Global Behaviors and Optimal Harvesting of a Class of Impulsive Periodic Logistic Single-Species System with Continuous Periodic Control Strategy

  • JinRong Wang1Email author,

    Affiliated with

    • X Xiang2 and

      Affiliated with

      • W Wei2

        Affiliated with

        Boundary Value Problems20092008:192353

        DOI: 10.1155/2008/192353

        Received: 30 September 2008

        Accepted: 17 December 2008

        Published: 8 February 2009

        Abstract

        Global behaviors and optimal harvesting of a class of impulsive periodic logistic single-species system with continuous periodic control strategy is investigated. Four new sufficient conditions that guarantee the exponential stability of the impulsive evolution operator introduced by us are given. By virtue of exponential stability of the impulsive evolution operator, we present the existence, uniqueness and global asymptotical stability of periodic solutions. Further, the existence result of periodic optimal controls for a Bolza problem is given. At last, an academic example is given for demonstration.

        1. Introduction

        In population dynamics, the optimal management of renewable resources has been one of the interesting research topics. The optimal exploitation of renewable resources, which has direct effect on their sustainable development, has been paid much attention [13]. However, it is always hoped that we can achieve sustainability at a high level of productivity and good economic profit, and this requires scientific and effective management of the resources.

        Single-species resource management model, which is described by the impulsive periodic logistic equations on finite-dimensional spaces, has been investigated extensively, no matter how the harvesting occurs, continuously [1, 4] or impulsively [57]. However, the associated single-species resource management model on infinite-dimensional spaces has not been investigated extensively.

        Since the end of last century, many authors including Professors Nieto and Hernández pay great attention on impulsive differential systems. We refer the readers to [822]. Particulary, Doctor Ahmed investigated optimal control problems [23, 24] for impulsive systems on infinite-dimensional spaces. We also gave a series of results [2534] for the first-order (second-order) semilinear impulsive systems, integral-differential impulsive system, strongly nonlinear impulsive systems and their optimal control problems. Recently, we have investigated linear impulsive periodic system on infinite-dimensional spaces. Some results [3537] including the existence of periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq1_HTML.gif -mild solutions and alternative theorem, criteria of Massera type, asymptotical stability and robustness against perturbation for a linear impulsive periodic system are established.

        Herein, we devote to studying global behaviors and optimal harvesting of the generalized logistic single-species system with continuous periodic control strategy and periodic impulsive perturbations:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ1_HTML.gif
        (1.1)

        On infinite-dimensional spaces, where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq2_HTML.gif denotes the population number of isolated species at time http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq3_HTML.gif and location http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq5_HTML.gif is a bounded domain and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq6_HTML.gif , operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq7_HTML.gif . The coefficients http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq9_HTML.gif are sufficiently smooth functions of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq10_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq11_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq15_HTML.gif . Denoting http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq17_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq18_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq19_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq20_HTML.gif is related to the periodic change of the resources maintaining the evolution of the population and the periodic control policy http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq21_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq22_HTML.gif is a suitable admissible control set. Time sequence http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq24_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq25_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq26_HTML.gif denote mutation of the isolate species at time http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq27_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq28_HTML.gif .

        Suppose http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq29_HTML.gif is a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq30_HTML.gif is a separable reflexive Banach space. The objective functional is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ2_HTML.gif
        (1.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq31_HTML.gif is Borel measurable, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq32_HTML.gif is continuous, and nonnegative and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq33_HTML.gif denotes the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq34_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq35_HTML.gif -mild solution of system (1.1) at location http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq36_HTML.gif and corresponding to the control http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq37_HTML.gif . The Bolza problem ( http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq38_HTML.gif ) is to find http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq39_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq40_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq41_HTML.gif

