Research Article | Open Access
Optimal Control of Linear Impulsive Antiperiodic Boundary Value Problem on Infinite Dimensional Spaces
A class of optimal control problems for infinite dimensional impulsive antiperiodic boundary value problem is considered. Using exponential stabilizability and discussing the impulsive evolution operators, without compactness and exponential stability of the semigroup governed by original principle operator, we present the existence of optimal controls. At last, an example is given for demonstration.
Antiperiodic and periodic motions arise naturally in the mathematical modeling of a variety of physical process. Many authors including us pay great attention to various classes of antiperiodic and periodic systems [1–6]. On the other hand, in order to describe dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases and, some authors have used impulsive differential systems to describe the model since the last century. For the basic theory on impulsive differential equations on finite dimensional spaces, the reader can refer to Lakshmikantham's book (see ).
Recently, we have begun to investigate impulsive periodic system on infinite dimensional spaces. The suitable impulsive evolution operator corresponding to homogenous impulsive periodic system was introduced and its properties (boundedness, periodicity, compactness, and exponential stability) were given. Some results including the existence of the periodic -mild solutions and alternative theorem, criteria of Massera type, asymptotical stability, and robustness by perturbation for linear impulsive periodic system were established. For semilinear impulsive periodic system and intergrodifferential impulsive periodic system, some fixed point theorems such as Horn fixed point theorem and Leary-Schauder fixed point theorem were applied to obtain the existence of the periodic -mild solutions, respectively. In order to do it, we had to construct Poincaré operator, discuss its properties, and derive some generalized Gronwall inequalities with impulse for the estimate of the -mild solutions [8–11].
However, to our knowledge, optimal control problems arising in systems governed by impulsive antiperiodic system on infinite dimensional spaces have not been extensively investigated. Herein, we study the following optimal control problem (P1):
subject to impulsive antiperiodic boundary problem on real Hilbert spaces and , where , , , is a fixed positive number, and denoted the number of impulsive points between and . The operator is the infinitesimal generator of a - semigroup on . Operator and . denotes the -antiperiodic -mild solution of system (1.2) corresponding to the control . We have the functions and . In this paper, using exponential stabilizability and discussing the impulsive evolution operators, without compactness and exponential stability of semigroup generated by original principle operator , we present the existence of antiperiodic optimal controls for problem (P1).
In order to study impulsive antiperiodic system on infinite dimensional spaces, we constructed the impulsive evolution operator associated with and which is very important in sequel. It can be seen from the discussion on linear impulsive antiperiodic system that the invertibility of is the key of the existence of antiperiodic -mild solution of system (1.2). For the invertibility of , compactness or exponential stability of generated by is needed. By virtue of concept of exponential stabilizibility, which is introduced by Barbu and Pavel in  to weaken the assumptions on the existence of antiperiodic -mild solutions, we replace the problem (P1) by problem (P2):
where , such that generates an exponentially stable semigroup. Discussing the impulsive evolution operator associated with operator and and giving some sufficient conditions for invertibility of , we prove that every antiperiodic -mild solution of (1.2) is an antiperiodic -mild solution of (1.4) with and vice versa. Therefore, the equivalence between problem (P1) and problem (P2) is shown. Utilizing some techniques of semigroup theory and functional analysis, we present the existence of antiperiodic optimal controls for problem (P2), which implies the existence of solutions for problem (P1).
The main result of this paper is the existence of optimal control for problem (P1) (given by Theorem 4.1). However, the novelty of this paper over other related results in literature consists in the fact that the invertibility of is replaced by weaker condition. In addition some sufficient conditions for invertibility of are presented.
This paper is organized as follows. In Section 2, impulsive evolution operator and its exponential stability are studied and some sufficient conditions guaranteeing are given. Section 3 is devoted to the equivalence of (P1) and (P2). In Section 4, the existence of optimal antiperiodic arcs for (P2) is presented. Hence, the existence of optimal controls for (P1) is obtained. At last, an example is given to demonstrate the applicability of our results.
2. Invertibility of
Let be a Hilbert space. denotes the space of linear operators in ; denotes the space of bounded linear operators in . is the Hilbert space with the usual supremum norm. Define . We introduce is continuous at , is continuous from left and has right hand limits at and Set
It can be seen that endowed with the norm , is a Hilbert space.
The basic hypotheses are the following Assumption [H1]. is the infinitesimal generator of a -semigroup in with domain . There exists such that . For each , and .
