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Stability analysis for optimal control problems governed by semilinear evolution equation
Advances in Difference Equations volumeÂ 2015, ArticleÂ number:Â 103 (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.
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.
Problem (P): Given \(T>0\), \(\hat{y}(\cdot)\in L^{2}(0,T;X)\), \(y_{0}\in X\), and \(y_{T}\in X\), find an optimal control \(\bar{u}\in \mathcal{U}[0,T]\) such that
where
subject to
where A is infinitesimal generator of \(C_{0}\)semigroup on a Banach space X, the set of admissible controls in the space \(L^{1}(0,T;E)\) is defined by
where E is another separable reflexive Banach space from which the controls u take values.
The paper is organized as follows. In SectionÂ 2, we give some properties of \(C_{0}\)semigroup and some results as regards compact sets. In SectionÂ 3, the existence of an optimal control is obtained. In SectionÂ 4, for the readerâ€™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])
There exist constants \(\omega\geq0\) and \(M\geq1\) such that
Denote \(L^{p}(0,T;X)\) the space of measurable functions from \([0,T]\) into X equipped with the norm
and \(C([0,T];X)\) the space of continuous functions from \([0,T]\) into X equipped with the norm
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])
Suppose that S is a subset of \(L^{p}(0,T;X)\). S is relatively compact in \(L^{p}(0,T;X)\) if and only if:

(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
It follows from the compactness of \(U \subseteq E\) that there exists a constant \(M>0\) such that
hence for any \(0< t_{1}<t_{2}<T\), we get
for all \(u\in\mathcal{U}[0,T]\), this implies that \(\int_{t_{1}}^{t_{2}}\ u(t)\_{E} \, dt\) is bounded in \(R^{1}\), so the set \(\{\int_{t_{1}}^{t_{2}}\ u(t)\_{E}\, dt \mid u\in\mathcal{U}[0,T] \}\) is relatively compact in \(R^{1}\). Since \(\int_{t_{1}}^{t_{2}}u(t)\, dt\in E\), by the HahnBanach theorem, there exists a bounded linear functional \(P\in E^{*}\) such that
By the continuity and linearity of P, we have
and
namely,
Due to \(\{\int_{t_{1}}^{t_{2}}\u(t)\_{E}\,dt \mid u\in\mathcal{U}[0,T] \}= \{P(\int_{t_{1}}^{t_{2}}u(t)\,dt) \mid u\in\mathcal{U}[0,T] \}\) is relatively compact in \(R^{1}\) and inequality (6), by LemmaÂ 2.2, we see that \(\{\int_{t_{1}}^{t_{2}}u(t)\,dt \mid u\in\mathcal {U}[0,T] \}\) is relatively compact in E, for any \(0< t_{1}<t_{2}<T\).
To this end, by LemmaÂ 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.
If \(\{u_{n}\}\subseteq\mathcal{U}[0,T]\) and \(u_{n}\rightarrow u\) in \(L^{1}(0,T; E)\), namely,
then \(\u(t)\_{E}\) is measurable in \([0,T]\) and there exists a subsequence of \(\{u_{n}\}\), denoted by \(\{u_{n_{k}}\}\), such that
i.e., \(\u_{n_{k}}(t)u(t)\_{E}\rightarrow0\) for all \(t\in A\), where A is some subset of \([0,T]\) and \(\mu([0,T]\backslash A)=0\).
Since U is compact, for any fixed \(t\in A\) and \(u_{n_{k}}(t)\in U\), there exists \(u(t)\in U\) such that
Without loss of generality, we define
where \(t_{0}\) can be chosen by any point of A, then \(u(t)=\hat{u}(t)\) a.e. \(t\in[0,T]\). It is clear that \(\hat{u}(t)\in U\) for all \(t\in[0,T]\). u and \(\hat{u}\) are equivalent in \(L^{1}(0,T; E)\). Namely,
then \(\mathcal{U}[0,T]\) is closed. From the above, we complete this proposition.â€ƒâ–¡
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 Î© is a bounded open subset of \(R^{n}\) and âˆ‚Î© 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 \(L^{p}(\Omega)\).
