Communications in Mathematics and Applications Volume 4 (213), Number 2, pp. 119 125 RGN Publications http://www.rgnpublications.com Pontryagin s Maximum Principle of Optimal Control Governed by A Convection Diffusion Equation Youjun Xu, Huilan Wang, and Yanqi Liu Abstract In this paper we analyze an optimal control problem governed by a convection diffusion equation. This problem with state constraints is discussed by adding penalty arguments involving the application of Ekeland s variational principle and finite codimensionality of certain sets. Necessary conditions for optimal control is established by the method of spike variation. 1. Introduction and Main Results Consider the following controlled convection diffusion equations: µ y+β y+σy=f(x,u) in, y= on (1.1) where the diffusity constantµ>and the advective fieldβ (L ()) 2 with β L (), the reactionσ L () withσ σ >. To simplify the notation throughoutσ=1. Here R 2 is a convex bounded polygonal domain with a smooth boundary, f : U R, with U being a separable metric space. Function u( ), called a control, is taken from the set ={w : U w( ) is measurable}. Under some mild conditions, for any u( ), (1.1) admits a unique weak solution y( ) y( ;u( )), which is called the state (corresponding to the control u( )). The performance of the control is measured by the cost functional J(u( ))= for some given map f : R U R. f (x, y(x),u(x))d x, (1.2) 21 Mathematics Subject Classification. 35K27; 49K2. Key words and phrases. Optimal control; Pontryagin s maximum principle; State constraint. The work is supported by NSFC grants 1671211 and 112617, and Hunan Provincial Natural Science Foundation of China (Grant No. 11JJ46) and Doctor priming Fund Project of University of South China (Grant No. 5-211-XQD-8).
12 Youjun Xu, Huilan Wang, and Yanqi Liu Our optimal control problem can be stated as follows. Problem C. Find a ū( ) such that J(ū( ))= inf J(u( )). (1.3) u( ) Any ū( ) satisfying the above is called an optimal control, and the corresponding ȳ( ) ȳ( ;u( )) is called an optimal state. The pair(ȳ( ),ū( )) is called an optimal pair. In this paper, we make the following assumptions. (H1) Set R 2 is a convex bounded polygonal domain with a smooth boundary. (H2) Set U is a separable metric space. (H3) The function f : U R has the following properties: f( ; u) is measurable on, and f(x, ) continuous on R U and for any R>, a constant M R >, such that f(x,u) M R, for all(x,u) U. (H4) Function f (x, y, v) is measurable in x and continuous in(y, v) R U for almost all x. Moreover, for any R>, there exists a K R > such that f (x, y, v) + f y (x, y, v) K R, a.e.(x, v) U, y R. (1.4) (H5) is a Banach space with strict convex dual, F : W 1,p () is continuously Fréchet differentiable, and W is closed and convex. (H6) F (ȳ)d r W has finite condimensionality in for some r >, where D r ={z ; z r}. Definition 1.1 ([2, 6]). Let X is a Banach space and X is a subspace of X. We say that X is finite codimensional in X if there exists x 1, x 2,, x n X such that span{x, x 1,, x n }=the space spanned by{x, x 1,, x n }=X. A subset S of X is said to be finite codimensional in X if for some x S, span(s {x }) the closed subspace spanned by{x x x S} is a finite codimensional subspace of X and cos the closed convex hull of S {x } has a nonempty interior in this subspace. Lemma 1.2. Let (H1)-(H4) hold. Then, for any u( ), (1.1) admits a unique weak solution y( ) W 1,p () L (). Furthermore, there exists a constant K>, independent of u( ), y( ;u) 1,p W () L () K. (1.5) The weak solution y V= H 1 () of the state equation (1.1) is determined by a(y, v)=(f, v), for all v V,