Under Assumption [H1], we consider the Cauchy problem
For Cauchy problem (2.2), if and is an invariant subspace of , using (, Theorem , page 144), step by step, one can verify that the Cauchy problem (2.2) has a unique classical solution represented by where
Definition 2.1. The operator given by (2.4) is called the impulsive evolution operator associated with operator and .
Lemma 2.2. Impulsive evolution operator has the following properties. (1)For , there exists a constant such that (2) For , , .(3)For and , (4) For and , .(5)For , there exits , such that
It is well known that if there exist constants and such that the semigroup generated by satisfies , the semigroup is said to be exponential stable. In general, a semigroup may not be exponential stable.
Let . The pair is said to be exponentially stabilizable, if there exists such that generates an exponentially stable -semigroup ; that is, there exist and such that
Remark 2.3. By [13, Theorem ], the following inequality implies that the exponential stability of .
Impulsive evolution operator plays an important role in the sequel. Here, we need to discuss the exponential stability and exponential stabilizability of impulsive evolution operator.
Definition 2.4. is called exponential stability if there exist and such that
Consider the Cauchy problem The impulsive evolution operator associated with operator and can be given by It is not difficult to verify that also satisfies the similar properties in Lemma 2.2.
Assumption [H2]: The pair is exponentially stabilizable.
Under Assumptions [H1] and [H2], by [14, Lemmas and ], we can give some sufficient conditions guaranteeing exponential stability of immediately.
Lemma 2.5. Assumptions [H1] and [H2] hold. There exists such that Then is exponentially stable.
Lemma 2.6. Assumptions [H1] and [H2] hold. Suppose If there exists such that where then is exponentially stable.
Corollary 2.7. Let Assumption [H1] and (2.12) hold. There exist , such that , . If there exists such that where then is exponential stable.
Now some sufficient conditions for the existence of inversion of can be given.
Proof. Consider the Under the assumptions of Lemma 2.5, is exponential stable. It comes from the periodicity of that Thus, we obtain Obviously, the series is convergent, thus operator . It comes from that
Further, we give a little big stronger condition which will guarantee exponential stability of . However, it is more easy to be demonstrated.
Corollary 2.9. Assumptions [H1] and [H2] hold. If then the impulsive evolution operator is strongly convergent to zero at infinity (i.e., as ). Further, the operator is inverse and .
Remark 2.10. If , then as and the operator is inverse and .
3. Optimal Control Problem of Impulsive Antiperiodic System
We study the following optimal control problem (P1):
Definition 3.1. A function is said to be a -antiperiodic -mild solution of the controlled system (3.2) if satisfies
If system (3.2) has a -antiperiodic -mild solution corresponding to , is said to be an admissible pair. Set
which is called admissible set. Problem (P1) can be rewritten as follows.
Find such that
In fact, if the condition
is satisfied, then for every the -antiperiodic -mild solution of system (3.2) can be given by
Under Assumptions [H1] and [H2], we can write system (3.2) formally in the form
and substitute so . Therefore, we led to the problem (P2):
It can be seen from the proof of Theorem 2.8 that if is exponentially stable, then exists and . Set
then given by
is the antiperiodic -mild solution of (3.11).
By Theorem 2.8, we have the following existence result immediately.
In order to show the equivalence of problem (P1) and problem (P2), we have to prove that every -mild solution of (3.2) is a -mild solution of (3.11) with and vice versa. It is not obvious for -mild solution. Here is the equivalence.
Theorem 3.3. Under Assumptions [H1] and [H2], if is exponentially stable, then every -mild solution of (3.2) is a -mild solution of (3.11) with and vice versa. Therefore, the problem (P1) is equivalent to problem (P2).
Proof. It is obvious that every strong solution of system (3.2) is a strong solution of system (3.11). We prove only that (3.3) implies
as the inverse statement will have the same proof. Therefore, let satisfy (3.3) and denote the Yosida approximation of by . Let be the strong solution of
Taking into account that it follows that for each but fixed, where the operator is the impulsive evolution operator associated with and .
In fact, for ,
For , step by step, as , uniformly in . Of course, we have
On the other hand, define then
Since by virtue of Majorized Convergence theorem, we obtain
This implies that in as .
However, (3.16) can be written as with .
Similarly, one can obtain that in (3.27) is also convergent to the solution of (3.14) with .
At the same time, it is easy to see that and problem (P1) is equivalent to problem (P2).
4. Existence of Optimal Controls
In this section, we present the existence of optimal controls for problem (P1) which is the main result of this paper.
We make the following assumptions.[H3]The function is convex and lower semicontinuous; , where . Moreover, has the the following growth properties:
[H4]The function is convex and lower semicontinuous; for arbitrary ,
for some and .
