# Existence of the Optimal Control for Stochastic Boundary Control Problems Governed by Semilinear Parabolic Equations

Authors: Weifeng Wang; Baobin Wang
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

We study an optimal control problem governed by a semilinear parabolic equation, whose control variable is contained only in the boundary condition. An existence theorem for the optimal control is obtained.


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## Body

## 1. Introduction

Control theory is a mathematical description of how to act optimally to gain future rewards. Since the necessary conditions of optimal problems were established for the deterministic control systems by Pontryagin’s group [1] in the 1950s and 1960s, a lot of work has been done not only on the deterministic case but also on the stochastic case. To understand the deterministic case of optimal control problems governed by partial differential equations, we can see the classical book written by Lions [2] in 1971. And to the stochastic case, we can see [3–8] and so forth. The boundary control problems of stochastic partial differential equations have been developing fast and very active in recent years, including the boundary exact controllability [9, 10] and the maximum principle for boundary control [11].In our paper, this kind of boundary control problem is extensively studied in many papers in the case of deterministic control systems. For example, we can see Chai [12]. But in the stochastic case, as we known, there are only a few results for this problem. One of the difficulties is that the properties of the solution of the state equation are not clear compared with the deterministic case.In many papers, the authors only assume that the optimal controls exist for their problems, and they have not proved the existence. In our life, we always only want to use the necessary conditions to find the optimal control and achieve our goal, but the existence for the control problems is also important. Some useful results have been obtained, for instance, in [13–16]. About the control problems governed by semilinear parabolic equations, we can see [17, 18]. In this paper, the necessary conditions for our problem have been worked out by a member of our team in the paper [19]. So, in this paper, we only consider the existence of the optimal control as a complement for [19].The rest of this paper is organized as follows. Section2 begins with a general formulation of our stochastic optimal control problem. In Section 3, we give two important lemmas of our problem. And we give the existence result of the optimal control in Section 4.


