2 a ɛ,i+ v i+ (a 2 ɛ,i+ 2 + a ɛ,i )u i + a 2 ɛ,i u i 2 h 2 + v i = f i, i =, 2, n + a ɛ,i )v i + a 2 ɛ,i v i 2 h 2 βu i = βu i, i =, 2, n u = u n = v = v n =, where for any function g(x), g i = g(x i ) and = x < x < < x n = is a uniform grid, with grid spacing x = h = /n. We will choose a ɛ from paper by Greéoire Allaire and Robert Brizzi (24). Given a ɛ = sin( 2πx ɛ ), β =, and U(x) = sin(2πx), f(x) = x2, we will look at several examples with different values of ɛ. (i) ɛ =., x = 2, the graphs of u and v are as follows,.8.6 u ε U(x)=sin(2πx) Figure 2. The graph of u ɛ and U(x) = sin(2πx), x, ɛ =. (ii) ɛ =., x = 2, the graphs of u and v are as follows, From graph 2. and 2.3, we could find that, the shapes of function u ɛ and U(x) = sin(2πx) are almost the same, when ɛ is small enough. This special case was studied by Kesavan and

15 Figure 2.2 The graph of v ɛ, x, ɛ =. Vanninathan. They assume that a ɛ is periodic. For the following problem, d (a du d ( ) + v (x) = f(x) x (, ), ) βu (x) = βu(x) x (, ) a dv u = v = x =,, where a is a constant and they proved that a was indeed the limit of a ɛ in the topology of H-convergence. Also for the periodic a ɛ of the one-dimensional case, they also gave its limit of H-convergence, which is a = [ ( )] m, a where m(h) = h(y) dy for a periodic function h on [,].

16 8.8.6 u ε U(x)=sin(2πx) Figure 2.3 The graph of u ɛ and U(x) = sin(2πx), x, ɛ = Figure 2.4 The graph of v ɛ, x, ɛ =.

17 9 CHAPTER 3. Compare the Results for a ɛ and a We will compare the relationship between d (a ɛ(x) duɛ ) + v ɛ(x) = f(x) x (, ), ( ) aɛ (x) dvɛ βuɛ (x) = βu(x) x (, ) d u ɛ = v ɛ = x =,, and (a du ) + v (x) = f(x) ( ) x (, ), βu (x) = βu(x) x (, ) d a dv u = v = x =,, with two numerical examples. Like the prior example, let s suppose a ɛ = sin( 2πx ɛ ), β =, and U(x) = sin(2πx), f(x) = x 2. Let u ɛ, v ɛ denote the numerical solutions of partial differential equations with a ɛ and u, v denote the numerical solutions of partial differential equations with a. Hence, [ ( )] [ ] a = m = (2 +.8 sin(2πy)) dy = a 2. We will give several graphs to illustrate the errors between u ɛ and u, v ɛ and v for different values of ɛ. (i) ɛ =., x = 2, the graphs of errors of u ɛ and v ɛ are as follows, (ii) ɛ =., x = 2, the graphs of errors of u ɛ and v ɛ are as follows, From figure 3.2 and figure 3.6, we can find the oscillation of the error of u ɛ and u. Therefore, we can find a test function, such that u ɛ is weak convergent to u. Analogously, the error of v ɛ and v also has such oscillation, which means that v ɛ is also weak convergent to v.

18 .8 u ε u Figure 3. The graph of u ɛ and u, ɛ = Figure 3.2 The graph of error between u ɛ and u, ɛ =.

19 25 2 v ε v Figure 3.3 The graph of v ɛ and v, ɛ = Figure 3.4 The graph of error between v ɛ and v, ɛ =.

20 2.8 u ε u Figure 3.5 The graph of u ɛ and u, ɛ = x Figure 3.6 The graph of error between u ɛ and u, ɛ =.

21 v ε v Figure 3.7 The graph of v ɛ and v, ɛ = Figure 3.8 The graph of error between v ɛ and v, ɛ =.

22 4 CHAPTER 4. Results of Optimal Control Problem For the optimal control problem d (a ɛ(x) du ɛ ) = f(x) + c(x), x (, ) for a given c(x), there will be a corresponding u ɛ. What we want to do is to find a pair of c(x) and u ɛ, such that L(u, c) = β 2 can attain its minimum. Since c(x) = v(x), L(u, c) = β 2 u ɛ U u ɛ U c 2 (x) v 2 (x). For the same given a ɛ = sin( 2πx ɛ ), β =, and U(x) = sin(2πx), f(x) = x2, after solving the equivalent partial differential equations, we have the following minimum of L(u ɛ, c). (i) Let ɛ =., x = 2. Then the minimum that L(u ɛ, c) attains is min L(u ɛ, c) = β 2 u ɛ U v 2 (x) = (i) Let ɛ =., x = 2. Then the minimum that L(u ɛ, c) attains is min L(u ɛ, c) = β 2 u ɛ U v 2 (x) =

23 5 BIBLIOGRAPHY S. Kesavan and J. Saint Jean Paulin (997). Homogenization of an optimal control problem. SIAM J. Control Optim., 35 (5), Greéoire Allaire and Robert Brizzi (24). A multiscale finite element method for numerical homogenization. J. L. Lions (968). Sur le contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris.