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Boundary Value Problems

, 2008:192353 | Cite as

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

Open Access
Research Article

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.

Keywords

Banach Space Periodic Solution Cauchy Problem Exponential Stability Mild Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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, 2, 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, 6, 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, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 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, 26, 27, 28, 29, 30, 31, 32, 33, 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, 36, 37] including the existence of periodic Open image in new window -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:

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

Suppose Open image in new window is a Banach space and Open image in new window is a separable reflexive Banach space. The objective functional is given by

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

Suppose that Open image in new window , Open image in new window , Open image in new window and Open image in new window is the least positive constant such that there are Open image in new window s in the interval Open image in new window and Open image in new window where Open image in new window , Open image in new window . The first equation of system (1.1) describes the variation of the population number Open image in new window 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 Open image in new window satisfy some properties (such as strongly elliptic) in Open image in new window and set Open image in new window (such as Open image in new window . For every Open image in new window define Open image in new window , Open image in new window is the infinitesimal generator of a Open image in new window -semigroup Open image in new window on the Banach space Open image in new window (such as Open image in new window ). Define x Open image in new window Open image in new window , Open image in new window and Open image in new window then system (1.1) can be abstracted into the following controlled system:
On the Banach space Open image in new window , and the associated objective functional

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

It can be seen that endowed with the norm Open image in new window , Open image in new window is a Banach space.

In order to investigate periodic solutions, we introduce the following two spaces:

It can be seen that endowed with the norm Open image in new window , Open image in new window is a Banach space.

We introduce assumption [H1].

[H1.1]: Open image in new window is the infinitesimal generator of a Open image in new window -semigroup Open image in new window on Open image in new window with domain Open image in new window .

[H1.2]:There exists Open image in new window such that Open image in new window where Open image in new window .

[H1.3]:For each Open image in new window , Open image in new window , Open image in new window .

Under the assumption [H1], consider
and the associated Cauchy problem
For every Open image in new window , Open image in new window is an invariant subspace of Open image in new window , 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 Open image in new window represented by Open image in new window , where Open image in new window given by

The operator Open image in new window is called impulsive evolution operator associated with Open image in new window and Open image in new window .

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

Lemma 2.1.

Let assumption [H1] Open image in new window hold. The impulsive evolution operator Open image in new window has the following properties.

(1)For Open image in new window , Open image in new window , there exists a Open image in new window such that Open image in new window

(2)For Open image in new window , Open image in new window , Open image in new window .

(3)For Open image in new window , Open image in new window , Open image in new window .

(4)For Open image in new window , Open image in new window , Open image in new window .

Proof.
  1. (1)
    By assumption [H1.1], there exists a constant Open image in new window such that Open image in new window . Using assumption [H1.3], it is obvious that Open image in new window , for Open image in new window . (2) By the definition of Open image in new window -semigroup and the construction of Open image in new window , 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 Open image in new window , Open image in new window , by virtue of (3) again and again, we arrive at
     
  1. (5)
    Without loss of generality, for Open image in new window ,
     

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 Open image in new window . We first give the definition of exponential stable for Open image in new window .

Definition 2.2.

Assumption [H2]: Open image in new window is exponentially stable, that is, there exist Open image in new window and Open image in new window such that

An important criteria for exponential stability of a Open image in new window -semigroup is collected here.

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

Let Open image in new window be a Open image in new window -semigroup on Open image in new window , and let Open image in new window be its infinitesimal generator. Then the following assertions are equivalent:

(1) Open image in new window is exponentially stable.

(2)For every Open image in new window there exits a positive constants Open image in new window such that

Next, four sufficient conditions that guarantee the exponential stability of impulsive evolution operator Open image in new window are given.

Lemma 2.4.

Assumptions [H1] Open image in new window and [H2] Open image in new window hold. There exists Open image in new window such that

Then, Open image in new window is exponentially stable.

Proof.

Without loss of generality, for Open image in new window , we have

Let Open image in new window and Open image in new window , then we obtain Open image in new window

Lemma 2.5.

Assume that assumption [H1] Open image in new window holds. Suppose
If there exists Open image in new window such that

Then, Open image in new window is exponentially stable.

Proof.

It comes from (2.17) that
Further,

where Open image in new window is denoted the number of impulsive points in Open image in new window .

For Open image in new window , by (2.16), we obtain the following two inequalities:
This implies
that is,
Thus, we obtain

By (5) of Lemma 2.1, let Open image in new window , Open image in new window , Open image in new window

Lemma 2.6.

Assume that assumption [H1] Open image in new window holds. The limit
Suppose there exists Open image in new window such that

Then, Open image in new window is exponentially stable.

Proof.

that there exits a Open image in new window enough small such that
that is,
From (2.27), we know that
Then, we have

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

Lemma 2.7.

Imply the exponential stability of Open image in new window .

Proof.

It comes from the continuity of Open image in new window , the inequality

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

and hence,

where Open image in new window and Open image in new window . Since Open image in new window , this shows that our result.

3. Periodic Solutions and Global Asymptotical Stability

Consider the following controlled system:
and the associated Cauchy problem

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

[H3]: Open image in new window is measurable and Open image in new window for Open image in new window .

[H4]:For each Open image in new window , there exists Open image in new window and Open image in new window , Open image in new window .

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

[H6]:Operator Open image in new window and Open image in new window , for Open image in new window . Obviously, Open image in new window .

Denote the set of admissible controls

Obviously, Open image in new window and Open image in new window , Open image in new window is bounded, convex, and closed.