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq45_HTML.gif is the least positive constant such that there are http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq46_HTML.gif s in the interval http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq48_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq49_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq50_HTML.gif . The first equation of system (1.1) describes the variation of the population number http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq51_HTML.gif of the species in periodically continuous controlled changing environment. The second equation of system (1.1) shows that the species are isolated. The third equation of system (1.1) reflects the possibility of impulsive effects on the population.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq52_HTML.gif satisfy some properties (such as strongly elliptic) in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq53_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq54_HTML.gif (such as http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq55_HTML.gif . For every http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq56_HTML.gif define http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq57_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq58_HTML.gif is the infinitesimal generator of a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq59_HTML.gif -semigroup http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq60_HTML.gif on the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq61_HTML.gif (such as http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq62_HTML.gif ). Define x http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq63_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq66_HTML.gif then system (1.1) can be abstracted into the following controlled system:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ3_HTML.gif
        (1.3)
        On the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq67_HTML.gif , and the associated objective functional
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ4_HTML.gif
        (1.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq68_HTML.gif denotes the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq69_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq70_HTML.gif -mild solution of system (1.3) corresponding to the control http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq71_HTML.gif . The Bolza problem (P) is to find http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq72_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq73_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq74_HTML.gif The investigation of the system (1.3) cannot only be used to discuss the system (1.1), but also provide a foundation for research of the optimal control problems for semilinear impulsive periodic systems. The aim of this paper is to give some new sufficient conditions which will guarantee the existence, uniqueness, and global asymptotical stability of periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq75_HTML.gif -mild solutions for system (1.3) and study the optimal control problems arising in the system (1.3).

        The paper is organized as follows. In Section 2, the properties of the impulsive evolution operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq76_HTML.gif are collected. Four new sufficient conditions that guarantee the exponential stability of the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq77_HTML.gif are given. In Section 3, the existence, uniqueness, and global asymptotical stability of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq78_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq79_HTML.gif -mild solution for system (1.3) is obtained. In Section 4, the existence result of periodic optimal controls for the Bolza problem (P) is presented. At last, an academic example is given to demonstrate our result.

        2. Impulsive Periodic Evolution Operator and It's Stability

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq80_HTML.gif be a Banach space, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq81_HTML.gif denotes the space of linear operators on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq82_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq83_HTML.gif denotes the space of bounded linear operators on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq84_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq85_HTML.gif is the Banach space with the usual supremum norm. Denote http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq86_HTML.gif and define http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq87_HTML.gif is continuous at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq88_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq89_HTML.gif is continuous from left and has right-hand limits at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq91_HTML.gif .

        Set
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ5_HTML.gif
        (2.1)

        It can be seen that endowed with the norm http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq92_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq93_HTML.gif is a Banach space.

        In order to investigate periodic solutions, we introduce the following two spaces:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ6_HTML.gif
        (2.2)
        Set
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ7_HTML.gif
        (2.3)

        It can be seen that endowed with the norm http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq94_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq95_HTML.gif is a Banach space.

        We introduce assumption [H1].

        [H1.1]: http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq96_HTML.gif is the infinitesimal generator of a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq97_HTML.gif -semigroup http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq98_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq99_HTML.gif with domain http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq100_HTML.gif .

        [H1.2]:There exists http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq101_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq102_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq103_HTML.gif .

        [H1.3]:For each http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq104_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq105_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq106_HTML.gif .

        Under the assumption [H1], consider
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ8_HTML.gif
        (2.4)
        and the associated Cauchy problem
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ9_HTML.gif
        (2.5)
        For every http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq107_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq108_HTML.gif is an invariant subspace of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq109_HTML.gif , using ([38, Theorem 5.2.2, page 144]), step by step, one can verify that the Cauchy problem (2.5) has a unique classical solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq110_HTML.gif represented by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq111_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq112_HTML.gif given by
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ10_HTML.gif
        (2.6)

        The operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq113_HTML.gif is called impulsive evolution operator associated with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq115_HTML.gif .

        The following lemma on the properties of the impulsive evolution operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq116_HTML.gif associated with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq118_HTML.gif is widely used in this paper.

        Lemma 2.1.

