Nonlinear impulsive evolution equations with nonlocal conditions and optimal controls
 Lanping Zhu^{1}Email author and
 Qianglian Huang
https://doi.org/10.1186/s1366201507150
© Zhu and Huang 2015
Received: 27 August 2015
Accepted: 1 December 2015
Published: 15 December 2015
Abstract
This paper is concerned with controlled nonlinear impulsive evolution equations with nonlocal conditions. The existence of PCmild solutions is proved, but the uniqueness cannot be obtained. By constructing approximating minimizing sequences of functions, the existence of optimal controls of systems governed by nonlinear impulsive evolution equations is also presented.
Keywords
MSC
1 Introduction
By introducing a reasonable mild solution for (1.1) that can be represented by integral equation, we show the existence of feasible pairs. Moreover, a limited Lagrange problem of system governed by (1.1) is investigated. Due to the lack of uniqueness of feasible pairs, we mainly apply the idea of constructing approximating minimizing sequences of functions twice and derive the existence of optimal controls. It is different from the usual approach that the Banach contraction mapping theory is applied directly to prove the existence of optimal controls if the uniqueness of feasible pairs cannot be obtained. Therefore our results essentially generalize and develop many previous ones in this field.
This paper is organized as follows. In Section 2, we give some associated notations and recall some concepts and facts about the measure of noncompactness, fixed point theorem and impulsive semilinear differential equations. In Section 3, the existence results of controlled nonlocal evolution equations are obtained. In Section 4, the existence of optimal controls for a Lagrange problem is established. Finally, an example is presented to illustrate our results in the last section.
2 Preliminaries
Let X be a real Banach space. \(\mathcal {L}(X)\) is the class of (not necessary bounded) linear operators in X. We denote by \(C([0,T];X)\) the Banach space of all continuous functions from \([0,T]\) to X with the norm \(\u\= \sup \{ \u(t)\,t\in [0,T]\}\) and by \(L^{1}([0,T];X)\) the Banach space of all Xvalued Bochner integrable functions defined on \([0,T]\) with the norm \(\u\_{1}=\int ^{T}_{0}\u(t)\\,dt\). Let \(PC([0,T];X)=\{u: [0,T]\rightarrow X: u(t)\mbox{ is continuous at } t\neq t_{i}\mbox{ and left continuous at }t=t_{i}\mbox{ and the right limit } u(t_{i}^{+})\mbox{ exists for }i=1,2,\ldots,q\}\). It is easy to check that \(PC([0,T];X)\) is a Banach space with the norm \(\u\_{PC}=\sup\{\u(t)\, t\in[0,T]\}\) and \(C([0,T];X)\subseteq PC([0,T];X)\subseteq L^{1}([0,T];X)\).
The map \(Q:D\subset E\rightarrow E\) is said to be αcontraction if there exists a positive constant \(k < 1\) such that \(\alpha(QD) \leq k\alpha(D)\) for any bounded closed subset \(D\subset E\), where E is a Banach space.
The following fixed point theorem plays a key role in the proof of our main results.
Lemma 2.1
([17]: DarboSadovskii)
If \(D\subset E\) is bounded closed and convex, the continuous map \(Q:D\rightarrow D\) is an αcontraction, then the map Q has at least one fixed point in D.
Let Y be another separable reflexive Banach space where controls u take values. Denoted \(P_{f}(Y)\) by a class of nonempty closed and convex subsets of Y. We suppose that the multivalued map \(w:[0,T]\rightarrow P_{f}(Y)\) is measurable, \(w(\cdot)\subset E\), where E is a bounded set of Y, and the admissible control set \(U_{ad}=S_{w}^{p}=\{u\in L^{p}(E)u(t)\in w(t), \mbox{ a.e.}\}\), \(p>1\). Then \(U_{ad}\neq \emptyset\), which can be found in [1].
 (A)
The linear operator \(A: D(A)\subseteq X\rightarrow X\) generates a compact \(C_{0}\)semigroup \(\{S(t):t\geq0\}\). Hence, there exists a positive number M such that \(M=\sup_{0\leq t\leq T}\S(t)\\).
For use in the sequel, we introduce the following definition.
Definition 2.1
In addition, let r be a finite positive constant, and set \(B_{r} :=\{x\in X: \x\\leq r\}\) and \(W_{r} :=\{y\in PC([0,T]; X): y(t)\in B_{r}, \forall t\in[0,T]\}\).
