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

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.

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:

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

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.

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

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.

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 unique-periodic -mild solution which is globally asymptotically stable.

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.

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.

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 age-specific 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.

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).

1. Clark, CW: Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Pure and Applied Mathematics,p. xi+352. John Wiley & Sons, New York, NY, USA (1976)

2. Song, X, Chen, L: Optimal harvesting and stability for a two-species competitive system with stage structure. Mathematical Biosciences. 170(2), 173–186 (2001). PubMed Abstract | Publisher Full Text

3. Marzollo R (ed.): Periodic Optimization, Springer, New York, NY, USA (1972)

4. Fan, M, Wang, K: Optimal harvesting policy for single population with periodic coefficients. Mathematical Biosciences. 152(2), 165–177 (1998). PubMed Abstract | Publisher Full Text

5. Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics,p. x+228. Longman Scientific & Technical, Harlow, UK (1993)

6. Xiao, YN, Cheng, DZ, Qin, HS: Optimal impulsive control in periodic ecosystem. Systems & Control Letters. 55(7), 558–565 (2006). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

7. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics,p. xii+273. World Scientific, Teaneck, NJ, USA (1989)

8. Nieto, JJ: An abstract monotone iterative technique. Nonlinear Analysis: Theory, Methods & Applications. 28(12), 1923–1933 (1997). PubMed Abstract | Publisher Full Text

9. 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. 40(5-6), 509–518 (2004). Publisher Full Text

10. 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. 29(5), 653–663 (2008). Publisher Full Text

11. 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. 11(1), 77–97 (2008). Publisher Full Text

12. 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)

13. Zhang, H, Chen, L, Nieto, JJ: A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Analysis: Real World Applications. 9(4), 1714–1726 (2008). Publisher Full Text

14. 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. 69(10), 3291–3298 (2008). PubMed Abstract | Publisher Full Text

15. 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. 67(10), 2815–2828 (2007). PubMed Abstract | Publisher Full Text

16. Li, Y: Global exponential stability of BAM neural networks with delays and impulses. Chaos, Solitons & Fractals. 24(1), 279–285 (2005). PubMed Abstract

17. 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. 321(2), 621–650 (2006). Publisher Full Text

18. Yu, L, Zhang, J, Liao, Y, Ding, J: Parameter estimation error bounds for Hammerstein nonlinear finite impulsive response models. Applied Mathematics and Computation. 202(2), 472–480 (2008). Publisher Full Text

19. Jankowski, T: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Applied Mathematics and Computation. 202(2), 550–561 (2008). Publisher Full Text

20. Hernández, EM, Pierri, M, Goncalves, G: Existence results for an impulsive abstract partial differential equation with state-dependent delay. Computers & Mathematics with Applications. 52(3-4), 411–420 (2006). PubMed Abstract | Publisher Full Text

21. Hernández, EM, Rabello, M, Henríquez, HR: Existence of solutions for impulsive partial neutral functional differential equations. Journal of Mathematical Analysis and Applications. 331(2), 1135–1158 (2007). Publisher Full Text

22. Hernández, EM, Sakthivel, R, Aki, ST: Existence results for impulsive evolution differential equations with state-dependent delay. Electronic Journal of Differential Equations. 2008(28), 1–11 (2008)

23. Ahmed, NU: Some remarks on the dynamics of impulsive systems in Banach spaces. Dynamics of Continuous, Discrete and Impulsive Systems. Series A. 8(2), 261–274 (2001)

24. Ahmed, NU, Teo, KL, Hou, SH: Nonlinear impulsive systems on infinite dimensional spaces. Nonlinear Analysis: Theory, Methods & Applications. 54(5), 907–925 (2003). PubMed Abstract | Publisher Full Text

25. Xiang, X, Ahmed, NU: Existence of periodic solutions of semilinear evolution equations with time lags. Nonlinear Analysis: Theory, Methods & Applications. 18(11), 1063–1070 (1992). PubMed Abstract | Publisher Full Text

26. Xiang, X: Optimal control for a class of strongly nonlinear evolution equations with constraints. Nonlinear Analysis: Theory, Methods & Applications. 47(1), 57–66 (2001). PubMed Abstract | Publisher Full Text

27. Sattayatham, P, Tangmanee, S, Wei, W: On periodic solutions of nonlinear evolution equations in Banach spaces. Journal of Mathematical Analysis and Applications. 276(1), 98–108 (2002). Publisher Full Text

28. Xiang, X, Wei, W, Jiang, Y: Strongly nonlinear impulsive system and necessary conditions of optimality. Dynamics of Continuous, Discrete & Impulsive Systems. Series A. 12(6), 811–824 (2005). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

29. 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. 63(5–7), 53–63 (2005). PubMed Abstract | Publisher Full Text

30. Wei, W, Xiang, X, Peng, Y: Nonlinear impulsive integro-differential equations of mixed type and optimal controls. Optimization. 55(1-2), 141–156 (2006). Publisher Full Text

31. Xiang, X, Wei, W: Mild solution for a class of nonlinear impulsive evolution inclusions on Banach space. Southeast Asian Bulletin of Mathematics. 30(2), 367–376 (2006)

32. Yu, X, Xiang, X, Wei, W: Solution bundle for a class of impulsive differential inclusions on Banach spaces. Journal of Mathematical Analysis and Applications. 327(1), 220–232 (2007). Publisher Full Text

33. 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. 16(3), 481–496 (2007)

34. Peng, Y, Xiang, X: Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. Journal of Industrial and Management Optimization. 4(1), 17–32 (2008)

35. Wang, J: Linear impulsive periodic system on Banach spaces. In: Proceedings of the 4th International Conference on Impulsive and Hybrid Dynamical Systems, July 2007, Nanning, China. 5, 20–25

36. Wang, J, Xiang, X, Wei, W: Linear impulsive periodic system with time-varying generating operators on Banach space. Advances in Difference Equations. 2007, (2007)

37. 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)

38. Ahmed, NU: Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series,p. x+282. Longman Scientific & Technical, Harlow, UK (1991)

39. Luo, Z-H, Guo, B-Z, Morgül, O: Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series,p. xiv+403. Springer, London, UK (1999)