Pontryagin s Maximum Principle of Optimal Control Governed by A Convection Diffusion Equation 121 using the bilinear form a : V V R given by a(y, v)= µ y vd x+ β y vd x+ σyvd x, for all v V. Existence and uniqueness of the solution to (1.1) follow from the above hypotheses on the problem data (see [1, 3]). Thus, a state constraint of the following type makes sense: F(y) W. (1.6) Let ad be the set of all pairs(y( ),u( )) satisfying (1.1) and (1.6) is called an admissible set. Any(y,u) ad is called an admissible pair. We refer to such a pair(ȳ,ū), if it exists, as an optimal pair and refer to ȳ and ū as an optimal state and control, respectively. Now, let(ȳ,ū) be an optimal pair of Problem C. Let z= z( ;u( )) W 1,p () be the unique solution of the following problem: µ z+β z+σz= f(x,u) f(x,ū) in, (1.7) z= on. And define the reachable set of variational system (1.7) ={z( ;u( ) u( ) }. (1.8) Now, let us state the necessary conditions of an optimal control to Problem C as follows. Theorem 1.3 (Pontryagin s Maximum Principle). Let (H1)-(H6) hold. Let (ȳ( ),ū( )) be an optimal pair of Problem C. Then there exists a triplet(ψ,ψ,ϕ ) R W 1,p with(ψ,ϕ ) such that ψ,η F(ȳ),, for allη W, (1.9) µ ψ β ψ+(σ β) ψ=ψ f y (x, ȳ,ū) F (ȳ) ϕ in, ψ= on, (1.1) H(x, ȳ(x),ū(x),ψ,ψ(x))= max H(x, ȳ(x),u(x),ψ,ψ(x)) a.e. x. (1.11) u( ) In the above, (1.9), (1.1), and (1.11) are called the transversality condition, the adjoint system (along the given optimal pair), and the maximum condition, respectively. Problem of the types studied here arise in originally from the optimal control of linear convection diffusion equation (cf.[3, 4, 5]). In the work mentioned above, the control set is convex. However, in many practical cases, the control set can not convex. This stimulates us to study Problem C. Necessary conditions for optimal control is established by the method of spike variation. In the next section, we will prove Pontryagin s maximum principle of optimal control of Problem C.
122 Youjun Xu, Huilan Wang, and Yanqi Liu 2. Proof of the Maximum Principle This section is devoted to the proof of the maximum principle. Proof of Theorem 1.3. First, let d(u( ),ū( ))= {x u( ) ū( )}, where D is the Lebesgue measure of D. We can easily prove that(u,ρ) is a complete metric space. Let(ȳ,ū) be an optimal pair of Problem C. For any y( ;u) be the corresponding state, emphasizing the dependence on the control. Without loss of generality, we may assume that J(ū) =. For any ε >, define J ε (u)={[(j(u)+ε) + ] 2 + d 2 Q (F(y( ;u)))}1/2. (2.1) Where d Q = inf x x, and ū is an optimal control. Clearly, this function is continuous on the (complete) metric space(, d). Also, we have x Q Jε (u)>, for all u, J ε (ū)=ε inf J (2.2) ε(u)+ε. u Hence, by Ekeland s variational principle, we can find a u ε, such that d(u,u ε ) ε, J ε (u) J ε (u ε ) ε d(u,u ε ), for allu. (2.3) We let v andε>be fixed and let y ε = y( ;u ε ), we know that for any ρ (,1), there exists a measurable set E ε ρ with the property Eε ρ =ρ, such that if we define u ε ρ (x)= u ε (x), if x \E ε ρ, v(x), if x E ε ρ and let y ε ρ = y( ;uε ρ ) be the corresponding state, then y ε ρ = yε +ρz ε + r ε ρ, J ε (u ε ρ )=J(uε )+ρz,ε + r,ε ρ, for allu, (2.4) where z ε and z,ε satisfying the following µ z ε +β z ε +σz ε = f(x, v) f(x,u ε ) in, with z ε = z,ε ρ = on, (2.5) f y (x, yε,u ε )z ε + h,ε (x) d x, (2.6) h,ε (x)= f (x, y ε, v) f (x, y ε,u ε ), 1 lim ρ ρ rε ρ 1 W 1,p = lim ρ ρ r,ε ρ =. (2.7)