## 2. Preliminaries

Let(Ω,F,P) be a complete probability space, equipped with a right-continuous filtration {Ft}t≥0, on which a d-dimensional mutually independent standard Brownian motion {w(t)}t≥0={(w1(t),…,wd(t))}t≥0 is defined. Moreover, we assume that Ft is the augmentation of σ{w(s)∣0≤s≤t} by all the P-null sets of F. For a fixed 0<T<+∞, we denote the σ-algebra of predictable sets on Ω×[0,T] by P. Let O be an open bounded subset in Rn with a smooth boundary Γ. Let Q=O×(0,T) and Σ=Γ×(0,T), and B(X) stands for the σ-algebra of all Borel subsets of X, where X is a topological space. L2(·) denotes the corresponding square-integral space, and H1(·) denotes the Sobolev space W1,2(·). The positive constants M and C can be different in different place throughout this paper.LetU be the set of u(x,t) satisfying the following:(1)
u(x,t):Ω×Γ×(0,T)→R is B(Rn-1)×B([0,T])-measurable and Ft-adapted;(2)
E∫Σ‍|u(x,t)|2dρ(x)dt<+∞, where ρ(x) denotes the usual (n-1)-dimensional Lebesgue measure on Γ.The set of admissible controlsUad is a convex, closed subset of U.Now we consider the following stochastic distributed control system withR-value control processes in Ω×Σ:
(1)dy(x,t)=[Ay(x,t)+f(x,t,y(x,t))]dt+[By(x,t)+g(x,t,y(x,t))]dw(t)inQ,∂νAy(x,t)=h(x,t,y(x,t),u(x,t))on⁡Σ,y(x,0)=y0(x)inO,
where y0∈L2(O) (P-a.s.) is F0-adapted, and the differential operators A, B are defined as follows:
(2)Ay=∑i,j=1n‍∂xj[aij∂xiy+bjy]+∑j=1n‍dj∂xjy+cy,Bky=∑i=1n‍σik∂xiy+ηky,k=1,…,d,∂νAy=∑j=1n‍{∑i=1n‍[∂xiy]+bjy}νj,
where ν(x) denotes the outward unit normal vector to Γ at the point x. We can see [12] for more details to the Neumann boundary condition in the trace sense. f, h are R-value functions and g is a Rd-value function.The cost functional is defined as follows:(3)J(yu(·,·),u(·,·))=E∫Q‍L(x,t,yu(x,t))dxdt+E∫Σ‍l(x,t,yu(x,t),u(x,t))dρ(x)dt+E∫O‍r(x,yu(x,T))dx,
where yu(x,t) denotes the adapted solution of the state equation (1) corresponding to u(x,t) and L, l, r are R-value functions.From the above content, our optimal control problem can be stated as follows.Problem (P). Find a u¯(·,·)∈Uad such that
(4)J(yu¯(·,·),u¯(·,·))=inf⁡u(·,·)∈UadJ(yu(·,·),u(·,·)).
Any u¯(·,·)∈Uad satisfying the above identity is called an optimal control, and the corresponding state yu¯(·,·) is called an optimal trajectory. (yu¯(·,·),u¯(·,·)) is called an optimal pair.We assume that the following conditions hold.(H1) The coefficients a=(aij), b=(bj), c, d=(dj), σ=(σij), and η=(ηk) are measurable in (x,t) with values in the sets of real symmetrical n×n matrices, Rn, R, Rn, Rn×d, and Rd, respectively. The real function y0(x) is F0×B(Rn)-measurable and E∥y0(x)∥L2(O)2≤K, where K is a positive constant. And the functions a=(aij), b=(bj), c, d=(dj), σ=(σij), and η=(ηk) are all bounded by K in absolute value. Furthermore, the matrix S=(aij-(1/2)∑k=1dσikσjk) is uniformly positive definite, which means
(5)ξ⊤Sξ≥λ|ξ|2,∀(x,t)∈Q,∀ξ∈Rn,
for a constant λ>0.(H2) Consider f:Ω×Rn×[0,T]×R→R, g:Ω×Rn×[0,T]×R→Rd, and h:Ω×Rn-1×[0,T]×R×R→R to be functions which satisfy the following properties:(i)
f is P×B(Rn)×B(R)/B(R)-measurable, g is P×B(Rn)×B(R)/B(Rd)-measurable, and h is P×B(Rn-1)×B(R)×B(R)/B(R)-measurable;(ii)
f(·,·,0),gk(·,·,0)∈L2(Q) and E∫0T‍∫Γ‍|h(x,t,0,u(x,t))|2dρ(x)dt<+∞(k=1,…,d,∀u∈U);(iii)
there exists a positive constantM>0 such that, for ∀y1,y2∈R, we have
(6)|f(x,t,y1)-f(x,t,y2)|+∑k=1d‍|gk(x,t,y1)-gk(x,t,y2)|≤M|y1-y2| uniformly in (ω,x,t)∈Ω×Q and
(7)|∂h∂y(x,t,y,u)|+|∂h∂u(x,t,y,u)|≤M
uniformly in (ω,x,t,u)∈Ω×Q×R and
(8)∂h∂y(x,t,y,u)≤0
for almost every (ω,x,t,u)∈Ω×Q×R.(H3) Consider L:Rn×[0,T]×R→R,  l:Rn-1×[0,T]×R×R→R, and r:Rn×R→R to be maps such that the following conditions are satisfied:(i)
L is B(Rn)×B([0,T])×B(R)/B(R)-measurable, l is B(Rn-1)×B([0,T])×B(R)×B(R)/B(R)-measurable, and r is B(Rn)×B(R)/B(R)-measurable;(ii)
L(·,·,0)∈L2(Q) and l(·,·,0,u(·,·))∈L2(Σ) for every u(·,·)∈U and r(·,0)∈L2(O);(iii)
L, r are continuous and continuously differentiable functions with respect to the state y(x,t) and l is a continuous and continuously differentiable function with respect to the state y(x,t) and the control variable u(x,t); moreover, there exists a constant M>0 such that
(9)|∂L∂y(x,t,y)|+|∂r∂y(x,y)|≤Muniformlyin(x,t)∈Q,|∂l∂y(x,t,y,u)|+|∂l∂u(x,t,y,u)|≤Muniformlyin(x,t,u)∈Q×R.