Suppose that \(D=\{t_{1},\ldots, t_{N}\}\subset[0,T]\) and \(t_{0}=0< t_{1}<t_{2}<\cdots<t_{N}<T=t_{N+1} \). We define
Without loss of generality, we may assume \(h>0\), then
It follows from the continuity of u in \((t_{i1},t_{i})\) (\(i=1,2,3,\ldots,N+1\)) that, for any fixed \(h>0\), \(\tilde{u}(\cdot)\) is uniform continuous in \([t_{i1},t_{i}h]\) (\(i=1,2,3,\ldots,N+1\)). That is, for any \(t\in [t_{i1},t_{i}h]\) and any \(\epsilon>0\), there exists \(\delta_{i}>0\) such that
holds whenever \(h<\delta_{i}\). Let \(\delta=\min_{1\leq i\leq N+1}\delta_{i}\), then when \(h<\delta\), we have
Hence
If we take \(h<\min\{\delta,\frac{\epsilon}{4MN}\}\), then
By PropositionÂ 2.4, \(\mathcal{U}[0,T]\) is compact in \(L^{1}(0,T; L^{p}(\Omega))\).â€ƒâ–¡
3 Existence of optimal control
In order to study the optimal control problem (P), we assume that:
 (H_{ u }):

The set \(U\subseteq E\) is compact, \(\mathcal{U}[0,T]\subset L^{1}(0,T;E)\), and
$$\\tau_{h}uu\_{L^{1}(0,Th;E)}\rightarrow0 \quad \mbox{as }h \rightarrow0, $$uniformly for \(u\in\mathcal{U}[0,T]\).
 (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 that
$$\begin{aligned}& \bigl\Vert f(t,x,u)f(t,y,u)\bigr\Vert _{X}\leq L(t)\xy \_{X}, \\& \bigl\Vert f(t,x,u)\bigr\Vert _{X}\leq C \end{aligned}$$for all \(x,y\in X\), \(t\in[0,T]\), \(u\in U\).
 (H_{ A }):

The operator \(A:D(A)\subseteq X\rightarrow X\) generates a \(C_{0}\)semigroup \(e^{At}\) on X.
Definition 3.1
For given \(u\in\mathcal{U}[0,T]\), a function \(y:[0,T]\rightarrow X\) is called a mild solution of (3) if \(y\in C([0,T];X)\) satisfies
This solution is denoted by \(y(\cdot,u(\cdot))\).
From PropositionÂ 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
Suppose \(y_{k}(t)\) and \(y(t)\) are mild solutions of the system (3) with respect to \(u_{k}\) and u, respectively. Then
and
Since
by the Chebyshev inequality of TheoremÂ 2.5.3 in [14], we have
and the function f is continuous w.r.t. u, it follows from CorollaryÂ 2.2.6 of [14] that
and combining the Riesz TheoremÂ 2.2.5(i) and TheoremÂ 2.2.3 of [14], then for any subsequence \(\{u_{k'}\}\) we can extract a further subsequence \(\{u_{k''}\}\) such that
then
Also from PropositionÂ 2.1, there exist constants \(\omega\geq 0\) and \(M\geq1\) such that
Due to assumption (H_{ f }) and the Lebesgue dominated convergence theorem, it follows that
Since the subsequence of \(\{u_{k'}\}\) is arbitrary, we can conclude that
namely, for any \(\epsilon>0\), there exists \(N>0\) such that
holds whenever \(k\geq N\). So
and using the Gronwall inequality, we obtain
That is, \(y(\cdot,u_{k})\rightarrow y(\cdot,u)\in C([0,T];X)\).â€ƒâ–¡
From PropositionÂ 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
Assume that there exists a minimizing sequence of \(\{u_{k}\}\subseteq\mathcal{U}[0,T]\) such that
Since \(\mathcal{U}[0,T]\) is compact, there exists a subsequence \(\{u_{k'}\}\) of \(\{u_{k}\}\) and \(\bar{u}\in \mathcal{U}[0,T]\) such that
From TheoremÂ 3.2 we have
where \(y_{u_{k'}}(\cdot,\cdot)\) and \(y_{\bar{u}}(\cdot,\cdot)\) are solutions of (3) with respect to \(u_{k'}\) and \(\bar{u}\), respectively. Hence
and
then
Thanks to (15), there exists a subsequence of \(\{u_{k'}\}\), denoted by \(\{u_{k''}\}\), such that
It follows from (16) and (17) that
Since \(u_{k''}(t)\in U\) (compact set in E) and \(y_{k''}(\cdot)\in C([0,T];X)\), the image of the compact set under y is compact (see TheoremÂ 2.56 of [15]). For all \(k''\), there exists a positive constant M, such that
From the Lebesgue dominated convergence theorem (see TheoremÂ 2.8.1 of [14]), we have
From all the above, we have
That is, \(\bar{u}\in\mathcal{U}[0,T]\) is an optimal control. This completes the proof of the theorem.â€ƒâ–¡
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 \(x\in Q\).
