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

- JinRong Wang
^{1}Email author, - X Xiang
^{2}and - W Wei
^{2}

**2008**:192353

**DOI: **10.1155/2008/192353

© JinRongWang et al. 2008

**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 [1–3]. 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 [5–7]. 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 [8–22]. Particulary, Doctor Ahmed investigated optimal control problems [23, 24] for impulsive systems on infinite-dimensional spaces. We also gave a series of results [25–34] 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 [35–37] including the existence of periodic -mild solutions and alternative theorem, criteria of Massera type, asymptotical stability and robustness against perturbation for a linear impulsive periodic system are established.

On infinite-dimensional spaces, where denotes the population number of isolated species at time and location , is a bounded domain and , operator . The coefficients , are sufficiently smooth functions of in , where , and , . Denoting , , then = . is related to the periodic change of the resources maintaining the evolution of the population and the periodic control policy , where is a suitable admissible control set. Time sequence and as , denote mutation of the isolate species at time where .

where is Borel measurable, is continuous, and nonnegative and denotes the -periodic -mild solution of system (1.1) at location and corresponding to the control . The Bolza problem ( ) is to find such that for all

Suppose that , , and is the least positive constant such that there are s in the interval and where , . The first equation of system (1.1) describes the variation of the population number 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.

where denotes the -periodic -mild solution of system (1.3) corresponding to the control . The Bolza problem (P) is to find such that for all 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 -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 are collected. Four new sufficient conditions that guarantee the exponential stability of the are given. In Section 3, the existence, uniqueness, and global asymptotical stability of -periodic -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 be a Banach space, denotes the space of linear operators on ; denotes the space of bounded linear operators on . is the Banach space with the usual supremum norm. Denote and define is continuous at , is continuous from left and has right-hand limits at and .

It can be seen that endowed with the norm , is a Banach space.

It can be seen that endowed with the norm , is a Banach space.

We introduce assumption [H1].

[H1.1]: is the infinitesimal generator of a -semigroup on with domain .

[H1.2]:There exists such that where .

[H1.3]:For each , , .

The operator is called impulsive evolution operator associated with and .

The following lemma on the properties of the impulsive evolution operator associated with and is widely used in this paper.

Lemma 2.1.

*Let assumption [H1]*
hold. The impulsive evolution operator
has the following properties.

(1)For , , there exists a such that

(2)For , , .

(3)For , , .

(4)For , , .

- (1)By assumption [H1.1], there exists a constant such that . Using assumption [H1.3], it is obvious that , for . (2) By the definition of -semigroup and the construction of , 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 , , by virtue of (3) again and again, we arrive at(2.8)

- (5)Without loss of generality, for ,(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 . We first give the definition of exponential stable for .

Definition 2.2.

An important criteria for exponential stability of a -semigroup is collected here.

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

*Let*
be a
-semigroup on
, and let
be its infinitesimal generator. Then the following assertions are equivalent:

(1) is exponentially stable.

Next, four sufficient conditions that guarantee the exponential stability of impulsive evolution operator are given.

Lemma 2.4.

Then, is exponentially stable.

Proof.

Let and , then we obtain

Lemma 2.5.

*Assume that assumption [H1]*holds. Suppose

*Then*,
is exponentially stable.

Proof.

where is denoted the number of impulsive points in .

By (5) of Lemma 2.1, let , ,

Lemma 2.6.

*Assume that assumption [H1]*holds. The limit

*Suppose there exists*such that

*Then*,
is exponentially stable.

Proof.

Here, we only need to choose small enough such that , by (5) of Lemma 2.1 again, let , , we have

Lemma 2.7.

*Assume that assumption [H1]*holds. For some , ,

*Imply the exponential stability of*
.

Proof.

and the boundedness of , are convergent, that for every and fixed . This shows that is bounded for each and fixed and hence, by virtue of uniform boundedness principle, there exists a constant such that for all . Let denote the operator given by , and is fixed. Clearly, is defined every where on and by assumption it maps and it is a closed operator. Hence, by closed graph theorem, it is a bounded linear operator from to . Thus, there exits a constant such that for all and , is fixed.

where and . Since , this shows that our result.