Pontryagin s Maximum Principle of Optimal Control Governed by A Convection Diffusion Equation 123 We takeu=u ε ρ. It follows that where ε J ε(u ε ρ ) J ε(u ε ) ρ ξ ε = = 1 J ε (u ε ρ )+ J ε(u ε ) [(J(u ε ρ )+ε)+ ] 2 [(J(u ε )+ε) + ] 2 + d2 Q (F(yε ρ )) d2 Q (F(yε )) ρ (J(uε )+ε) + dq (F(y ε ))ξ J ε (u ε z,ε ε + ) J ε (u ε, F (y ε )z, ε (ρ ), (2.8) ) dq (F(y ε )), if F(y ε )/ Q,, if F(y ε ) Q. d Q ( ) denotes the subdifferential of d Q ( ). Next, we define(ϕ,ε,ϕ ε ) [,1] as follows: Then ϕ,ε = (J(uε )+ε) + J ε (u ε ) ρ, ϕ ε = d Q(F(y ε ))ξ ε J ε (u ε ). (2.9) ε ϕ,ε z,ε + ϕ ε, F (y ε )z ε, (2.1) ϕ,ε 2 + ϕ ε 2 = 1. (2.11) On the other hand, we have Then Consequently, ϕ ε,η F(y ε ) X,X, for allη W. (2.12) y ε ȳ 1,p W (), (ε ). (2.13) then lim ε F (y ε ) F (ȳ) (W 1,p (),) = (2.14) ϕ ε,η F(y ε ) X,X ϕ ε, F(y ε ) F(ȳ) X,X, for allη W. (2.15) By taking the limit forε in (2.15), we get (1.9). From (2.5) and (2.6), we have z ε z, in W 1,p (), z,ε z, (ε )
124 Youjun Xu, Huilan Wang, and Yanqi Liu where z is the solution of system (1.6) and z = f y (x, ȳ,ū)z(x)d x+ From (2.1) and (2.12), we have f (x, ȳ, v) f (x, ȳ,ū) d x. (2.16) ϕ,ε z,ε (v)+ ϕ ε, F (y ε )z ε ( ; v) η+ F(y ε ) δ ε, for all v,η W, (2.17) withδ ε asε. Because F (ȳ)d r W has finite condimensionality in, we can extract some subsequence, still denoted by itself, such that (ϕ,ε,ϕ ε ) (ϕ,ϕ). From (2.17), we have ϕ z (v)+ ϕ, F (ȳ)z( ; v) η+ F(ȳ), for all v,η Q. (2.18) Now, letψ = ϕ [ 1,]. Then(ψ,ϕ). Then we have ψ z (v)+ ϕ,η F(ȳ) F (ȳ) ϕ,z( ; v), for all u,η W. (2.19) Take v= ū, we obtain (1.9). Next, we letη= F(ȳ) to get ψ z (v) F (ȳ) ϕ,z( ; v), for all v. (2.2) Because F (ȳ) ϕ W 1,p (), for the givenψ, there exists a unique solution ψ W 1,p () of the adjoint equation (1.1). Then, from (1.6), (2.16) and (2.2), we have ψ z (v) F (ȳ) ϕ,z( ; v) =ψ f y (x, ȳ,ū)z(x)d x+ψ f (x, ȳ, v) f (x, ȳ,ū) d x + µ ψ β ψ+(σ β) ψ ψ f y (x, ȳ,ū),z = ψ f (x, ȳ, v) f (x, ȳ,ū) + ψ, f(x, v) f(x,ū) d x = H(x, ȳ(x), v(x),ψ,ψ(x)) H(x, ȳ(x),ū(x),ψ,ψ(x)) d x. (2.21) There, (1.11) follows. Finally, by (1.1), if (ψ,ψ) =, then F (ȳ) ϕ =. Thus, in the case where(f (ȳ) )={}, we must have(ψ,ψ), because (ψ,ϕ).
Pontryagin s Maximum Principle of Optimal Control Governed by A Convection Diffusion Equation 125 References [1] J.T. Oden and J.N. Reddy, Variational Methods in Theoretical Mechanics, Springer, Berlin and Heidelberg, 1983. [2] X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Birkhäuser, Boston, 1995. [3] R. Becker and B. Vexler, Optimal control of the convection-diffusion equation using stabilized finite element methods, Numer. Math. 16(3) (27), 349 367. [4] S. Micheletti and S. Perotto, An anisotropic mesh adaptation procedure for an optimal control problem of the advection-diffusion-reaction equation, MOX-Report No. 15, 28. [5] N. Yan and Z. Zhou, A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation, Journal of Computational and Applied Mathematics 223(1) (29), 198 217. [6] H. Lou and J. Yong, Optimal controls for semilinear elliptic equations with leaing term containing controls, SIAM J. Control Optim. 48(4) (29), 2366 2387. Youjun Xu, School of Mathematics and Physics, University of South China, Hengyang, Hunan 4211, P.R. China. E-mail: youjunxu@163.com Huilan Wang, School of Mathematics and Physics, University of South China, Hengyang, Hunan 4211, P.R. China. E-mail: huilan7@163.com Yanqi Liu, School of Mathematics and Physics, University of South China, Hengyang, Hunan 4211, P.R. China. E-mail: 5378564@99.com Received June 5, 212 Accepted August 11, 212