Theorem 4.1. In addition to assumptions of Theorem 3.3, Assumptions [H3] and [H4] hold. Then problem (P1) has at least one optimal control pair .
Proof. By virtue of Theorem 3.3, it is sufficient to show the existence of optimal controls for problem (P2). Set
If , there is nothing to prove. By Assumptions [H3] and [H4], we know .
Let with and be a minimizing sequence for problem (P2). This means
Set It is obvious that (4.4) implies that
Let be any measurable subset of and . Clearly, with and .
It can be seen from Assumption [H3] that there exists such that where
By standard argument, we have This implies that the set is uniformly integrable on . In view of the Dunford-Petties theorem, (4.9) implies that is sequentially weakly compact in . Say weakly in .
Moreover, (4.2) and (4.4) imply Taking into account that the pair satisfies It comes from (4.11) and (4.10) that which deduce that there exists such that that is, is bounded in Banach space . By Alaoglu theorem, we have weakly star convergent in .
Set and , then There exists a function such that with Clearly, One can verify weakly convergent in . This implies that . Hence is the -antiperiod -mild solution of system (3.11) corresponding to the control given by with
Letting in (4.4), using Assumptions [H3] and [H4] again, by [15, Theorem ], we can obtain Thus, we can conclude that . In fact, let ; is the optimal pair for problem (P1).
5. An Example
Let and let , , be an orthogonal basis for .
related by the following antiperiodic boundary value problem with impulse:
Let and : satisfy (4.1) and Assumptions [H3] and [H4]. The operator is defined as follows:
and is asymptotically stable but not exponentially stable.
Let , then generates the -semigroup given by
Obviously, is exponentially stable. By Lemma 2.5, there exists a
This work is supported by the National Natural Science Foundation of China (no. 10961009), Introducing Talents Foundation for the Doctor of Guizhou University (2009, no. 031) and Youth Teachers Natural Science Foundation of Guizhou University (2009, no. 083).
- H. Okochi, “On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 9, pp. 771–783, 1990.
- S. Aizicovici and N. H. Pavel, “Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space,” Journal of Functional Analysis, vol. 99, no. 2, pp. 387–408, 1991.
- Y. Chen, X. Wang, and H. Xu, “Anti-periodic solutions for semilinear evolution equations,” Journal of Mathematical Analysis and Applications, vol. 273, no. 2, pp. 627–636, 2002.
- X. Xiang and N. U. Ahmed, “Existence of periodic solutions of semilinear evolution equations with time lags,” Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 11, pp. 1063–1070, 1992.
- P. Sattayatham, S. Tangmanee, and W. Wei, “On periodic solutions of nonlinear evolution equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 98–108, 2002.
- W. Wei and X. Xiang, “Anti-periodic solutions for first and second order nonlinear evolution equations in Banach spaces,” Journal of Systems Science and Complexity, vol. 17, no. 1, pp. 96–108, 2004.
- V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
- J. Wang, X. Xiang, and W. Wei, “Linear impulsive periodic system with time-varying generating operators on Banach space,” Advances in Difference Equations, vol. 2007, Article ID 26196, 16 pages, 2007.
- J. Wang, X. Xiang, and Q. Chen, “Existence and global asymptotical stability of periodic solution for the -periodic logistic system with time-varying generating operators and -periodic impulsive perturbations on banach space,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 524945, 16 pages, 2008.
- J. Wang, X. Xiang, W. Wei, and Q. Chen, “Bounded and periodic solutions of semilinear impulsive periodic system on Banach spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 401947, 14 pages, 2008.
- J. Wang, X. Xiang, and W. Wei, “Periodic solutions of a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space,” Electronic Journal of Qualitative Theory of Differential Equations, no. 4, pp. 1–17, 2009.
- V. Barbu and N. H. Pavel, “Periodic optimal control in Hilbert space,” Applied Mathematics and Optimization, vol. 33, no. 2, pp. 169–188, 1996.
- N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, vol. 246 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1991.
- J. Wang, X. Xiang, and W. Wei, “Global behaviors and optimal harvesting of a class of impulsive periodic logistic single-species system with continuous periodic control strategy,” Boundary Value Problems, vol. 2008, Article ID 192353, 21 pages, 2008.
- E. J. Balder, “Necessary and sufficient conditions for -strong-weak lower semicontinuity of integral functionals,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 12, pp. 1399–1404, 1987.
Copyright © 2010 JinRong Wang and YanLong Yang. 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.