Stability analysis for optimal control problems governed by semilinear evolution equation
 Hongyong Deng^{1} and
 Wei Wei^{1, 2}Email author
https://doi.org/10.1186/s1366201504435
© Deng and Wei; licensee Springer. 2015
Received: 15 October 2014
Accepted: 16 March 2015
Published: 28 March 2015
Abstract
In this paper, the stability of solutions of optimal control for the distributed parameter system governed by a semilinear evolution equation with compact control set in the space \(L^{1}(0,T;E)\) is discussed. The stability results for optimal control problems with respect to the righthand side functions are obtained by the theory of setvalued mapping and the definition of essential solutions for optimal control problems.
Keywords
1 Introduction
The stability analysis of systems governed by differential equations in modern mathematics is very important for practical applications (see [1, 2]), especially in numerical computation (see [3, 4]).
In recent years, there has been growing interest in stability analysis of optimal control problems for the ODE system or PDE system (see [5, 6]). But most of the results are based on the existence and uniqueness of an optimal control for problems, there are only a small number of articles discussing a stability analysis for optimal control problems without the uniqueness in view of setvalued analysis. In recent years, Yu et al. discussed the stability of optimal controls with respect to the righthand side function based on setvalued mapping (see [7]). But all results are established under the control admissible set \(\mathcal{U} [0,T]\) assumed as the compact set of \(C([0,T];R^{m})\) in the optimal control problem for an ODE system. Moreover, we have done some work concerning the control admissible set \(\mathcal{U} [0,T]\) assumed as the compact set of \(L^{1}(0,T;R^{m})\) in order to include many cases of practical situations (see [8]). Therefore, it is natural for us to ask whether these results are still valid for infinite dimensional controlled systems.
This paper studies the existence and stability properties of solutions of optimal control problems governed by a semilinear evolution equation, and it is stated as follows.
The paper is organized as follows. In Section 2, we give some properties of \(C_{0}\)semigroup and some results as regards compact sets. In Section 3, the existence of an optimal control is obtained. In Section 4, for the reader’s convenience, the setvalued mapping theory is recalled, then we show stability results for the optimal control problem in the sense of the Baire category. In the last section, some examples demonstrate the applicability of our results.
2 Preliminaries
Throughout the paper, constants \(1\leq p <+\infty\) and \(T>0\) are given. Let X be a Banach space and \(e^{At}\) the \(C_{0}\)semigroup generated by a linear operator A. In this section, we recall some related results about semigroups and compact sets from [9, 10] and [11] which will be used in this paper.
Proposition 2.1
([9])
Lemma 2.2
([11])
A set V is called totally bounded if for every \(\epsilon>0\), there exist some \(\delta>0\), a metric space W, and a mapping \(\Psi: V\rightarrow W\) such that \(\Psi(V)\) is totally bounded, and \(d(x,y)<\epsilon\) if \(d(\Psi(x),\Psi(y))<\delta\) whenever \(x,y\in V\).
Denote \((\tau_{h}u)(t)=u(t+h)\) for \(h>0\), the compactness of \(L^{p}(0,T;X)\) is recalled as follows.
Lemma 2.3
([12])
 (1)
The set \(\{\int_{t_{1}}^{t_{2}}u(t)\, dt \mid u\in S \}\) is relatively compact in X, for any \(0< t_{1}<t_{2}<T\).
 (2)
The limit \(\\tau_{h}uu\_{L^{p}(0,Th;X)}\rightarrow0\) as \(h\rightarrow0\), uniformly for \(u\in S\).
Now we consider the compactness of the admissible set (4).
Proposition 2.4
Suppose that \(U\subseteq E\) is compact and \(\\tau_{h}uu\ _{L^{1}(0,Th;E)}\rightarrow0\) as \(h\rightarrow0\) uniformly for \(u\in \mathcal{U}[0,T]\). Then \(\mathcal{U}[0,T]\) is compact in \(L^{1}(0,T; E)\).
Proof
To this end, by Lemma 2.3, \(\mathcal{U}[0,T]\) is relatively compact in \(L^{1}(0,T; E)\). We only need next to prove that \(\mathcal {U}[0,T]\) is closed.