        Let assumption [H1] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq119_HTML.gif hold. The impulsive evolution operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq120_HTML.gif has the following properties.

        (1)For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq121_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq122_HTML.gif , there exists a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq123_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq124_HTML.gif

        (2)For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq125_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq127_HTML.gif .

        (3)For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq130_HTML.gif .

        (4)For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq131_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq133_HTML.gif .

        (5)For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq134_HTML.gif , there exits http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq136_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ11_HTML.gif
        (2.7)
        Proof.
        1. (1)
          By assumption [H1.1], there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq137_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq138_HTML.gif . Using assumption [H1.3], it is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq139_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq140_HTML.gif . (2) By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq141_HTML.gif -semigroup and the construction of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq142_HTML.gif , one can verify the result immediately. (3) By assumptions [H1.2], [H1.3], and elementary computation, it is easy to obtain the result. (4) For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq143_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq144_HTML.gif , by virtue of (3) again and again, we arrive at
          http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ12_HTML.gif
          (2.8)
           
        1. (5)
          Without loss of generality, for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq145_HTML.gif ,
          http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ13_HTML.gif
          (2.9)
           

        This completes the proof.

        In order to study the asymptotical properties of periodic solutions, it is necessary to discuss the exponential stability of the impulsive evolution operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq146_HTML.gif . We first give the definition of exponential stable for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq147_HTML.gif .

        Definition 2.2.

        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq149_HTML.gif is called exponentially stable if there exist http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq151_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ14_HTML.gif
        (2.10)
        Assumption [H2]: http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq152_HTML.gif is exponentially stable, that is, there exist http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq154_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ15_HTML.gif
        (2.11)

        An important criteria for exponential stability of a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq155_HTML.gif -semigroup is collected here.

        Lemma 2.3 (see [38, Lemma 7.2.1]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq156_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq157_HTML.gif -semigroup on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq158_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq159_HTML.gif be its infinitesimal generator. Then the following assertions are equivalent:

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq160_HTML.gif is exponentially stable.

        (2)For every http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq161_HTML.gif there exits a positive constants http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq162_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ16_HTML.gif
        (2.12)

        Next, four sufficient conditions that guarantee the exponential stability of impulsive evolution operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq163_HTML.gif are given.

        Lemma 2.4.

        Assumptions [H1] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq164_HTML.gif and [H2] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq165_HTML.gif hold. There exists http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq166_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ17_HTML.gif
        (2.13)

        Then, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq167_HTML.gif is exponentially stable.

        Proof.

        Without loss of generality, for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq168_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ18_HTML.gif
        (2.14)
        Suppose http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq169_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq170_HTML.gif Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ19_HTML.gif
        (2.15)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq171_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq172_HTML.gif , then we obtain http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq173_HTML.gif

        Lemma 2.5.

        Assume that assumption [H1] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq174_HTML.gif holds. Suppose
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ20_HTML.gif
        (2.16)
        If there exists http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq175_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ21_HTML.gif
        (2.17)
        for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq176_HTML.gif where
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ22_HTML.gif
        (2.18)

        Then, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq177_HTML.gif is exponentially stable.

        Proof.

        It comes from (2.17) that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ23_HTML.gif
        (2.19)
        Further,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ24_HTML.gif
        (2.20)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq178_HTML.gif is denoted the number of impulsive points in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq179_HTML.gif .

        For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq180_HTML.gif , by (2.16), we obtain the following two inequalities:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ25_HTML.gif
        (2.21)
        This implies
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ26_HTML.gif
        (2.22)
        that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ27_HTML.gif
        (2.23)
        Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ28_HTML.gif
        (2.24)
        Thus, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ29_HTML.gif
        (2.25)

        By (5) of Lemma 2.1, let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq181_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq182_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq183_HTML.gif

        Lemma 2.6.

        Assume that assumption [H1] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq184_HTML.gif holds. The limit
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ30_HTML.gif
        (2.26)
        Suppose there exists http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq185_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ31_HTML.gif
        (2.27)

        Then, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq186_HTML.gif is exponentially stable.

