Open Access

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

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 https://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:
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq3_HTML.gif and location https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq5_HTML.gif is a bounded domain and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq6_HTML.gif , operator https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq7_HTML.gif . The coefficients https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq9_HTML.gif are sufficiently smooth functions of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq10_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq11_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq13_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq15_HTML.gif . Denoting https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq17_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq18_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq19_HTML.gif . https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq21_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq24_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq25_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq27_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq28_HTML.gif .

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq29_HTML.gif is a Banach space and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ2_HTML.gif
(1.2)

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

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq43_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq44_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq46_HTML.gif s in the interval https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq48_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq49_HTML.gif , https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq53_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq54_HTML.gif (such as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq55_HTML.gif . For every https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq56_HTML.gif define https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq58_HTML.gif is the infinitesimal generator of a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq59_HTML.gif -semigroup https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq60_HTML.gif on the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq61_HTML.gif (such as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq62_HTML.gif ). Define x https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq63_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq64_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq65_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ3_HTML.gif
(1.3)
On the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq67_HTML.gif , and the associated objective functional
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq68_HTML.gif denotes the https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq69_HTML.gif -periodic https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq72_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq73_HTML.gif for all https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq78_HTML.gif -periodic https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq80_HTML.gif be a Banach space, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq82_HTML.gif ; https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq84_HTML.gif . https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq86_HTML.gif and define https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq87_HTML.gif is continuous at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq88_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq91_HTML.gif .

Set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq92_HTML.gif , https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ6_HTML.gif
(2.2)
Set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq94_HTML.gif , https://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]: https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq96_HTML.gif is the infinitesimal generator of a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq97_HTML.gif -semigroup https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq98_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq99_HTML.gif with domain https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq100_HTML.gif .

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

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

Under the assumption [H1], consider
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ9_HTML.gif
(2.5)
For every https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq107_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq108_HTML.gif is an invariant subspace of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq110_HTML.gif represented by https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq111_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq112_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ10_HTML.gif
(2.6)

The operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq114_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq116_HTML.gif associated with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq117_HTML.gif and https://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] https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq119_HTML.gif hold. The impulsive evolution operator https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq120_HTML.gif has the following properties.

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

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

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

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

(5)For https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq134_HTML.gif , there exits https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq135_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq136_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq137_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq139_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq140_HTML.gif . (2) By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq141_HTML.gif -semigroup and the construction of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq143_HTML.gif , https://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
    https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq145_HTML.gif ,
    https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq147_HTML.gif .

Definition 2.2.

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq148_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq151_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ14_HTML.gif
(2.10)
Assumption [H2]: https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq154_HTML.gif such that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq156_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq157_HTML.gif -semigroup on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq158_HTML.gif , and let https://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) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq160_HTML.gif is exponentially stable.

(2)For every https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq161_HTML.gif there exits a positive constants https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq162_HTML.gif such that
https://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 https://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] https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq164_HTML.gif and [H2] https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq165_HTML.gif hold. There exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq166_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ17_HTML.gif
(2.13)

Then, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq168_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ18_HTML.gif
(2.14)
Suppose https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq169_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq170_HTML.gif Then,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ19_HTML.gif
(2.15)

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq171_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq172_HTML.gif , then we obtain https://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] https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq174_HTML.gif holds. Suppose
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ20_HTML.gif
(2.16)
If there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq175_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ21_HTML.gif
(2.17)
for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq176_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ22_HTML.gif
(2.18)

Then, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ23_HTML.gif
(2.19)
Further,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ24_HTML.gif
(2.20)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq179_HTML.gif .

For https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ25_HTML.gif
(2.21)
This implies
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ26_HTML.gif
(2.22)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ27_HTML.gif
(2.23)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ28_HTML.gif
(2.24)
Thus, we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq182_HTML.gif , https://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] https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq184_HTML.gif holds. The limit
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ30_HTML.gif
(2.26)
Suppose there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq185_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ31_HTML.gif
(2.27)

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

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq187_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq188_HTML.gif . It comes from
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ32_HTML.gif
(2.28)
that there exits a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq189_HTML.gif enough small such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ33_HTML.gif
(2.29)
that is,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ35_HTML.gif
(2.31)
Then, we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ36_HTML.gif
(2.32)
Hence,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq190_HTML.gif small enough such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq192_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq193_HTML.gif , we have https://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] https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq195_HTML.gif holds. For some https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq196_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq197_HTML.gif ,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq199_HTML.gif , the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ39_HTML.gif
(2.35)

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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq236_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq237_HTML.gif . Since https://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:
https://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
https://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]: https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq239_HTML.gif is measurable and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq240_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq241_HTML.gif .

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

[H5]: https://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, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq247_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq248_HTML.gif are bounded, where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq250_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq251_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq252_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq253_HTML.gif .

