Global behaviors and optimal harvesting of a class of impulsive periodic logistic singlespecies 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.
Singlespecies resource management model, which is described by the impulsive periodic logistic equations on finitedimensional spaces, has been investigated extensively, no matter how the harvesting occurs, continuously [1, 4] or impulsively [5–7]. However, the associated singlespecies resource management model on infinitedimensional 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 infinitedimensional spaces. We also gave a series of results [25–34] for the firstorder (secondorder) semilinear impulsive systems, integraldifferential impulsive system, strongly nonlinear impulsive systems and their optimal control problems. Recently, we have investigated linear impulsive periodic system on infinitedimensional 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 singlespecies system with continuous periodic control strategy and periodic impulsive perturbations:
On infinitedimensional 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
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:
On the Banach space , and the associated objective functional
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 righthand limits at and .
Set
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:
Set
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
and the associated Cauchy problem
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
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
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
(5) Without loss of generality, for ,
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
Assumption [H2]: is exponentially stable, that is, there exist and such that
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
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
Then, is exponentially stable.
Proof.
Without loss of generality, for , we have
Suppose and let Then,
Let and , then we obtain
Lemma 2.5.
Assume that assumption [H1]holds. Suppose
If there exists such that
for where
Then, is exponentially stable.
Proof.
It comes from (2.17) that
Further,
where is denoted the number of impulsive points in .
For , by (2.16), we obtain the following two inequalities:
This implies
that is,
Then,
Thus, we obtain
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.
Let with . It comes from
that there exits a enough small such that
that is,
From (2.27), we know that
Then, we have
Hence,
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.
It comes from the continuity of , the inequality
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
Then,
and hence,
Thus, for ,
where . Fix . Then, for any we can write for some and and we have
where and . Since , 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]: 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
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
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 ,
Further,
is a bounded linear operator and
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.
Consider the operator . By (4) of Lemma 2.1 and the stability of , we have
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
Let , where
Note that
it is not difficult to verify that the mild solution of the Cauchy problem (3.2) corresponding to initial value given by
is just the unique periodic of system (3.1).
It is obvious that is linear. Next, verify the estimation (3.8). In fact, for ,
On the other hand,
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,
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
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 uniqueperiodic 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 , theperiodic 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.
Since and are the periodic mild solution corresponding to and , respectively, then we have
where
Further,
where . This completes the proof.
Lemma 4.A.
Suppose is a strong continuous operator. The operator , given by
is strongly continuous.
Proof.
Without loss of generality, for ,
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
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
[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
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
where
Define
then by Lemma 4.2, we have
as weakly convergence in .
Next, we show that
In fact, for , we have
By elementary computation, we arrive at
Consider the time interval , similarly we obtain
In general, given any , and the , , prior to the jump at time , we immediately follow the jump as
the associated interval , we also similarly obtain
Step by step, we repeat the procedures till the time interval is exhausted. Let , , , , , , thus we obtain
that is,
with strongly convergence as .
Since , using the assumption [H7]and Balder's theorem, we can obtain
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:
where denotes time, denotes age, is called age density function, and are positive constants, is a bounded measurable function, that is, . denotes the agespecific death rate, denotes the age density of migrants, and denotes the control. The admissible control set .
A linear operator defined on by
where the domain of is given by
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:
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:
Hence, Lemma 2.3 leads to the exponential stability of . That is, there exist and such that
Let
Define , , , , . Thus system (5.1) can be rewritten as
with the cost function
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.
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. 152).
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