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq187_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq188_HTML.gif . It comes from
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ32_HTML.gif
        (2.28)
        that there exits a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq189_HTML.gif enough small such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ33_HTML.gif
        (2.29)
        that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ34_HTML.gif
        (2.30)
        From (2.27), we know that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ35_HTML.gif
        (2.31)
        Then, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ36_HTML.gif
        (2.32)
        Hence,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ37_HTML.gif
        (2.33)

        Here, we only need to choose http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq190_HTML.gif small enough such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq191_HTML.gif , by (5) of Lemma 2.1 again, let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq192_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq193_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq194_HTML.gif

        Lemma 2.7.

        Assume that assumption [H1] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq195_HTML.gif holds. For some http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq196_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq197_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ38_HTML.gif
        (2.34)

        Imply the exponential stability of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq198_HTML.gif .

        Proof.

        It comes from the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq199_HTML.gif , the inequality
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ39_HTML.gif
        (2.35)

        and the boundedness of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq200_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq201_HTML.gif are convergent, that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq202_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq203_HTML.gif and fixed http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq204_HTML.gif . This shows that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq205_HTML.gif is bounded for each http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq206_HTML.gif and fixed http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq207_HTML.gif and hence, by virtue of uniform boundedness principle, there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq208_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq209_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq210_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq211_HTML.gif denote the operator given by http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq212_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq214_HTML.gif is fixed. Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq215_HTML.gif is defined every where on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq216_HTML.gif and by assumption it maps http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq217_HTML.gif and it is a closed operator. Hence, by closed graph theorem, it is a bounded linear operator from http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq218_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq219_HTML.gif . Thus, there exits a constant http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq220_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq221_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq222_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq223_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq224_HTML.gif is fixed.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq225_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq226_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq227_HTML.gif and define http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq228_HTML.gif as
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ40_HTML.gif
        (2.36)
        Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ41_HTML.gif
        (2.37)
        and hence,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ42_HTML.gif
        (2.38)
        Thus, for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq229_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ43_HTML.gif
        (2.39)
        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq230_HTML.gif . Fix http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq231_HTML.gif . Then, for any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq232_HTML.gif we can write http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq233_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq234_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq235_HTML.gif and we have
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ44_HTML.gif
        (2.40)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq236_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq237_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq238_HTML.gif , this shows that our result.

        3. Periodic Solutions and Global Asymptotical Stability

        Consider the following controlled system:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ45_HTML.gif
        (3.1)
        and the associated Cauchy problem
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ46_HTML.gif
        (3.2)

        In addition to assumption [H1], we make the following assumptions:

        [H3]: http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq239_HTML.gif is measurable and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq240_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq241_HTML.gif .

        [H4]:For each http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq242_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq243_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq244_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq245_HTML.gif .

        [H5]: http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq246_HTML.gif has bounded, closed, and convex values and is graph measurable, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq247_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq248_HTML.gif are bounded, where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq249_HTML.gif is a separable reflexive Banach space.

        [H6]:Operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq251_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq252_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq253_HTML.gif .

        Denote the set of admissible controls
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ47_HTML.gif
        (3.3)

        Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq254_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq256_HTML.gif is bounded, convex, and closed.

        We introduce http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq257_HTML.gif -mild solution of Cauchy problem (3.2) and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq258_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq259_HTML.gif -mild solution of system (3.1).

        Definition 3.1.

        A function http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq260_HTML.gif , for finite interval http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq261_HTML.gif , is said to be a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq262_HTML.gif -mild solution of the Cauchy problem (3.2) corresponding to the initial value http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq263_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq264_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq265_HTML.gif is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ48_HTML.gif
        (3.4)

        A function http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq266_HTML.gif is said to be a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq267_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq268_HTML.gif -mild solution of system (3.1) if it is a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq269_HTML.gif -mild solution of Cauchy problem (3.2) corresponding to some http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq270_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq271_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq272_HTML.gif .

        Theorem 3.2 .A.