We give an example of the compact set in \(L^{1}(0,T;E)\) in the following.
Example 2.1
Let \(u \in\mathcal{U}[0,T] \subset L^{1}(0,T;W^{1,p}(\Omega))\). For any fixed \(x\in\Omega\), \(u(t,x)\) is a piecewise continuous with respect to t containing only a finite number of discontinuous points. \(S(x)=\{u(t,x) \mid t\in[0,T]\}\), \(S(x)\) is a bounded closed set in \(W^{1,p}(\Omega)\), where Ω is a bounded open subset of \(R^{n}\) and ∂Ω is \(C^{1}\). Then \(\mathcal{U}[0,T]\) is compact in \(L^{1}(0,T; L^{p}(\Omega))\).
Proof
From the compact embedding theorem of [13], that is, \(W^{1,p}(\Omega)\) is compactly embedded in \(L^{p}(\Omega)\), written \(W^{1,p}(\Omega)\subset\subset L^{p}(\Omega)\), since \(S(x)\) is a bounded closed set in \(W^{1,p}(\Omega)\), then \(S(x)\) is compact in \(L^{p}(\Omega)\).
3 Existence of optimal control
 (H_{ u }):

The set \(U\subseteq E\) is compact, \(\mathcal{U}[0,T]\subset L^{1}(0,T;E)\), anduniformly for \(u\in\mathcal{U}[0,T]\).$$\\tau_{h}uu\_{L^{1}(0,Th;E)}\rightarrow0 \quad \mbox{as }h \rightarrow0, $$
 (H_{ f }):

The function \(f:[0,T]\times X \times U \rightarrow X\) is continuous w.r.t. t and u. There exist a function \(L(t)>0\) and constant \(C>0\) such thatfor all \(x,y\in X\), \(t\in[0,T]\), \(u\in U\).$$\begin{aligned}& \bigl\Vert f(t,x,u)f(t,y,u)\bigr\Vert _{X}\leq L(t)\xy \_{X}, \\& \bigl\Vert f(t,x,u)\bigr\Vert _{X}\leq C \end{aligned}$$
 (H_{ A }):

The operator \(A:D(A)\subseteq X\rightarrow X\) generates a \(C_{0}\)semigroup \(e^{At}\) on X.
Definition 3.1
From Proposition 5.3 of [10], we have the following theorem.
Theorem 3.1
Suppose assumptions (H_{ f }), (H_{ A }), and (H_{ u }) hold. Then, for any \(u \in\mathcal{U}[0,T]\), the Cauchy problem (3) has a unique mild solution \(y\in C([0,T];X)\).
Moreover, we also have the following result.
Theorem 3.2
Suppose assumptions (H_{ f }), (H_{ A }), and (H_{ u }) hold. \(y\in C([0,T];X)\) is the mild solution of system (3), then the map \(u(\cdot)\rightarrow y(\cdot,u(\cdot))\) is continuous from \(L^{1}(0,T; E)\) into \(C([0,T];X)\).
Proof
From Proposition 2.4, we have the following lemma.
Lemma 3.3
If assumption (H_{ u }) holds, \(\mathcal{U}[0,T]\) is compact in \(L^{1}(0,T; E)\).
Now, we discuss the existence of an optimal control for problem (P).
Theorem 3.4
Suppose assumptions (H_{ u }), (H_{ A }), and (H_{ f }) hold, then problem (P) admits at least one optimal control.
Proof
4 Stability analysis of optimal control problems
In this section, we will use the setvalued mapping theory to study the stability of the optimal control. At first, we recall some definitions and a lemma on setvalued mappings for convenience of the reader (see [7, 8, 16]). Let W and Z be metric spaces.
Definition 4.1
A setvalued mapping \(F:W\rightarrow2^{Z}\) is called upper (respectively, lower) semicontinuous at \(x\in W\) if and only if for each open set G in Z with \(G\supset F(x)\) (respectively, \(G\cap F(x)\neq\emptyset\)), there exists \(\delta>0\) such that \(G\supset F(x')\) (respectively, \(G\cap F(x')\neq\emptyset\)) for any \(x'\in W\) with \(\rho(x,x')<\delta\). It is said to be upper (respectively, lower) semicontinuous in W if and only if it is upper (respectively, lower) semicontinuous at any point of W.
