Abstract and Applied Analysis

Volume 2010 (2010), Article ID 213503, 16 pages

http://dx.doi.org/10.1155/2010/213503

## Second-Order Nonlinear Impulsive Integro-Differential Equations of Mixed Type and Optimal Controls in Fractional Power Spaces

Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

Received 23 September 2010; Revised 20 November 2010; Accepted 27 November 2010

Academic Editor: Irena Lasiecka

Copyright © 2010 Y. Peng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A class of second-order nonlinear impulsive integro-differential equations of mixed type whose principal part is given by time-varying generating operators in fractional power spaces is considered. We introduce the reasonable PC--mild solution of second-order nonlinear impulsive integro-differential equations of mixed type and prove its existence. The existence of optimal controls for a Lagrange problem of systems governed by second-order nonlinear impulsive integro-equations of mixed type is also presented. An example is given for demonstration.

#### 1. Introduction

Some interesting models of mathematical biology or population, mechanics of materials, nuclear physics, and so forth, can be written in terms of second-order nonlinear partial integro-differential equations. This is the case of the model proposed to describe viscoelastic problems with memory. The system is given by where generates an evolution system in the parabolic case in Banach spaces (see [1–3]). , are nonlinear integral operators given by

, , are nonlinear maps, and , . This represents the jump in the state , at time , respectively, with , determining the size of the jump at time .

In fact, since the end of last century, impulsive evolution equations on infinite-dimensional spaces have been investigated by many authors including us. Particularly, Ahmed and we considered optimal control problems of systems governed by first-order impulsive evolution equations and first-order impulsive integro-differential equations [4–7]. Recently, we discussed the second-order impulsive evolution equations and the second-order impulsive integro-differential equations and their optimal controls in general Banach spaces [8–11]. In addition, to our knowledge, the second-order impulsive functional differential equations and the second-order impulsive integro-differential equations whose principal operator is bounded have been deeply studied by many authors [12–16]. However, the second-order impulsive integro-differential evolutions equations of mixed type whose principle operator is unbounded in infinite dimensional fractional power spaces and corresponding optimal control problems have not been extensively considered in the literature.

Reducing the second-order evolution equations to the first-order evolution equations, we introduce a family of unbounded linear matrix operators , and prove that generates an evolution system which can be represented by , . Based on the evolution system , we introduce a reasonable PC-*α*-mild solution of (1.1). Using the interpolation space technique, we can overcome the difficulty brought by fractional power spaces. Next, by a virtue of the generalized Gronwall lemma with singularity, impulse, and integrals of mixed type given by us, one can overcome the difficulty brought by operator to get a priori estimate of PC-*α*-mild solution. By compactness condition of space and Leray-Schauder fixed point theory, we can obtain the existence of PC-*α*-mild solution for (1.1). Particularly, introducing new norm, we use the contraction mapping principle to give the uniqueness of *α*-mild solution for the second-order nonlinear integro-differential equation (1.1) without impulses (see Remark 3.4). A Lagrange problem of a system governed by (1.1) whose cost functional includes both and is investigated. By the structure of and compactness of , the existence of optimal controls is verified.

The rest of the paper is organized as follows. In Section 2, we give some associated notations and important lemmas. In Section 3, the existence of PC-*α*-mild solution for (1.1) is presented. In Section 4, we consider a Lagrange problem of system governed by (1.1) and prove the existence of optimal controls. At last, an example demonstrates the applicability of our results.

#### 2. Preliminaries

Let , denote a pair of Banach spaces. If is continuously embedded in , we write ; if is compactly embedded in , we write . Set , is the class of (not necessarily bounded) linear operators in . stands for the family of bounded linear operators in . For , let denote the resolvent set and the resolvent corresponding to .

*Assumption A. *(P_{1}) Let be a family of closed linear operators in , the domain of , and dense in and independent of .

(P_{2}) For , the resolvent of exists for all with and there exists a constant such that

(P_{3}) There exist constants and such that

Let where . is a Banach space and . More generally, in a usual way we introduce the fractional power operator (), which satisfies and for . Let for and denote the Banach space as . Then it is clear that for . Define as continuous at ; is continuous from left and has right-hand limits at , . It can be seen that and are Banach spaces, respectively, with the norms

For the initial value problem
it is well known that (2.4) has a unique classical solution . Moreover, provided . Further, there exists a unique evolution operator , , such that every solution of (2.4) can be represented in the form

Consider the following second-order initial value problem:
Set and ; (2.6) can be rewritten as
where . By [9, Theorem 2.A], we have the following theorem.

Theorem 2.1. *Under the Assumption A, (2.7) has a unique evolution system given by
**In order to derive a priori estimates on the PC- α-mild solution of integro-differential equation of mixed type, we need the following generalized Gronwall inequality with impulses.*

Theorem 2.2. *Let and satisfy the following inequality:
**for all , where , , and , , , , are constants, and . Then there exists constant such that
*

*Proof. *By the inequality (2.9) and [7, Lemma 2.1], there exist , such that
By [17, Lemma 1.7.1], we have
Using the argument method [10, Lemma 3.1], there exists constant such that
This completes the proof.

Next, we extend the Ascoli-Arzela Theorem from to .