        Assumptions [H1], [H3], [H4], [H5], and [H6] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq273_HTML.gif hold. Suppose http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq274_HTML.gif is exponentially stable, for every http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq275_HTML.gif , system (3.1) has a unique http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq276_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq277_HTML.gif -mild solution:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ49_HTML.gif
        (3.5)
        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq278_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ50_HTML.gif
        (3.6)
        Further,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ51_HTML.gif
        (3.7)
        is a bounded linear operator and
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ52_HTML.gif
        (3.8)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq279_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq280_HTML.gif .

        Further, for arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq281_HTML.gif , the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq282_HTML.gif -mild solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq283_HTML.gif of the Cauchy problem (3.2) corresponding to the initial value http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq284_HTML.gif and control http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq285_HTML.gif , satisfies the following inequality:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ53_HTML.gif
        (3.9)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq286_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq287_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq288_HTML.gif -mild solution of system (3.1), http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq289_HTML.gif is not dependent on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq290_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq291_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq292_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq293_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq294_HTML.gif can be approximated to the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq295_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq296_HTML.gif -mild solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq297_HTML.gif according to exponential decreasing speed.

        Proof.

        Consider the operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq298_HTML.gif . By (4) of Lemma 2.1 and the stability of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq299_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ54_HTML.gif
        (3.10)
        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq300_HTML.gif . Obviously, the series http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq301_HTML.gif is convergent, thus operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq302_HTML.gif . It comes from http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq303_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq304_HTML.gif It is well known that system (3.1) has a periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq305_HTML.gif -mild solution if and only if http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq306_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq307_HTML.gif is invertible, we can uniquely solve
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ55_HTML.gif
        (3.11)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq308_HTML.gif , where
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ56_HTML.gif
        (3.12)
        Note that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ57_HTML.gif
        (3.13)
        it is not difficult to verify that the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq309_HTML.gif -mild solution of the Cauchy problem (3.2) corresponding to initial value http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq310_HTML.gif given by
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ58_HTML.gif
        (3.14)

        is just the unique http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq311_HTML.gif -periodic of system (3.1).

        It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq312_HTML.gif is linear. Next, verify the estimation (3.8). In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq313_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ59_HTML.gif
        (3.15)
        On the other hand,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ60_HTML.gif
        (3.16)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq314_HTML.gif , next the estimation (3.8) is verified.

        System (3.1) has a unique http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq315_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq316_HTML.gif -mild solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq317_HTML.gif given by (3.5) and (3.6). The http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq318_HTML.gif -mild solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq319_HTML.gif of the Cauchy problem (3.2) corresponding to initial value http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq320_HTML.gif and control http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq321_HTML.gif can be given by (3.4). Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ61_HTML.gif
        (3.17)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq322_HTML.gif , one can obtain (3.9) immediately.

        Definition 3.3.

        The http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq323_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq324_HTML.gif -mild solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq325_HTML.gif of the system (3.1) is said to be globally asymptotically stable in the sense that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ62_HTML.gif
        (3.18)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq326_HTML.gif is any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq327_HTML.gif -mild solutions of the Cauchy problem (3.2) corresponding to initial value http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq328_HTML.gif and control http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq329_HTML.gif .

        By Theorem 3.2 and the stability of the impulsive evolution operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq330_HTML.gif in Section 2, one can obtain the following results.

        Corollary 3.A.

        Under the assumptions of Theorem 3.2 , the system ( 3.1 ) has a unique http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq331_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq332_HTML.gif -mild solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq333_HTML.gif which is globally asymptotically stable.

        4. Existence of Periodic Optimal Harvesting Policy

        In this section, we discuss existence of periodic optimal harvesting policy, that is, periodic optimal controls for optimal control problems arising in systems governed by linear impulsive periodic system on Banach space.

        By the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq334_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq335_HTML.gif -mild solution expression of system (3.1) given in Theorem 3.2, one can obtain the result.

        Theorem 4.A.

