Abstract
In this paper, we are concerned with a class of optimal control problem governed by nonlinear first order dynamic equation on time scales. By imposing some suitable conditions on the related functions, for any given control policy, we first obtain the existence of a unique solution for the nonlinear controlled system. Then, we study the existence of an optimal solution for the optimal control problem.
1. Introduction
The theory of time scales was introduced by Hilger in [1] in order to unify discrete and continuous analysis. Some foundational definitions and results from the calculus on time scales will be defined in Section 2. For more details, one can see [2–4].
In recent years, the calculus of variations and optimal control problems on time scales have attracted the attention of some researchers. For example, [5–8] discussed the calculus of variations on time scales and [9–12] studied some maximum principles on time scales, while [13–16] investigated the existence of optimal solutions or the necessary conditions of optimality for some optimal control problems on time scales.
In 2017, Guo [17] studied the projective synchronization problem of a class of chaotic systems in arbitrary dimensions. Firstly, a necessary and sufficient condition for the existence of the projective synchronization problem was presented. Secondly, an algorithm was proposed to obtain all the solutions of the projective synchronization problem. Thirdly, a simple and physically implementable controller was designed to ensure the realization of the projective synchronization. Finally, some numerical examples were provided to verify the effectiveness and the validity of the proposed results.
In 2020, Xu and Zhang [18] investigated general mean-field linear-quadratic (LQ) games of stochastic large-population system, where the individual diffusion coefficient could depend on both the state and the control of the agent, and the control weight in the cost functional could be indefinite. The asymptotic suboptimality property of the decentralized strategies for the LQ games was derived through the consistency condition. A pricing problem was also studied, for which the decentralized suboptimal price was obtained.
Throughout this paper, we always assume that is a time scale, is fixed, and . For each interval of , we denote by .
Suppose that there is a flock of sheep in a pasture. We consider the changes in the number of sheep during a time interval . It is well known that the supply of herbage, which influences growth rate and reproductive ability of sheep, is one of the main ways to control the number of sheep. Now, we define some related functions as follows: is the number of sheep at time is the number of births per unit of time at time is the number of sales per sheep per unit of time at time is the amount of herbage supplied at time is the number of sheep converted by per unit of herbage supplied per unit of time at time
Let be the admissible control set. Then, for any given control policy , it is easy to know that the changes in the number of sheep can be described by the following linear dynamic equation:
At the same time, in order to keep steady development, we may assume that the number of sheep at the beginning is equal to that at the end, that is,
Suppose that is the solution of the controlled systems (1) and (2) corresponding to the control policy and is the desired value. Recently, the authors [19] considered the optimal control problem . Find a such thatwhereis the quadratic cost functional.
Motivated greatly by the abovementioned works, in this paper, we suppose that the controlled system is governed by the following more general nonlinear periodic boundary value problem:
First, by imposing some suitable conditions on , and , for any given control policy , we obtain the existence of a unique solution for the nonlinear controlled system (5). Then, we study the optimal control problem . Find a such thatwherewhere is the desired value and is continuous.
2. Preliminaries
In this section, we will provide some foundational definitions and results from the calculus on time scales.
Definition 1. We define the forward jump operator bywhile the backward jump operator is defined byIn this definition, we put and , where denotes the empty set. If , we say that is right-scattered, while if , we say that is left-scattered. Also, if and , then is called right-dense, and if and , then is called left-dense. If has a left-scattered maximum , then we define , otherwise . Finally, the graininess function is defined by
Definition 2. Assume is a function and let . Then, we define to be the number (provided it exists) with the property that given any , there is a neighborhood of such thatWe call the delta derivative of at .
Moreover, we say that is delta differentiable (or in short, differentiable) on provided exists for all . The function is then called the (delta) derivative of on . A function is called an antiderivative of providedIf is an antiderivative of , then we define the Cauchy integral by
Definition 3. A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in .
Definition 4. We say that a function is regressive providedholds. The set of all regressive and rd-continuous functions will be denoted by . We define the set of positively regressive functions as the set consisting of those satisfying
Lemma 1. Let , , and be the exponential function on . Then,(i)(ii)Moreover, if , then
Lemma 2. Let be a continuous function on that is differentiable on . Then, is increasing, decreasing, nondecreasing, and nonincreasing on if , , , and for all , respectively.
Lemma 3. Let be a sequence of integrable functions on and suppose that uniformly on for a function defined on . Then, is integrable from to and
In the remainder of this paper, we always assume that Banach spaceis equipped with the norm , is rd-continuous, and denote
Then, it is easy to see that .
Lemma 4 (see [20]). For any , the following first order linear periodic boundary value problemhas a unique solution
3. Main Results
First, we list the following two conditions which we shall use in the sequel. is continuous and there exists such that and there exists such that
From now on, we always suppose that the control space is and the admissible control set is a compact subset of .
Lemma 5. Assume that conditions and are satisfied. Then, for any given control policy , the nonlinear controlled system (5) has a unique solution and
Proof. For any fixed , we define an operator as follows:Obviously, is a solution of the nonlinear controlled system (5) if and only if is a fixed point of in .
Let . Then, in view of Lemmas 1 and 2 and , we havesowhich together with implies that is a contraction mapping.
Therefore, it follows from Banach contraction principle that has a unique fixed point . This indicates that the nonlinear controlled system (5) has a unique solution and
Theorem 1. Assume that conditions and are satisfied and is continuous. Then, the optimal control problem has an optimal solution .
Proof. First, it follows from Lemma 5 that, for any given control policy , the nonlinear controlled system (5) has a unique solution andNext, in view of , it is obvious that exists. Thus, by the definition of infimum, we know that there exists a sequence such thatOn the one hand, since is a compact subset of and , has a convergent subsequence in . Without loss of generality, we may assume that converges in , that is, there exists such thatOn the other hand, in view of Lemmas 1 and 2, and , for any , we haveso, for any , we obtainwhich together with (31) implies thatThus, in view of Lemma 3, (31), and (34), we obtainwhich together with (30) indicates thatTherefore, for all . This shows that is an optimal solution of the optimal control problem .
Example 1. Let . We suppose that the controlled system is governed by the following nonlinear periodic boundary value problemwhereIn view of , and for , it is not difficult to obtain thatSince for and for , it is obvious that is continuous and is satisfied. Moreover, if we choose , then and it follows from Lagrange mean value theorem thatThis shows that is fulfilled.
For any given constant , let . Then, it is easy to verify that is a compact subset of .
By Lemma 5, we know that, for any given control policy , the nonlinear controlled system (37) has a unique solution .
Now, we consider the optimal control problem . Find a such thatwherewhere is the desired value.
Since for , is continuous, thus, all the conditions of Theorem 1 are satisfied. Therefore, it follows from Theorem 1 that the optimal control problem has an optimal solution .
4. Conclusions
In this paper, we consider a class of optimal control problem governed by nonlinear first order dynamic equation on time scales. First, by imposing some suitable conditions on the related functions and applying Banach contraction principle, for any given control policy, we obtain the existence of a unique solution for the nonlinear controlled system. Next, we prove that the optimal control problem has an optimal solution in the admissible control set. Finally, an example is also given to illustrate the main result of this paper.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 11661049).