## 3. Some Basic Lemmas

First, we give the definition of solution of the state equation (1).Definition 1.
We say a functiony(x,t)∈LF2(Ω×(0,T);H1(O)) is a weak solution of the state equation (1), if it satisfies
(10)∫O‍y(t)ϕdx=∫O‍y0ϕdx-∑i,j=1n‍∫0t‍∫O‍aij∂xiy∂xjϕdxds-∑j=1n‍∫0t‍∫O‍bjy∂xjϕdxds+∑j=1n‍∫0t‍∫O‍dj∂xjyϕdxds+∫0t‍∫O‍cyϕdxds+∫0t‍∫O‍f(x,s,y)ϕdxds×∫0t‍∫Γ‍h(x,s,y)ϕdρ(x)ds+∑i=1n‍∫0t‍∫O‍σi∂xiyϕdxdw(s)×∫0t‍∫O‍ηyϕdxdw(s)+∫0t‍∫O‍g(x,s,y)ϕdxdw(s),P-a.s.
for every ϕ∈H1(O) and almost every t∈(0,T].Lemma 2 (Yu and Liu [19]).
Let the conditions (H1) and (H2) hold; then the state equation (1) has a unique (weak) solution y(x,t)∈LF2(Ω×(0,T);H1(O)) for every control u(x,t)∈U. Moreover, there exists a constant C>0 such that
(11)sup⁡0≤t≤TE∥y(t)∥L2(O)2+∥y∥LF2(Ω×(0,T);H1(O))2≤C.Remark 3.
We can see that, by Lemma2, the cost functional J(yu(·,·),u(·,·)) can be denoted by J(u(·,·)) for simplicity.Lemma 4.
Let one designate byym(x,t) the solution of the state equation (1) corresponding to um(x,t)∈Uad and suppose that um(x,t)→u¯(x,t) weakly in L2(Σ). Then, for any fixed ω∈Ω, one has
(12)ym(x,t)⟶yu¯(x,t)inL2((0,T);L2(O))asm⟶+∞.Proof.
From the state equation (1), we have
(13)ym(x,t)-yu¯(x,t)=∫0t‍[A(ym(x,s)-yu¯(x,s))+f(x,s,ym(x,s))mmmmm-f(x,s,yu¯(x,s))]ds+∫0t‍[B(ym(x,s)-yu¯(x,s))+g(x,s,ym(x,s))mmmmmm-g(x,s,yu¯(x,s))]dw(s).
By the definition of the operators A, B, we apply Itô’s formula to (ym(x,t)-yu¯(x,t))2, and then we can get
(14)E∫O‍|ym(x,t)-yu¯(x,t)|2dx≜E∫O‍|ζ(x,t)|2dx=2E∫0t‍∫Γ‍[h(x,s,ym(x,s),um(x,s))mmmmmmim-h(x,s,yu¯(x,s),u¯(x,s))]ζ(x,s)dρ(x)ds-2E∑i,j=1n‍∫0t‍∫O‍aij∂xiζ(x,s)∂xjζ(x,s)dxds+2E∑j=1n‍∫0t‍∫O‍[dj-bj]∂xjζ(x,s)ζ(x,s)dxds+2E∫0t‍∫O‍{cζ2(x,s)mmmmmmmmm+[f(x,s,ym(x,s))-f(x,s,yu¯(x,s))]mmmmmmmmm×ζ(x,s)cζ2(x,s)}dxds+E∑i,j=1n‍∫0t‍∫O‍σiσj∂xiζ(x,s)∂xjζ(x,s)dxds+E∫0t‍∫O‍|ηζ(x,s)|2+(g(x,s,ym(x,s))-g(x,s,yu¯(x,s)))2dxds+2E∑i=1n‍∫0t‍∫O‍σiη∂xiζ(x,s)ζ(x,s)dxds+2E∑i=1n‍∫0t‍∫O‍σiη∂xiζ(x,s)×(g(x,s,ym(x,s))mmmmmmmmmmm-g(x,s,yu¯(x,s)))dxds+2E∫0t‍∫O‍ηζ(x,s)×(g(x,s,ym(x,s))-g(x,s,yu¯(x,s)))dxds.
By (H2), it follows that
(15)∫0t‍∫Γ‍[h(x,s,ym(x,s),um(x,s))-h(x,s,yu¯(x,s),u¯(x,s))]ζ(x,s)dρ(x)ds≤M∫0t‍∫Γ‍[|um(x,s)-u¯(x,s)|+|ζ(x,s)|]×ζ(x,s)dρ(x)ds.
For any fixed ω∈Ω,
(16)∫0t‍∫Γ‍|um(x,s)-u¯(x,s)|×|ym(x,s)-yu¯(x,s)|dρ(x)ds≤C∥um(x,s)-u¯(x,s)∥H-1(Σ)×∥ym(x,s)-yu¯(x,s)∥H1(Σ).
Thanks to Lemma 2 and because the imbedding
(17)L2(Σ)↪H-1(Σ)
is compact, we can obtain
(18)E∫0t‍∫Γ‍|um(x,s)-u¯(x,s)|×|ym(x,s)-yu¯(x,s)|dρ(x)ds⟶0mmmmmmmmmmmmmmmasm⟶+∞.
From the conditions (H1)-(H2), using Young’s inequality, it follows that
(19)E∥ζ(x,t)∥L2(O)2+λE∫0t‍∥ζ(x,s)∥H1(O)2ds≤CE∫0t‍∥ζ(x,s)∥L2(O)2dsasm⟶+∞,
where λ>0 is a constant.Hence, our conclusion follows.