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

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq254_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq255_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq258_HTML.gif -periodic https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq260_HTML.gif , for finite interval https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq261_HTML.gif , is said to be a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq263_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq264_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq265_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ48_HTML.gif
(3.4)

A function https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq266_HTML.gif is said to be a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq267_HTML.gif -periodic https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq270_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq271_HTML.gif for https://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] https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq273_HTML.gif hold. Suppose https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq274_HTML.gif is exponentially stable, for every https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq275_HTML.gif , system (3.1) has a unique https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq276_HTML.gif -periodic https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq277_HTML.gif -mild solution:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ49_HTML.gif
(3.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq278_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ50_HTML.gif
(3.6)
Further,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ52_HTML.gif
(3.8)

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

Further, for arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq281_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq282_HTML.gif -mild solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq284_HTML.gif and control https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq285_HTML.gif , satisfies the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ53_HTML.gif
(3.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq286_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq287_HTML.gif -periodic https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq288_HTML.gif -mild solution of system (3.1), https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq289_HTML.gif is not dependent on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq290_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq291_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq292_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq293_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq294_HTML.gif can be approximated to the https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq295_HTML.gif -periodic https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq296_HTML.gif -mild solution https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq299_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ54_HTML.gif
(3.10)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq300_HTML.gif . Obviously, the series https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq301_HTML.gif is convergent, thus operator https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq302_HTML.gif . It comes from https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq303_HTML.gif that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq305_HTML.gif -mild solution if and only if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq306_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq307_HTML.gif is invertible, we can uniquely solve
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ55_HTML.gif
(3.11)
Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq308_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ56_HTML.gif
(3.12)
Note that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq310_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ58_HTML.gif
(3.14)

is just the unique https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq313_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ59_HTML.gif
(3.15)
On the other hand,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ60_HTML.gif
(3.16)

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq315_HTML.gif -periodic https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq316_HTML.gif -mild solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq318_HTML.gif -mild solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq320_HTML.gif and control https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ61_HTML.gif
(3.17)

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq323_HTML.gif -periodic https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq324_HTML.gif -mild solution https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ62_HTML.gif
(3.18)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq326_HTML.gif is any https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq328_HTML.gif and control https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq331_HTML.gif -periodic https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq332_HTML.gif -mild solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq334_HTML.gif -periodic https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq336_HTML.gif -periodic https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq338_HTML.gif , that is, let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq339_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq340_HTML.gif -periodic https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq342_HTML.gif . There exists constant https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq343_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ63_HTML.gif
(4.1)

Proof.

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

where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq352_HTML.gif , given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq353_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq354_HTML.gif , boundedness of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq355_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq356_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq357_HTML.gif is strongly continuous.

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq358_HTML.gif denote the https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq359_HTML.gif -periodic https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq362_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq363_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq364_HTML.gif , where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq365_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq366_HTML.gif .

Assumption [H7].

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

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

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

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

[H7.5] https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq383_HTML.gif The functional https://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] https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq387_HTML.gif By assumption [H7], we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ71_HTML.gif
(4.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq388_HTML.gif is a constant. Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq390_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq391_HTML.gif

Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq392_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq393_HTML.gif , there exists a subsequence, relabeled as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq394_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq395_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq396_HTML.gif weakly convergence in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq397_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq398_HTML.gif Because of https://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, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq400_HTML.gif . Suppose https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq401_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq402_HTML.gif are the https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq403_HTML.gif -periodic https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq405_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq406_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq407_HTML.gif , respectively, then https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq408_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq409_HTML.gif can be given by
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ72_HTML.gif
(4.10)
where
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ73_HTML.gif
(4.11)
Define
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ75_HTML.gif
(4.13)

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

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

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

Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq427_HTML.gif , using the assumption [H7] https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ84_HTML.gif
(4.22)

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq429_HTML.gif attains its minimum at https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ85_HTML.gif
(5.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq432_HTML.gif denotes time, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq433_HTML.gif denotes age, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq434_HTML.gif is called age density function, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq435_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq436_HTML.gif are positive constants, https://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, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq438_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq439_HTML.gif denotes the age-specific death rate, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq442_HTML.gif .

A linear operator https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq443_HTML.gif defined on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq444_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ86_HTML.gif
(5.2)
where the domain of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq445_HTML.gif is given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq446_HTML.gif is an infinitesimal generator of a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq448_HTML.gif is an infinitesimal generator of a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq449_HTML.gif -semigroup since the operator https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq451_HTML.gif defines a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq452_HTML.gif -semigroup and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq454_HTML.gif -semigroup https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq455_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq456_HTML.gif , then there exits a constant https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq457_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq458_HTML.gif a.e. https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq459_HTML.gif . For an arbitrary function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq461_HTML.gif , the following inequality holds:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq462_HTML.gif . That is, there exist https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq463_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq464_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq465_HTML.gif

Let
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ90_HTML.gif
(5.6)
Define https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq466_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq467_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq468_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq469_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_Equ91_HTML.gif
(5.7)
with the cost function
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq471_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq473_HTML.gif -periodic https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq474_HTML.gif -mild solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq476_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F192353/MediaObjects/13661_2008_Article_792_IEq477_HTML.gif for all https://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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