We introduce Open image in new window -mild solution of Cauchy problem (3.2) and Open image in new window -periodic Open image in new window -mild solution of system (3.1).

Definition 3.1.

A function Open image in new window , for finite interval Open image in new window , is said to be a Open image in new window -mild solution of the Cauchy problem (3.2) corresponding to the initial value Open image in new window and Open image in new window if Open image in new window is given by

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

Theorem 3.2 .A.

Assumptions [H1], [H3], [H4], [H5], and [H6] Open image in new window hold. Suppose Open image in new window is exponentially stable, for every Open image in new window , system (3.1) has a unique Open image in new window -periodic Open image in new window -mild solution:
is a bounded linear operator and

where Open image in new window and Open image in new window .

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

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

Proof.

Consider the operator Open image in new window . By (4) of Lemma 2.1 and the stability of Open image in new window , we have
Thus, Open image in new window . Obviously, the series Open image in new window is convergent, thus operator Open image in new window . It comes from Open image in new window that Open image in new window It is well known that system (3.1) has a periodic Open image in new window -mild solution if and only if Open image in new window . Since Open image in new window is invertible, we can uniquely solve
Note that
it is not difficult to verify that the Open image in new window -mild solution of the Cauchy problem (3.2) corresponding to initial value Open image in new window given by

is just the unique Open image in new window -periodic of system (3.1).

It is obvious that Open image in new window is linear. Next, verify the estimation (3.8). In fact, for Open image in new window ,
On the other hand,

Let Open image in new window , next the estimation (3.8) is verified.

System (3.1) has a unique Open image in new window -periodic Open image in new window -mild solution Open image in new window given by (3.5) and (3.6). The Open image in new window -mild solution Open image in new window of the Cauchy problem (3.2) corresponding to initial value Open image in new window and control Open image in new window can be given by (3.4). Then,

Let Open image in new window , one can obtain (3.9) immediately.

Definition 3.3.

The Open image in new window -periodic Open image in new window -mild solution Open image in new window of the system (3.1) is said to be globally asymptotically stable in the sense that

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

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

Proof.

where Open image in new window . This completes the proof.

Lemma 4.A.

Suppose Open image in new window is a strong continuous operator. The operator Open image in new window , given by

is strongly continuous.

Proof.

Without loss of generality, for Open image in new window ,

By virtue of strong continuity of Open image in new window , boundedness of Open image in new window , Open image in new window , Open image in new window is strongly continuous.

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

We introduce the following assumption on Open image in new window and Open image in new window .

Assumption [H7].

[H7.1] Open image in new window The functional Open image in new window is Borel measurable.

[H7.2] Open image in new window Open image in new window is sequentially lower semicontinuous on Open image in new window for almost all Open image in new window .

[H7.3] Open image in new window Open image in new window is convex on Open image in new window for each Open image in new window and almost all Open image in new window .

[H7.5] Open image in new window The functional Open image in new window 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] Open image in new window holds. Under the assumptions of Theorem 3.2, the problem (P) has a unique solution.

Proof.

If Open image in new window there is nothing to prove.

We assume that Open image in new window By assumption [H7], we have

where Open image in new window is a constant. Hence Open image in new window .

By the definition of infimum there exists a sequence Open image in new window , such that Open image in new window

Since Open image in new window is bounded in Open image in new window , there exists a subsequence, relabeled as Open image in new window , and Open image in new window such that Open image in new window weakly convergence in Open image in new window and Open image in new window Because of Open image in new window is the closed and convex set, thanks to the Mazur lemma, Open image in new window . Suppose Open image in new window and Open image in new window are the Open image in new window -periodic Open image in new window -mild solution of system (3.1) corresponding to Open image in new window ( Open image in new window ) and Open image in new window , respectively, then Open image in new window and Open image in new window can be given by
then by Lemma 4.2, we have

as Open image in new window weakly convergence in Open image in new window .

Next, we show that
By elementary computation, we arrive at
Consider the time interval Open image in new window , similarly we obtain
In general, given any Open image in new window , Open image in new window and the Open image in new window , Open image in new window , prior to the jump at time Open image in new window , we immediately follow the jump as
the associated interval Open image in new window , we also similarly obtain
Step by step, we repeat the procedures till the time interval is exhausted. Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , thus we obtain
that is,

with strongly convergence as Open image in new window .

Since Open image in new window , using the assumption [H7] Open image in new window and Balder's theorem, we can obtain

This shows that Open image in new window attains its minimum at Open image in new window . This completes the proof.

5. Example

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

Let Open image in new window and consider the following population evolution equation with impulses:

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

where the domain of Open image in new window is given by

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

Now let us consider the following operators family:
It is not difficult to verify that Open image in new window defines a Open image in new window -semigroup and Open image in new window is just the infinitesimal generator of the Open image in new window -semigroup Open image in new window . Since Open image in new window , then there exits a constant Open image in new window such that Open image in new window a.e. Open image in new window . For an arbitrary function Open image in new window , by using the expression (5.4) of the semigroup Open image in new window , the following inequality holds:

Hence, Lemma 2.3 leads to the exponential stability of Open image in new window . That is, there exist Open image in new window and Open image in new window such that Open image in new window

with the cost function

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

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.

Notes

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).

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© JinRongWang et al. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.College of Computer Science and TechnologyGuizhou UniversityGuiyangChina
  2. 2.College of ScienceGuizhou UniversityGuiyangChina

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