Now, we consider the stability of optimal controls. Denote
For every \(f_{1},f_{2}\in Y\), we define
Then the space \((Y,\rho)\) is a complete metric space [7].
Let
Then the correspondence \(f\rightarrow S(f)\) is a setvalued mapping \(S:Y\rightarrow2^{\mathcal{U}[0,T]}\).
From TheoremÂ 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
By DefinitionÂ 3.1 of a mild solution of Cauchy problem (3) we have
and
for every \(t \in[0,T]\).
It follows from \(f_{k}\rightarrow f\) in Y that for any \(\epsilon>0\) there exists a constant \(N_{1}>0\) such that
whenever \(k\geq N_{1}\). The inequality (8) implies that
The \(u_{k} \rightarrow u\) in \(L^{1}(0,T;E)\) and (10) imply that there exists a constant \(N_{2}>0\) such that
whenever \(k\geq N_{2}\).
Let \(N=\max\{N_{1},N_{2}\}\), then
holds for any \(k\geq N\), and it follows from the Gronwall inequality that
This implies \(y_{k}\rightarrow y\) in \(C([0,T]; X)\), which completes the proof of the theorem.â€ƒâ–¡
We denote
One can easily obtain the following proposition from PropositionÂ 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
By the compactness of \(\mathcal{U}[0,T]\) and LemmaÂ 4.1, we only need to show that
is closed. Let \(\{f_{k}\}\subset Y\) with \(f_{k}\rightarrow f\) in Y and \(\{\bar{u}_{k}\}\subset S(f_{k})\) with \(\bar {u}_{k}\rightarrow\bar{u}\) in \(L^{1}(0,T; E)\). Now we need to show that \(\bar{u}\in S(f)\).
Due to \(\bar{u}_{k}\in S(f_{k})\), we have
PropositionÂ 4.5 yields
and
Combining (25), (26), and (24), we have
namely, \(\bar{u}\in S(f)\). The proof is completed.â€ƒâ–¡
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 LemmaÂ 4.2 and TheoremÂ 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]. LemmaÂ 4.8 and TheoremÂ 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 TheoremÂ 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.
Problem (P1): Given \(T>0\), \(\hat{y}(t,x)\), and \(y_{T}(x)\), find an optimal control \(\bar{u}\in\mathcal{U}[0,T]\) such that
where
subject to
where \(\Omega\subset R^{n}\) is a bounded domain with a smooth boundary âˆ‚Î©, let \(X=L^{2}(\Omega)\) and \(A=\Delta= \frac{\partial^{2}}{\partial x_{1}^{2}}+\cdots +\frac{\partial^{2}}{\partial x_{n}^{2}}\) with \(D(A)=W^{2,2}(\Omega)\cap W_{0}^{1,2}(\Omega)\). Then A is infinitesimal generator of a \(C_{0}\)semigroup \(e^{At}\) on the infinite dimensional Banach space X. Suppose that \(y_{0}\in X\) and the set of admissible controls in the space \(L^{1}(0,T; E)\) is defined by (4) where \(E=L^{2}(\Omega)\). With the assumptions (H_{ u }) and (H_{ f }) holding, the stability results are valid.