3 Controlled impulsive differential equations
 (F)
 (1)
\(f:[0,T]\times X\rightarrow X\) is a Carathéodory function, i.e., for all \(x\in X\), \(f(\cdot,x): [0,T]\rightarrow X\) is measurable and for a.e. \(t\in[0,T]\), \(f(t,\cdot):X\rightarrow X\) is continuous.
 (2)For finite positive constant \(r>0\), there exists a function \(\varphi_{r}\in L^{1}(0,T;R)\) such thatfor a.e. \(t\in[0, T]\) and \(x\in B_{r}\).$$\bigl\ f(t,x)\bigr\ \leq\varphi_{r}(t) $$
 (1)
 (B)
\(B:[0,T]\rightarrow \mathcal {L}(Y,X)\) is essentially bounded, i.e., \(B\in L^{\infty}([0,T],\mathcal {L}(Y,X))\).
 (G)
\(g: PC([0,T]; X)\rightarrow X\) is a continuous and compact operator.
 (J)\(J_{i}:X\rightarrow X\) satisfies the following Lipschitz condition:for some constants \(h_{i}>0\) and \(x_{1},x_{2}\in PC([0,T]; X)\).$$\bigl\ J_{i}(x_{1})J_{i}(x_{2})\bigr\ \leq h_{i}\x_{1}x_{2}\,\quad i=1,2,\ldots,q, $$
 (R)
\(M\{\x_{0}\+\sup_{x\in W_{r}}\g(x)\+\Bu\_{L^{1}}+\sum_{1\leq i\leq q}[\J_{i}(0)\+h_{i}r]+\\varphi_{r}\_{L^{1}}\}\leq r\).
Remark 3.1
From the assumption (B) and the definition of \(U_{ad}\), it is also easy to verify that \(Bu\in L^{p}([0,T];X)\) with \(p>1\) for all \(u\in U_{ad}\). Therefore, \(Bu\in L^{1}([0,T];X)\) and \(\Bu\_{L^{1}}<+\infty\).
Remark 3.2
Under the assumption (G), \(\sup_{x\in W_{r}}\g(x)\<+\infty\).
Theorem 3.1
Assume that there exists a constant \(r > 0\) such that the conditions (A), (F), (B), (G), (J), and (R) are satisfied. Then the nonlocal problem (3.1) has at least one PCmild solution on \([0,T]\).
In order to obtain the existence of solutions for controlled impulsive evolution equations and optimal controls, we need the following important lemma, which can easily be proved (refer to Lemma 3.2 and Corollary 3.3 in Chapter 3 of [4]).
Lemma 3.1
Proof of Theorem 3.1
The proof is divided into the following three steps.
Remark 3.3
In [7], the authors obtain the solvability of impulsive integrodifferential equations of mixed type with given initial value on an infinite dimensional Banach space. However, they need the locally Lipschitz continuity of f. Here, without the Lipschitz assumption of f, we make full use of the technique of the compactness as regards the solution operator to obtain the solvability of the controlled nonlocal impulsive equation (3.1). Therefore, our results improve those in [7, 13] and the references therein, and they have broader applications.
Remark 3.4
The uniqueness of solution of the controlled impulsive differential equation (3.1) cannot be obtained. Therefore, we can denote by \(\operatorname{Sol}(u)\) all solutions of system (3.1) in \(W_{r}\), for any \(u\in U_{ad}\).
 (G′):

\(g(x)=g(s_{1},\ldots,s_{m},x(s_{1}),\ldots,x(s_{m}))=\sum_{j=1}^{m}c_{j} x(s_{j})\), where \(c_{j}\), \(j=1,2,\ldots,m\), are given constants, and \(0< s_{1}< s_{2}<\cdots< s_{m}<T\).
Corollary 3.1
Proof
It is easy to see that if \(g(x)=g(s_{1},\ldots,s_{m},x(s_{1}),\ldots,x(s_{m}))=\sum_{j=1}^{m}c_{j} x(s_{j})\), then the condition (G) holds. Thus all the conditions in Theorem 3.1 are satisfied. Then the nonlocal impulsive equation (3.1) has at least one PCmild solution on \([0,T]\). This completes the proof. □
4 Existence of optimal controls
In the section, we give the existence of optimal controls for system (3.1).