## 3. Periodic Solutions and Global Asymptotical Stability

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

[H3]: is measurable and for .

[H4]:For each , there exists and , .

[H5]: has bounded, closed, and convex values and is graph measurable, and are bounded, where is a separable reflexive Banach space.

[H6]:Operator and , for . Obviously, .

Obviously, and , is bounded, convex, and closed.

We introduce -mild solution of Cauchy problem (3.2) and -periodic -mild solution of system (3.1).

Definition 3.1.

A function is said to be a -periodic -mild solution of system (3.1) if it is a -mild solution of Cauchy problem (3.2) corresponding to some and for .

Theorem 3.2 .A.

*Assumptions [H1], [H3], [H4], [H5], and [H6]*hold. Suppose is exponentially stable, for every , system (3.1) has a unique -periodic -mild solution:

where and .

*Further, for arbitrary*, the -mild solution of the Cauchy problem (3.2) corresponding to the initial value and control , satisfies the following inequality:

where is the -periodic -mild solution of system (3.1), is not dependent on , , , and . That is, can be approximated to the -periodic -mild solution according to exponential decreasing speed.

Proof.

is just the unique -periodic of system (3.1).

Let , next the estimation (3.8) is verified.

Let , one can obtain (3.9) immediately.

Definition 3.3.

where is any -mild solutions of the Cauchy problem (3.2) corresponding to initial value and control .

By Theorem 3.2 and the stability of the impulsive evolution operator 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*
-periodic
-mild solution
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 -periodic -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*-periodic -mild solution of system (3.1) continuously depends on the control on , that is, let be -periodic -mild solution of system (3.1) corresponding to . There exists constant such that

Proof.

where . This completes the proof.

Lemma 4.A.

*Suppose*is a strong continuous operator. The operator , given by

is strongly continuous.

Proof.

By virtue of strong continuity of , boundedness of , , is strongly continuous.

Let denote the -periodic -mild solution of system (3.1) corresponding to the control , we consider the Bolza problem (P).

We introduce the following assumption on and .

Assumption [H7].

[H7.1] The functional is Borel measurable.

[H7.2] is sequentially lower semicontinuous on for almost all .

[H7.3] is convex on for each and almost all .

[H7.5] The functional 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]*
holds. Under the assumptions of Theorem 3.2, the problem (P) has a unique solution.

Proof.

If there is nothing to prove.

where is a constant. Hence .

By the definition of infimum there exists a sequence , such that

as weakly convergence in .

with strongly convergence as .

This shows that attains its minimum at . This completes the proof.

## 5. Example

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

where denotes time, denotes age, is called age density function, and are positive constants, is a bounded measurable function, that is, . denotes the age-specific death rate, denotes the age density of migrants, and denotes the control. The admissible control set .

By the fact that the operator is an infinitesimal generator of a -semigroup (see [39, Example 2.21]) and [38, Theorem 4.2.1], then is an infinitesimal generator of a -semigroup since the operator is bounded.

Hence, Lemma 2.3 leads to the exponential stability of . That is, there exist and such that

By Lemma 2.4, for , 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 -periodic -mild solution which is globally asymptotically stable and there exists a periodic control such that for all