A setvalued mapping \(F:W\rightarrow2^{Z}\) is continuous at x if it is both upper semicontinuous and lower semicontinuous at x, and that it is continuous if and only if it is continuous at every point of W.
Definition 4.2
A setvalued mapping \(F:W\rightarrow2^{Z}\) is called compact upper semicontinuous (called USCO) if \(F(x)\) is nonempty compact, for each \(x\in W\), and F is upper semicontinuous.
Definition 4.3
A setvalued mapping \(F:W\rightarrow2^{Z}\) is called closed if \(\operatorname{Graph}(F)\) is closed, where \(\operatorname{Graph}(F)=\{(x,z)\in W\times Z \mid z\in F(x)\} \) is the graph of F.
Lemma 4.1
If the setvalued mapping \(F:W\rightarrow2^{Z}\) is closed and Z is compact, then F is an USCO mapping.
Definition 4.4
A subset \(Q\subset W\) is called a residual set if it contains a countable intersection of open dense subsets of W.
If W is a complete metric space, any residual subset of W must be dense in W.
Lemma 4.2
Let W be a complete metric space, and \(F:W\rightarrow2^{Z}\) be an USCO mapping, then there exists a dense residual subset Q of W such that F is lower semicontinuous at each \(x\in Q\).
Then the space \((Y,\rho)\) is a complete metric space [7].
Then the correspondence \(f\rightarrow S(f)\) is a setvalued mapping \(S:Y\rightarrow2^{\mathcal{U}[0,T]}\).
From Theorem 3.4, we have the following theorem.
Theorem 4.3
If assumptions (H_{ u }), (H_{ A }), and (H_{ f }) hold, \(S(f)\neq\emptyset\) for each \(f\in Y\).
The following proposition is important in studying the stability of optimal controls.
Proposition 4.4
Let \(\{f_{k}\}\) be any sequence of Y such that \(f_{k}\rightarrow f\) in Y and \(\{u_{k}\}\) any sequence of \(\mathcal{U}[0,T]\) such that \(u_{k}\rightarrow u\) in \(L^{1}(0,T; E)\), then \(y_{f_{k}}(\cdot,u_{k}(\cdot))\rightarrow y_{f}(\cdot,u(\cdot))\) in \(C([0,T];X)\) as \(k\rightarrow+\infty\).
Proof
One can easily obtain the following proposition from Proposition 4.4.
Proposition 4.5
Let \(\{f_{k}\}\) be any sequence of Y such that \(f_{k}\rightarrow f\) in Y and \(\{u_{k}\}\) any sequence of \(\mathcal{U}[0,T]\) such that \(u_{k}\rightarrow u\) in \(L^{1}(0,T; E)\). Then \(J_{f_{k}}(u_{k})\rightarrow J_{f}(u)\) as \(k\rightarrow+\infty\).
Theorem 4.6
Suppose that (H_{ u }), (H_{ A }), and (H_{ f }) hold. Then \(S:Y\rightarrow 2^{\mathcal{U}[0,T]}\) is an USCO mapping.
Proof
We introduce the following definition for considering the stability of solutions of an optimal control problem.
Definition 4.5
\(u\in S(f)\) is called an essential solution iff for any \(\epsilon>0\) there exists \(\delta>0\) such that for any \(f'\in Y\) with \(\rho(f',f)<\delta\), there is \(u'\in S(f')\) with \(\ uu'\_{L^{1}(0,T;E)}<\epsilon\). The optimal control problem (P) associated with f is called essential iff its solutions are all essential.
From [7], we can obtain the following theorem.
Theorem 4.7
The optimal control problem (P) associated with f is essential if and only if \(S:Y\rightarrow2^{\mathcal{U}[0,T]}\) is lower semicontinuous at \(f\in Y\).
Now we need to consider that S is lower semicontinuous in Y. From Lemma 4.2 and Theorem 4.6, we have the following lemma.