        Under the assumptions of Theorem 3.2 , the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq336_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq337_HTML.gif -mild solution of system (3.1) continuously depends on the control on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq338_HTML.gif , that is, let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq339_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq340_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq341_HTML.gif -mild solution of system (3.1) corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq342_HTML.gif . There exists constant http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq343_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ63_HTML.gif
        (4.1)

        Proof.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq344_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq345_HTML.gif are the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq346_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq347_HTML.gif -mild solution corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq348_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq349_HTML.gif , respectively, then we have
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ64_HTML.gif
        (4.2)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ65_HTML.gif
        (4.3)
        Further,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ66_HTML.gif
        (4.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq350_HTML.gif . This completes the proof.

        Lemma 4.A.

        Suppose http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq351_HTML.gif is a strong continuous operator. The operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq352_HTML.gif , given by
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ67_HTML.gif
        (4.5)

        is strongly continuous.

        Proof.

        Without loss of generality, for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq353_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ68_HTML.gif
        (4.6)

        By virtue of strong continuity of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq354_HTML.gif , boundedness of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq355_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq356_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq357_HTML.gif is strongly continuous.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq358_HTML.gif denote the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq359_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq360_HTML.gif -mild solution of system (3.1) corresponding to the control http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq361_HTML.gif , we consider the Bolza problem (P).

        Find http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq362_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq363_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq364_HTML.gif , where
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ69_HTML.gif
        (4.7)

        We introduce the following assumption on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq365_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq366_HTML.gif .

        Assumption [H7].

        [H7.1] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq367_HTML.gif The functional http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq368_HTML.gif is Borel measurable.

        [H7.2] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq369_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq370_HTML.gif is sequentially lower semicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq371_HTML.gif for almost all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq372_HTML.gif .

        [H7.3] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq373_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq374_HTML.gif is convex on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq375_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq376_HTML.gif and almost all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq377_HTML.gif .

        [H7.4] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq378_HTML.gif There exist constants http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq379_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq380_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq381_HTML.gif is nonnegative and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq382_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ70_HTML.gif
        (4.8)

        [H7.5] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq383_HTML.gif The functional http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq384_HTML.gif is continuous and nonnegative.

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

        Theorem 4.B.

        Suppose C is a strong continuous operator and assumption [H7] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq385_HTML.gif holds. Under the assumptions of Theorem 3.2, the problem (P) has a unique solution.

        Proof.

        If http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq386_HTML.gif there is nothing to prove.

        We assume that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq387_HTML.gif By assumption [H7], we have
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ71_HTML.gif
        (4.9)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq388_HTML.gif is a constant. Hence http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq389_HTML.gif .

        By the definition of infimum there exists a sequence http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq390_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq391_HTML.gif

        Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq392_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq393_HTML.gif , there exists a subsequence, relabeled as http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq394_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq395_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq396_HTML.gif weakly convergence in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq397_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq398_HTML.gif Because of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq399_HTML.gif is the closed and convex set, thanks to the Mazur lemma, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq400_HTML.gif . Suppose http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq401_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq402_HTML.gif are the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq403_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq404_HTML.gif -mild solution of system (3.1) corresponding to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq405_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq406_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq407_HTML.gif , respectively, then http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq408_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq409_HTML.gif can be given by
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ72_HTML.gif
        (4.10)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ73_HTML.gif
        (4.11)
        Define
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ74_HTML.gif
        (4.12)
        then by Lemma 4.2, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ75_HTML.gif
        (4.13)

        as http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq410_HTML.gif weakly convergence in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq411_HTML.gif .