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

## References

- Clark CW:
*Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Pure and Applied Mathematics*. John Wiley & Sons, New York, NY, USA; 1976:xi+352. - Song X, Chen L:
**Optimal harvesting and stability for a two-species competitive system with stage structure.***Mathematical Biosciences*2001,**170**(2):173-186. 10.1016/S0025-5564(00)00068-7MathSciNetView ArticleMATH - Marzollo R (Ed):
*Periodic Optimization*. Springer, New York, NY, USA; 1972.MATH - Fan M, Wang K:
**Optimal harvesting policy for single population with periodic coefficients.***Mathematical Biosciences*1998,**152**(2):165-177. 10.1016/S0025-5564(98)10024-XMathSciNetView ArticleMATH - Bainov DD, Simeonov PS:
*Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics*.*Volume 66*. Longman Scientific & Technical, Harlow, UK; 1993:x+228.MATH - Xiao YN, Cheng DZ, Qin HS:
**Optimal impulsive control in periodic ecosystem.***Systems & Control Letters*2006,**55**(7):558-565. 10.1016/j.sysconle.2005.12.003MathSciNetView ArticleMATH - Lakshmikantham V, Bainov DD, Simeonov PS:
*Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics*.*Volume 6*. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View Article - Nieto JJ:
**An abstract monotone iterative technique.***Nonlinear Analysis: Theory, Methods & Applications*1997,**28**(12):1923-1933. 10.1016/S0362-546X(97)89710-6MathSciNetView ArticleMATH - Yan J, Zhao A, Nieto JJ:
**Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems.***Mathematical and Computer Modelling*2004,**40**(5-6):509-518. 10.1016/j.mcm.2003.12.011MathSciNetView ArticleMATH - Jiao J-J, Chen L-S, Nieto JJ, Angela T:
**Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey.***Applied Mathematics and Mechanics*2008,**29**(5):653-663. 10.1007/s10483-008-0509-xMathSciNetView ArticleMATH - Zeng G, Wang F, Nieto JJ:
**Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response.***Advances in Complex Systems*2008,**11**(1):77-97. 10.1142/S0219525908001519MathSciNetView ArticleMATH - Wang W, Shen J, Nieto JJ:
**Permanence and periodic solution of predator-prey system with Holling type functional response and impulses.***Discrete Dynamics in Nature and Society*2007,**2007:**-15. - Zhang H, Chen L, Nieto JJ:
**A delayed epidemic model with stage-structure and pulses for pest management strategy.***Nonlinear Analysis: Real World Applications*2008,**9**(4):1714-1726. 10.1016/j.nonrwa.2007.05.004MathSciNetView ArticleMATH - Ahmad B, Nieto JJ:
**Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(10):3291-3298. 10.1016/j.na.2007.09.018MathSciNetView ArticleMATH - Jiang D, Yang Y, Chu J, O'Regan D:
**The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(10):2815-2828. 10.1016/j.na.2006.09.042MathSciNetView ArticleMATH - Li Y:
**Global exponential stability of BAM neural networks with delays and impulses.***Chaos, Solitons & Fractals*2005,**24**(1):279-285.MathSciNetView ArticleMATH - De la Sen M:
**Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces.***Journal of Mathematical Analysis and Applications*2006,**321**(2):621-650. 10.1016/j.jmaa.2005.08.038MathSciNetView ArticleMATH - Yu L, Zhang J, Liao Y, Ding J:
**Parameter estimation error bounds for Hammerstein nonlinear finite impulsive response models.***Applied Mathematics and Computation*2008,**202**(2):472-480. 10.1016/j.amc.2008.01.002MathSciNetView ArticleMATH - Jankowski T:
**Positive solutions to second order four-point boundary value problems for impulsive differential equations.***Applied Mathematics and Computation*2008,**202**(2):550-561. 10.1016/j.amc.2008.02.040MathSciNetView ArticleMATH - Hernández EM, Pierri M, Goncalves G:
**Existence results for an impulsive abstract partial differential equation with state-dependent delay.***Computers & Mathematics with Applications*2006,**52**(3-4):411-420. 10.1016/j.camwa.2006.03.022MathSciNetView ArticleMATH - Hernández EM, Rabello M, Henríquez HR:
**Existence of solutions for impulsive partial neutral functional differential equations.