Lemma 4.8
There exists a dense residual subset \(Q\subset Y\) such that S is lower semicontinuous at each \(f\in Q\), namely, S is continuous at each \(f\in Q\).
Since S is continuous at each \(f\in Q\) and Y is a complete metric space, Q is a second category [17]. Lemma 4.8 and Theorem 4.7 yield the generic stability in the sense of the Baire category.
Theorem 4.9
There exists a dense residual subset Q of Y such that for any \(f\in Q\), \(S(f)\) is stable in the sense of Hausdorff metric and \(J_{f}\) is robust with respect to \(f\in Q\). So every optimal control problem associated \(f \in Y\) can be closely approximated arbitrarily by an essential optimal control problem.
We need to note that when the solution \(S(f)\) is a singleton, the result also holds. By Theorem 4.9 we can also see that any \(f\in Y \) can be closely approximated by an essential optimal control problem.
Remark 4.1
In this paper, we only discuss the stability results as regards the optimal control problem with quadratic cost functional. In fact, the results also hold for the optimal control problem with general cost functional such as Bolza problems under some assumptions.
5 Example
Our main result can be applied to the controlled systems of heat equations and wave equations with the \(C_{0}\)semigroup. We will state optimal control problems with parabolic controlled systems and hyperbolic controlled systems, respectively, in the following.
The following example is provided to show that not all optimal control problems are essential.
Example 5.1
In order to simplify the calculation, let \(v_{0}=0\).
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11261011).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
 Dontchev, AL, Hager, WW: Lipschitzian stability for state constrained nonlinear optimal control. SIAM J. Control Optim. 36, 698718 (1998) View ArticleMATHMathSciNetGoogle Scholar
 Hermant, A: Stability analysis of optimal control problems with a secondorder constraint. SIAM J. Control Optim. 20, 104129 (2009) View ArticleMATHMathSciNetGoogle Scholar
 Teo, KL, Goh, CJ, Wong, KH: A Unified Computational Approach to Optimal Control Problems. Wiley, New York (1991) MATHGoogle Scholar
 Lin, Q, Loxton, R, Teo, KL: The control parameterization method for nonlinear optimal control: a survey. J. Ind. Manag. Optim. 10, 275309 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Casas, E, Kunisch, K: Optimal control of semilinear elliptic equations in measure spaces. SIAM J. Control Optim. 52, 339364 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Corona, D, Giua, A, Seatzu, C: Stabilization of switched systems via optimal control. Nonlinear Anal. Hybrid Syst. 11, 110 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Yu, J, Liu, ZX, Peng, DT: Existence and stability analysis of optimal control. Optim. Control Appl. Methods 35, 721729 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Deng, HY, Wei, W: Existence and stability analysis for nonlinear optimal control problems with 1mean equicontinuous controls. J. Ind. Manag. Optim. 11, 14091422 (2015). doi:10.3934/jimo.2015.11.1409 View ArticleMATHMathSciNetGoogle Scholar
 Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) MATHView ArticleGoogle Scholar
 Li, XJ, Yong, JM: Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston (1995) View ArticleGoogle Scholar
 HancheOlsen, H, Holden, H: The KolmogorovRiesz compactness theorem. Expo. Math. 28, 385394 (2010) View ArticleMATHMathSciNetGoogle Scholar
 Simon, J: Compact sets in the space \(L^{p}(0,T;B)\). Ann. Mat. Pura Appl. 146, 6596 (1986) View ArticleMATHMathSciNetGoogle Scholar
 Evans, LC: Partial Differential Equations. Am. Math. Soc., Providence (1998) MATHGoogle Scholar
 Bogachev, VI: Measure Theory, vol. I. Springer, Berlin (2007) View ArticleMATHGoogle Scholar
 Kreyszig, E: Introductory Functional Analysis with Applications. Wiley, New York (1978) MATHGoogle Scholar
 Aubin, JP, Frankowska, H: SetValued Analysis. Birkhäuser, Boston (1990) MATHGoogle Scholar
 Rudin, W: Functional Analysis, 2nd edn. McGrawHill, New York (1991) MATHGoogle Scholar