Let \(x^{u}\in W_{r}\) denote the PCmild solution of system (3.1) corresponding to the control \(u\in U_{ad}\), we consider the following limited Lagrange problem (P):
 (L)
 (1)
The function \(l:[0,T]\times X\times Y\rightarrow R\cup \infty\) is Borel measurable;
 (2)
\(l(t,\cdot,\cdot)\) is sequentially lower semicontinuous on \(X\times Y\) for a.e. \(t\in[0,T]\);
 (3)
\(l(t,x,\cdot)\) is convex on Y for each \(x\in X\) and a.e. \(t\in[0,T]\);
 (4)there are two constants \(c\geq0\), \(d>0\) and \(\phi\in L^{1}([0,T];R)\) such that$$l(t,x,u)\geq\phi(t)+c\x\+d\u\_{Y}^{p}. $$
 (1)
Remark 4.1
A pair \((x(\cdot),u(\cdot))\) is said to feasible if it satisfies system (3.1) for \(x(\cdot)\in W_{r}\).
Remark 4.2
If \((x^{u},u)\) is a feasible pair, then \(x^{u}\in \operatorname{Sol}(u)\subset W_{r}\).
Theorem 4.1
Assume that condition (L) is satisfied. Under the conditions of Theorem 3.1, the problem (P) has at least one optimal feasible pair.
Proof
The proof is divided into the following four steps.
If there are finite elements in \(\operatorname{Sol}(u)\), there exists some \(\overline{x}^{u}\in \operatorname{Sol}(u)\), such that \(J(\overline{x}^{u},u)=\inf_{x^{u}\in \operatorname{Sol}(u)} J(x^{u},u)=J(u)\).
If there are infinite elements in \(\operatorname{Sol}(u)\), there is nothing to prove in the case of \(J(u)=\inf_{x^{u}\in \operatorname{Sol}(u)} J(x^{u},u)=+\infty\).
We assume that \(J(u)=\inf_{x^{u}\in \operatorname{Sol}(u)} J(x^{u},u)<+\infty\). By assumption (L), one has \(J(u)>\infty\).
By the definition of the infimum there exists a sequence \(\{x^{u}_{n}\} _{n=1}^{\infty}\subseteq \operatorname{Sol}(u)\), such that \(J(x^{u}_{n},u)\rightarrow J(u)\) as \(n\rightarrow \infty\).
Step 4. Find \(u_{0}\in U_{ad}\) such that \(J(u_{0})\leq J(u)\), for all \(u\in U_{ad}\).
If \(\inf_{u\in U_{ad}}J(u)=+\infty\), there is nothing to prove.
Assume that \(\inf_{u\in U_{ad}}J(u)<+\infty\). Similar to Step 1, we can prove that \(\inf_{u\in U_{ad}}J(u)>\infty\), and there exists a sequence \(\{u_{n}\} _{n=1}^{\infty}\subseteq U_{ad}\) such that \(J(u_{n})\rightarrow\inf_{u\in U_{ad}}J(u)\) as \(n\rightarrow \infty\). We use that \(\{u_{n}\}_{n=1}^{\infty}\subseteq U_{ad}\), \(\{u_{n}\} _{n=1}^{\infty}\) is bounded in \(L^{p}([0,T]; Y)\). Moreover, \(L^{p}([0,T]; Y)\) is reflexive Banach space. Thus there exists a subsequence, and without loss of generality we may suppose that \(\{u_{n}\}_{n=1}^{\infty}\) converges weakly to some \(u_{0}\in L^{p}([0,T]; Y)\) as \(n\rightarrow\infty\).
Remark 4.3
Constructing approximating minimizing sequences of functions twice plays a key role in the proof of looking for optimal controls, which enable us to deal with the multiple solution problem of feasible pairs. More importantly, this will allow us to study more extensive and complex evolution equations and optimal controls problems. Moreover, we have the following consequences.
Corollary 4.1
Proof
If condition (G′) holds, then we conclude to the existence of feasible pairs on \([0,T]\) from Corollary 3.1. Similar to the proof of Theorem 4.1, we can obtain the optimal feasible pair on \([0,T]\). This completes the proof. □
In particular, if \(g(x)=x_{0}\), we have the following result.
Corollary 4.2
 (J′):

\(J_{i}:X\rightarrow X\), \(i=1,2,\ldots,q\), are continuous and compact mappings.