***Journal of Mathematical Analysis and Applications*2007,**331**(2):1135-1158. 10.1016/j.jmaa.2006.09.043MathSciNetView ArticleMATH - Hernández EM, Sakthivel R, Aki ST:
**Existence results for impulsive evolution differential equations with state-dependent delay.***Electronic Journal of Differential Equations*2008,**2008**(28):1-11.MathSciNetMATH - Ahmed NU:
**Some remarks on the dynamics of impulsive systems in Banach spaces.***Dynamics of Continuous, Discrete and Impulsive Systems. Series A*2001,**8**(2):261-274.MathSciNetMATH - Ahmed NU, Teo KL, Hou SH:
**Nonlinear impulsive systems on infinite dimensional spaces.***Nonlinear Analysis: Theory, Methods & Applications*2003,**54**(5):907-925. 10.1016/S0362-546X(03)00117-2MathSciNetView ArticleMATH - Xiang X, Ahmed NU:
**Existence of periodic solutions of semilinear evolution equations with time lags.***Nonlinear Analysis: Theory, Methods & Applications*1992,**18**(11):1063-1070. 10.1016/0362-546X(92)90195-KMathSciNetView ArticleMATH - Xiang X:
**Optimal control for a class of strongly nonlinear evolution equations with constraints.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(1):57-66. 10.1016/S0362-546X(01)00156-0MathSciNetView ArticleMATH - Sattayatham P, Tangmanee S, Wei W:
**On periodic solutions of nonlinear evolution equations in Banach spaces.***Journal of Mathematical Analysis and Applications*2002,**276**(1):98-108. 10.1016/S0022-247X(02)00378-5MathSciNetView ArticleMATH - Xiang X, Wei W, Jiang Y:
**Strongly nonlinear impulsive system and necessary conditions of optimality.***Dynamics of Continuous, Discrete & Impulsive Systems. Series A*2005,**12**(6):811-824.MathSciNetMATH - Wei W, Xiang X:
**Necessary conditions of optimal control for a class of strongly nonlinear impulsive equations in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2005,**63**(5–7):53-63.View Article - Wei W, Xiang X, Peng Y:
**Nonlinear impulsive integro-differential equations of mixed type and optimal controls.***Optimization*2006,**55**(1-2):141-156. 10.1080/02331930500530401MathSciNetView ArticleMATH - Xiang X, Wei W:
**Mild solution for a class of nonlinear impulsive evolution inclusions on Banach space.***Southeast Asian Bulletin of Mathematics*2006,**30**(2):367-376.MathSciNetMATH - Yu X, Xiang X, Wei W:
**Solution bundle for a class of impulsive differential inclusions on Banach spaces.***Journal of Mathematical Analysis and Applications*2007,**327**(1):220-232. 10.1016/j.jmaa.2006.03.075MathSciNetView ArticleMATH - Peng Y, Xiang X, Wei W:
**Nonlinear impulsive integro-differential equations of mixed type with time-varying generating operators and optimal controls.***Dynamic Systems and Applications*2007,**16**(3):481-496.MathSciNetMATH - Peng Y, Xiang X:
**Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls.***Journal of Industrial and Management Optimization*2008,**4**(1):17-32.MathSciNetView ArticleMATH - Wang J:
**Linear impulsive periodic system on Banach spaces.***Proceedings of the 4th International Conference on Impulsive and Hybrid Dynamical Systems, July 2007, Nanning, China***5:**20-25. - Wang J, Xiang X, Wei W:
**Linear impulsive periodic system with time-varying generating operators on Banach space.***Advances in Difference Equations*2007,**2007:**-16. - Wang J, Xiang X, Wei W, Chen Q:Existence and global asymptotical stability of periodic solution for the
-periodic logistic system with time-varying generating operators and
-periodic impulsive perturbations on Banach spaces
*Discrete Dynamics in Nature and Society*2008,**2008:**-16. - Ahmed NU:
*Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series*.*Volume 246*. Longman Scientific & Technical, Harlow, UK; 1991:x+282. - Luo Z-H, Guo B-Z, Morgül O:
*Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series*. Springer, London, UK; 1999:xiv+403.View ArticleMATH

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