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

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.

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:

(1.1)

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 .

Suppose is a Banach space and is a separable reflexive Banach space. The objective functional is given by

(1.2)

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.

Let satisfy some properties (such as strongly elliptic) in and set (such as . For every define , is the infinitesimal generator of a -semigroup on the Banach space (such as ). Define x, and then system (1.1) can be abstracted into the following controlled system:

(1.3)

On the Banach space , and the associated objective functional

(1.4)

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 .

Set

(2.1)

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

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

(2.2)

Set

(2.3)

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 , , .

Under the assumption [H1], consider

(2.4)

and the associated Cauchy problem

(2.5)

For every , is an invariant subspace of , 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 represented by , where given by

(2.6)

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 , , .

(5)For , there exits , such that

(2.7)

Proof.

• (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.

, is called exponentially stable if there exist and such that

(2.10)

Assumption [H2]: is exponentially stable, that is, there exist and such that

(2.11)

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.

(2)For every there exits a positive constants such that

(2.12)

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

Lemma 2.4.

Assumptions [H1] and [H2]hold. There exists such that

(2.13)

Then, is exponentially stable.

Proof.

Without loss of generality, for , we have

(2.14)

Suppose and let Then,

(2.15)

Let and , then we obtain

Lemma 2.5.

Assume that assumption [H1]holds. Suppose

(2.16)

If there exists such that

(2.17)

for where

(2.18)

Then, is exponentially stable.

Proof.

It comes from (2.17) that

(2.19)

Further,

(2.20)

where is denoted the number of impulsive points in .

For , by (2.16), we obtain the following two inequalities:

(2.21)

This implies

(2.22)

that is,

(2.23)

Then,

(2.24)

Thus, we obtain

(2.25)

By (5) of Lemma 2.1, let , ,

Lemma 2.6.

Assume that assumption [H1]holds. The limit

(2.26)

Suppose there exists such that

(2.27)

Then, is exponentially stable.

Proof.

Let with . It comes from

(2.28)

that there exits a enough small such that

(2.29)

that is,

(2.30)

From (2.27), we know that

(2.31)

Then, we have

(2.32)

Hence,

(2.33)

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 , ,

(2.34)

Imply the exponential stability of.

Proof.

It comes from the continuity of , the inequality

(2.35)

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.

Let , and and define as

(2.36)

Then,

(2.37)

and hence,

(2.38)

Thus, for ,

(2.39)

where . Fix . Then, for any we can write for some and and we have

(2.40)

where and . Since , this shows that our result.

3. Periodic Solutions and Global Asymptotical Stability

Consider the following controlled system:

(3.1)

and the associated Cauchy problem

(3.2)

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, .

Denote the set of admissible controls

(3.3)

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 , for finite interval , is said to be a -mild solution of the Cauchy problem (3.2) corresponding to the initial value and if is given by

(3.4)

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:

(3.5)

where ,

(3.6)

Further,

(3.7)

is a bounded linear operator and

(3.8)

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:

(3.9)

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.

Consider the operator . By (4) of Lemma 2.1 and the stability of , we have

(3.10)

Thus, . Obviously, the series is convergent, thus operator . It comes from that It is well known that system (3.1) has a periodic -mild solution if and only if . Since is invertible, we can uniquely solve

(3.11)

Let , where

(3.12)

Note that

(3.13)

it is not difficult to verify that the -mild solution of the Cauchy problem (3.2) corresponding to initial value given by

(3.14)

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

It is obvious that is linear. Next, verify the estimation (3.8). In fact, for ,

(3.15)

On the other hand,

(3.16)

Let , next the estimation (3.8) is verified.

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

(3.17)

Let , one can obtain (3.9) immediately.

Definition 3.3.

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

(3.18)

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

(4.1)

Proof.

Since and are the -periodic -mild solution corresponding to and , respectively, then we have

(4.2)

where

(4.3)

Further,

(4.4)

where . This completes the proof.

Lemma 4.A.

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

(4.5)

is strongly continuous.

Proof.

Without loss of generality, for ,

(4.6)

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).

Find such that for all , where

(4.7)

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 .

[H7.4]There exist constants , , is nonnegative and such that

(4.8)

[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.

We assume that By assumption [H7], we have

(4.9)

where is a constant. Hence .

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

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

(4.10)

where

(4.11)

Define

(4.12)

then by Lemma 4.2, we have

(4.13)

as weakly convergence in .

Next, we show that

(4.14)

In fact, for , we have

(4.15)

By elementary computation, we arrive at

(4.16)

Consider the time interval , similarly we obtain

(4.17)

In general, given any , and the , , prior to the jump at time , we immediately follow the jump as

(4.18)

the associated interval , we also similarly obtain

(4.19)

Step by step, we repeat the procedures till the time interval is exhausted. Let , , , , , , thus we obtain

(4.20)

that is,

(4.21)

with strongly convergence as .

Since , using the assumption [H7]and Balder's theorem, we can obtain

(4.22)

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

5. Example

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

Let and consider the following population evolution equation with impulses:

(5.1)

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 .

A linear operator defined on by

(5.2)

where the domain of is given by

(5.3)

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.

Now let us consider the following operators family:

(5.4)

It is not difficult to verify that defines a -semigroup and is just the infinitesimal generator of the -semigroup . Since , then there exits a constant such that a.e. . For an arbitrary function , by using the expression (5.4) of the semigroup , the following inequality holds:

(5.5)

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

Let

(5.6)

Define , , , , . Thus system (5.1) can be rewritten as

(5.7)

with the cost function

(5.8)

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.