 (R′):

\(M\{\x_{0}\+\sup_{x\in W_{r}}\g(x)\+\Bu\_{L^{1}}+\sup_{x\in W_{r}}\sum_{i=1}^{q}\J_{i}(x(t_{i}))\+\\varphi_{r}\_{L^{1}}\}\leq r\).
We may apply Schauder’s second fixed point theorem to obtain the existence of PCmild solutions. Note that H is a continuous mapping from \(W_{r}\) to \(W_{r}\). We need to prove that H is a compact mapping. In fact, we already proved that \(\Lambda_{1}\) and \(\Lambda_{2}\) are both compact operators in Theorem 3.1. The same idea can be used to prove the compactness of \(\Lambda_{3}\) due to the assumption (J′) and the AscoliArzela theorem. The rest of the proof is similar to that of Theorem 4.1. So we can obtain the following result.
Corollary 4.3
Let (A), (F), (B), (G), (J′), (R′), and (L) be satisfied. Then the problem (P) has at least one optimal control on \([0,T] \).
5 An example
In this section, we shall give one example to illustrate our theory.
Example 5.1
Suppose that \(\omega:[0,T]\times[0,1]\times[0,1]\times R\rightarrow R\) satisfies the Carathéodory condition, that is, \(\omega (t,y,\xi,r)\) is a continuous function about r for a.e. \((t,y,\xi)\in [0,T]\times[0,1]\times[0,1]\); \(\omega(t,y,\xi,r)\) is measurable about \((t,y,\xi)\) for each fixed \(r\in R\).
 (i)
\(\omega(t,y,\xi,r)\omega(t,y',\xi,r)\leq g_{k}(t,y,y',\xi)\) for all \((t,y,\xi,r), (t,y',\xi,r)\in[0,T]\times [0,1]\times [0,1]\times R\) with \(r\leq k\), where \(g_{k}\in L^{1}([0,T]\times[0,1]\times[0,1]\times R;R^{+})\) satisfies \(\lim_{y\rightarrow y'}\int_{0}^{1}\int_{0}^{T} g_{k}(t,y,y',\xi)\,dt\,d\xi=0\), uniformly in \(y'\in[0,1]\).
 (ii)
\(\omega(t,y,\xi,r)\leq \frac{\delta}{T}r+\zeta(t,y,\xi)\) for all \(r\in R\), where \(\zeta\in L^{2}([0,T]\times[0,1]\times[0,1]; R^{+})\) and \(\delta>0\).
 (1)\(f:[0,T]\times X\rightarrow X\) is a continuous function defined byMoreover, for given \(r>0\), there exists an integrable function \(\phi_{r}: [0,T]\rightarrow R\) such that \(\f(t,z)\\leq \phi_{r}(t)\) for \(t\in[0, T]\), \(z\in B_{r}\).$$f(t,z) (y)=F\bigl(t,z(y)\bigr), \quad 0\leq t\leq T, 0\leq y\leq1. $$
 (2)\(g:PC([0,T];X)\rightarrow X\) is defined byFrom Theorem 4.2 in [19], we directly see that g is well defined and it is a continuous and compact map by the above conditions (i) and (ii) about the function ω.$$g(x) (y)= \int_{0}^{1} \int_{0}^{T} \omega\bigl(t,y,\xi,x(t,\xi)\bigr)\,dt\,d\xi,\quad y\in[0,1]. $$
 (3)\(J_{i}:X\rightarrow X\) is a continuous function for each \(i=1,2,\ldots,q\), defined byHere we take \(J_{i}(x(y))=(\alpha_{i}x(y)+t_{i})^{1}\), \(\alpha _{i}>0\), \(i=1,2,\ldots,q\), \(0< t_{1}< t_{2}<\cdots<t_{q}<T\), \(y\in[0,1]\). Then \(J_{i}\) is Lipschitz continuous with constant \(h_{i}=\alpha_{i}/t_{i}^{2}\), \(i=1,2,\ldots,q\), i.e., the assumption (J) is satisfied.$$J_{i}(x) (y)=J_{i}\bigl(x(y)\bigr). $$
Declarations
Acknowledgements
The authors are grateful to the editor and anonymous reviewers for their valuable comments and suggestions. Moreover, this research was supported by the Natural Science Foundation of China (11201410) and the Natural Science Foundation of Jiangsu Province (BK2012260 and BK20141271).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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