Problem (P2): Given \(T>0\), \(\hat{y}(t,x)\), and \(y_{T}(x)\), find an optimal control \(\bar{u}\in\mathcal{U}[0,T]\) such that
where
subject to
where \(\Omega\subset R^{n}\) is a bounded domain with a smooth boundary âˆ‚Î© and \(X=W^{1,2}(\Omega)\times L^{2}(\Omega)\). Let \((y_{0},y_{1})\in X\), \(\Delta= \frac{\partial^{2}}{\partial x_{1}^{2}}+\cdots+\frac{\partial^{2}}{\partial x_{n}^{2}}\), \(A=\bigl({\scriptsize\begin{matrix} 0 & I \cr \Delta& 0\end{matrix}} \bigr)\) with \(D(A)= (W^{2,2}(\Omega)\cap W_{0}^{1,2}(\Omega) )\times W^{1,2}(\Omega)\). Then A is the infinitesimal generator of a \(C_{0}\)semigroup \(e^{At}\) on the infinite dimensional Banach space X, the set of admissible controls in the space \(L^{1}(0,T; E)\) is defined by (4) where \(E=L^{2}(\Omega)\). With the assumptions (H_{ u }) and (H_{ f }) holding, the stability results are also valid.
The following example is provided to show that not all optimal control problems are essential.
Example 5.1
Let \(H=[0,1]\), \(\Omega=[0,\pi]\subset R^{1}\). Consider the optimal problem
subject to the distributed parameter system
where \(u \in\mathcal{U}[0,1]\), \(X=L^{2}(\Omega)\), \(v_{0}\in X\), \(A=\frac {\partial^{2}}{\partial x^{2}}\), and \(D(A)=W^{2,2}(\Omega)\cap W_{0}^{1,2}(\Omega)\). Then the A generate a \(C_{0}\)semigroup \(e^{At}\). The mild solution can be given by
Let
here we set
Similarly to ExampleÂ 2.1, \(\mathcal{U}[0,1]\) is compact.
In order to simplify the calculation, let \(v_{0}=0\).
(1) Let \(f(t,v(t,x),u(t,x))=u(t,x)\in\mathcal{U}[0,1]\), we see that the A generate a \(C_{0}\)semigroup \(T(t)=e^{t}\); then the mild solution is
we have \(v(1,x)=0\), for any \(u_{k}\), \(k=1,2, \ldots,N\), hence \(J(u_{k})=0\), \(k=1,2, \ldots, N \). So
(2) Let \(f_{m}(t, v(t,x), u(t,x))=u(t,x)+\frac{1}{m}\sin x\), \(t\in H\), \(x\in \Omega\), for every \(m=1,2, \ldots \)â€‰. Then \(v(1,x)=0\), and \(J(u_{m})=0\), \(m=1,2, \ldots \)â€‰, we get
We see \(\rho(f, f_{m})=\frac{\sqrt{2\pi}}{2m} \rightarrow0\) as \(m\rightarrow+\infty\). For any \(u_{k} \in S(f)\), we have
where \(H_{1}\) is the union of intervals in which \(u_{k}\) takes the value sinx and \(H_{2}\) is the union of intervals in which \(u_{k}\) takes the value \(e^{\frac{1}{2k}}\sin x\), which shows that \(u_{k}\) is not essential. If let \(u^{0}(t,x)\equiv0\), we have
Then all solutions in \(S(f)\) except \(u^{0}(t,x)\equiv0\) are not essential.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No.Â 11261011).
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Deng, H., Wei, W. Stability analysis for optimal control problems governed by semilinear evolution equation. Adv Differ Equ 2015, 103 (2015). https://doi.org/10.1186/s1366201504435
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DOI: https://doi.org/10.1186/s1366201504435
Keywords
 optimal control
 stability
 semilinear evolution equation
 \(C_{0}\)semigroup
 setvalued mapping