        Next, we show that
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ76_HTML.gif
        (4.14)
        In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq412_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ77_HTML.gif
        (4.15)
        By elementary computation, we arrive at
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ78_HTML.gif
        (4.16)
        Consider the time interval http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq413_HTML.gif , similarly we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ79_HTML.gif
        (4.17)
        In general, given any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq414_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq415_HTML.gif and the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq416_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq417_HTML.gif , prior to the jump at time http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq418_HTML.gif , we immediately follow the jump as
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ80_HTML.gif
        (4.18)
        the associated interval http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq419_HTML.gif , we also similarly obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ81_HTML.gif
        (4.19)
        Step by step, we repeat the procedures till the time interval is exhausted. Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq420_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq421_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq422_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq423_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq424_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq425_HTML.gif , thus we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ82_HTML.gif
        (4.20)
        that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ83_HTML.gif
        (4.21)

        with strongly convergence as http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq426_HTML.gif .

        Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq427_HTML.gif , using the assumption [H7] http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq428_HTML.gif and Balder's theorem, we can obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ84_HTML.gif
        (4.22)

        This shows that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq429_HTML.gif attains its minimum at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq430_HTML.gif . This completes the proof.

        5. Example

        Last, an academic example is given to illustrate our theory.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq431_HTML.gif and consider the following population evolution equation with impulses:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ85_HTML.gif
        (5.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq432_HTML.gif denotes time, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq433_HTML.gif denotes age, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq434_HTML.gif is called age density function, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq435_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq436_HTML.gif are positive constants, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq437_HTML.gif is a bounded measurable function, that is, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq438_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq439_HTML.gif denotes the age-specific death rate, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq440_HTML.gif denotes the age density of migrants, and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq441_HTML.gif denotes the control. The admissible control set http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq442_HTML.gif .

        A linear operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq443_HTML.gif defined on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq444_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ86_HTML.gif
        (5.2)
        where the domain of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq445_HTML.gif is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ87_HTML.gif
        (5.3)

        By the fact that the operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq446_HTML.gif is an infinitesimal generator of a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq447_HTML.gif -semigroup (see [39, Example 2.21]) and [38, Theorem 4.2.1], then http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq448_HTML.gif is an infinitesimal generator of a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq449_HTML.gif -semigroup since the operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq450_HTML.gif is bounded.

        Now let us consider the following operators family:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ88_HTML.gif
        (5.4)
        It is not difficult to verify that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq451_HTML.gif defines a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq452_HTML.gif -semigroup and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq453_HTML.gif is just the infinitesimal generator of the http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq454_HTML.gif -semigroup http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq455_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq456_HTML.gif , then there exits a constant http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq457_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq458_HTML.gif a.e. http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq459_HTML.gif . For an arbitrary function http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq460_HTML.gif , by using the expression (5.4) of the semigroup http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq461_HTML.gif , the following inequality holds:
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ89_HTML.gif
        (5.5)

        Hence, Lemma 2.3 leads to the exponential stability of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq462_HTML.gif . That is, there exist http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq463_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq464_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq465_HTML.gif

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ90_HTML.gif
        (5.6)
        Define http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq466_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq467_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq468_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq469_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq470_HTML.gif . Thus system (5.1) can be rewritten as
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ91_HTML.gif
        (5.7)
        with the cost function
        http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ92_HTML.gif
        (5.8)

        By Lemma 2.4, for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq471_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq472_HTML.gif is exponentially stable. Now, all the assumptions are met in Theorems 3.2 and 4.3, our results can be used to system (5.1). Thus, system (5.1) has a unique http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq473_HTML.gif -periodic http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq474_HTML.gif -mild solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq475_HTML.gif which is globally asymptotically stable and there exists a periodic control http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq476_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq477_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq478_HTML.gif

        The results show that the optimal population level is truly the periodic solution of the considered system, and hence, it is globally asymptotically stable. Meanwhile, it implies that we can achieve sustainability at a high level of productivity and good economic profit by virtue of scientific, effective, and continuous management of the resources.

        Declarations

        Acknowledgments

        This work is supported by Natural Science Foundation of Guizhou Province Education Department (no. 2007008). This work is also supported by the undergraduate carve out project of Department of Guiyang Science and Technology (2008, no. 15-2).

        Authors’ Affiliations

        (1)
        College of Computer Science and Technology, Guizhou University
        (2)
        College of Science, Guizhou University

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