Theorem 2.3. *Suppose is a subset. If the following conditions are satisfied:*(1)* is uniformly bounded subset of ,*(2)* is equicontinuous in , , where , ,*(3)*its -sections , , , , , and are relatively compact subsets of , **then is a relatively compact subset of .*

*Proof. *Let satisfy assumptions of Theorem 2.3 and any sequence of . Define . By Ascoli-Arzela Theorem, is a relatively compact subset of . Then, there exists a subsequence of , labeled , and , such that
Define and . It is not difficult to see that, due to Ascoli-Arzela Theorem again, is a relatively compact subset of . There exists a subsequence of , labeled , and such that
In general, define and . Similarly, is a relatively compact subset of . There exists a subsequence of , labeled , and such that
Setting
then
Thus, the set is a relatively compact set.

#### 3. Solution of Second-Order Nonlinear Integro-Differential Equations of Mixed Type

We introduce reasonable mild solution for (1.1) and show the existence of PC-*α*-mild solution.

*Definition 3.1. *A function is said to be a PC-*α*-mild solution of (1.1) if satisfies the following integral equation:

We introduce the following assumptions.

*Assumption F. *(1) The functions , , and are measurable in and locally Lipschitz continuous, that is, for all , , , , , , satisfying , , , , , , , , we have
(2) There exist a constant and a function such that
(3) , .

*Assumption J. *There exists a constant such that maps and satisfy

Theorem 3.2. *Suppose that has a compact resolvent, , . Under the Assumptions A, F and J, the second-order impulsive integro-differential equation (1.1) has a PC- α-mild solution .*

*Proof. *Let be fixed; define the operator on given by
By virtue of the properties of evolution system and Assumption J, for , we have and
This means that for .

For and , in which is a constant. Using Assumptions F(1) and J, we have
where ,
Hence is a continuous operator.

Let be a bounded subset; there exists a constant such that for all . By Assumption F(2) and the properties of evolution operators, there exists constant such that
and is a bounded subset of . Define
Clearly, and are compact, and hence, it is only necessary to consider . Since is a compact operator, for . For , define
By the properties of , one can verify that
where , , and
This means that the set can be approximated to an arbitrary degree of accuracy by a relatively compact set for . Hence itself is relatively compact in for . Define
Using relative compactness of and () and Assumption J, one can show that and are relative compacts in .

By the same procedure, the compactness of and () can also be proved.

Further, using representations of and , properties of (see Theorem 2.1), and those above, one can show that is piecewise equicontinuous. Summarily, is a compact operator in .

By virtue of Gronwall inequality (see Theorem 2.2), one can verify that is a bounded subset of . According to Leray-Schauder fixed point theorem, has a fixed point in . It can be given by the representation (3.1) and for .

*Remark 3.3. *Similarly, under suitable assumptions, the following second-order nonlinear impulsive integro-differential equation of mixed type
has a PC-*α*-mild solution given by

*Remark 3.4. *Assume that has a compact resolvent, , , and .

Under the Assumptions A and F, the second-order nonlinear integro-differential equation of mixed type
has a unique *α*-mild solution given by

#### 4. Existence of Optimal Controls

We suppose that is a reflexive Banach space from which the controls take the values. We denote a class of nonempty closed and convex subsets of by . Assume that the multifunction is measurable and where is a bounded set of , the admissible control set (see [18, Proposition 1.7, page 142, Lemma 3.2, page 175]).

Consider the following controlled system:

*Assumption B. *.

It is easy to see that for all . Define . It is obvious that satisfies the Assumption F.

Theorem 4.1. *Suppose has a compact resolvent. Under Assumptions A, B, F, and J, for every , the system (4.1) has a PC- α-mild solution corresponding to provided .*

Define as a PC-*α*-mild solution of system (4.1) corresponding to the control . is called a feasible pair. We consider the Lagrange problem (P).

Find such that where

We introduce some assumptions on .

*Assumption L. *(1) The functional is Borel measurable.

(2) is sequentially lower semicontinuous on for almost all .

(3) is convex on for each and almost all .

(4) There exist constants , , , and such that

Now we can give the following result on the existence of optimal controls for problem (P).

Theorem 4.2. *Under assumptions of Theorem 4.1 and assumption L, the problem (P) has a solution.*

*Proof. *If , there is nothing to prove. Assume that . By Assumption L, we have . By definition of infimum, there exists a sequence such that . Since is bounded in , there exists a subsequence, relabeled as , and such that
Since is closed and convex, from the Mazur Lemma, we have .

Suppose is the PC-*α*-mild solution of (4.1) corresponding to (). Then satisfies the following impulsive integral equation:
Setting , by Assumptions F(2) and J, and Lemma 3.1, we obtain that (). Furthermore, there exists a subsequence, relabeled as , and such that in as .

Define operators , given by
By [7, Lemma 4.1], we have

Consider the following impulsive differential equation:
By Theorem 3.2, we know that (4.9) have a PC-*α*-mild solution
Define
then, in as . Using Assumption J, we obtain
Using the Gronwall lemma with impulse (see [17, Lemma 1.7.1]), we have
that is, in as . Further,
By the uniqueness of limit, we have
Furthermore,
Thus, is a PC-*α*-mild solution of (4.1) corresponding to .

Since , using the Assumption L, we can obtain
This means that attains its minimum at .

At the end of this section, an example is given to illustrate our theory. We consider the following problem: where , , is bounded domain, and .

Define , , and for ; is a nonempty bounded closed subset of . Let

Define , , , , , , ; then satisfies Assumption F. Thus the problem (4.18) can be rewritten as with the cost functionObviously, satisfying all the assumptions given in our former Theorem 4.2, our results can be used in (4.20).

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 10961009, the Fok Ying Tung Education Foundation under Grant no. 121104 and Introducing Talents Foundation for the Doctor of Guizhou University.

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