## 4. The Existence of the Optimal Control

In this section, we give our main result based on the above lemmas.Theorem 5 (existence of the optimal control).
One assumes(H1)–(H3) hold together with the following additional conditions: (i)
the functionl(x,t,yu(x,t),u(x,t)) is convex for each (x,t,y,·)∈Q×R;(ii)
there exists au0(·,·)∈Uad such that the set {u(·,·)∈Uad:J(u(·,·))≤J(u0(·,·))} is bounded in L2(Σ).Then there exists at least one optimal control for Problem (P).Proof.
Thanks to the conditions and Lemma2, it is easy to verify that J(u(·,·)) is finite for each u(·,·)∈Uad.
Now we fix aω∈Ω, and we can find a minimizing sequence {um(·,·)}m∈N⊂Uad for J(u(·,·)). Let {ym(·,·)}m∈N be the corresponding states, that is, the solutions of (1) corresponding to {um(·,·)}m∈N. By condition (ii), there exists a constant C>0 such that
(20)∥um(·,·)∥L2(Σ)≤C,∀m∈N.
Hence, we can assume that
(21)um(·,·)⟶u¯(·,·)weaklyinL2(Σ),
for some u¯(·,·)∈Uad (because Uad is a closed set ).
Combining the conclusions of Lemmas2 and 4, we can deduce that
(22)E∥ym(t)∥L2(O)2+∥ym∥LF2(Ω×(0,T);H1(O))2≤C∀m∈N,ym(·,·)⟶yu¯(·,·)inL2((0,T);L2(O))asm⟶+∞.
On the other hand, by Mazur's theorem (see, for instance, [20]), we can find a sequence of convex combinations {vm(·,·)}m∈N; that is,
(23)vm(·,·)=∑l=mN(m)‍λmlul(·,·)with∑l=mN(m)‍λml=1,λml≥0,m≤l≤N(m),
such that
(24)vm(·,·)⟶u¯(·,·)inL2(Σ)asm⟶+∞.
Now, using the convexity ofl(x,t,yu(x,t),u(x,t)) with respect to u(x,t), by the dominated convergence theorem and condition (H3), we obtain
(25)J(u¯(·,·))=E∫Q‍L(x,t,yu¯(x,t))dxdt+E∫Σ‍l(x,t,yu¯(x,t),u¯(x,t))dρ(x)dt+E∫O‍r(x,yu¯(x,T))dx=lim⁡m→+∞E∫Σ‍l(x,t,yu¯(x,t),vm(x,t))dρ(x)dt+E∫Q‍L(x,t,yu¯(x,t))dxdt+E∫O‍r(x,yu¯(x,T))dx=lim⁡m→+∞∑l=mN(m)‍λml×(+E∫Σ‍(l(x,t,yu¯(x,t),ul(x,t))J(ul(x,t))+E∫Σ‍(l(x,t,yu¯(x,t),ul(x,t))-l(x,t,yul(x,t),ul(x,t))(x,t,yu¯(x,t),ul(x,t)))dρ(x)dt+E∫Q‍(L(x,t,yu¯(x,t))-L(x,t,yul(x,t)))dxdt+E∫O‍(r(x,yu¯(x,T))-r(x,yul(x,T)))dx)≤inf⁡u(·,·)∈UadJ(u(·,·))+lim⁡m→+∞∑l=mN(m)‍(E∫Q‍M|yul-yu¯|dxdt+E∫Q‍M|yul-yu¯|dxdt+E∫O‍M|yul-yu¯|dxE∫Q‍M|yul-yu¯|)≤inf⁡u(·,·)∈UadJ(u(·,·))+Clim⁡m→+∞∑l=mN(m)‍∥yul-yu¯∥L2((0,T);L2(O))=inf⁡u(·,·)∈UadJ(u(·,·)).
That means u¯(·,·) is an optimal control for Problem (P).Remark 6.
In this paper, we give a new method to study the existence of the optimal control for stochastic control problems. And we also can see that, compared to the necessary conditions, we need to give some additional conditions to obtain the existence result. So the existence problems are harder to be considered.



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