Necessary Conditions for a Class of Optimal Control Problems on Time Scales

Abstract

Based on the Gateaux differential on time scales, we investigate and establish necessary conditions for Lagrange optimal control problems on time scales. Moreover, we present an economic model to demonstrate the effectiveness of our results.

1. Introduction

In this paper, we consider optimal control problem (P). Find such that where is the cost functional given by and is a solution corresponding to the control of the following equation: where is a bounded time scale, and The admissible control set is Here, the control set is a bounded, closed, and convex subset of .

Time scale calculus was initiated by Hilger in his Ph.D. thesis in 1988 [1] in order to unite two existing approaches of dynamic models-difference and differential equations into a general framework, which can be used to model dynamic processes whose time domains are more complex than the set of integers (difference equations) or real numbers (differential equation). There are many potential applications for this relatively new theory. The optimal control problems on time scales are also an interesting topic, and many researchers are working in this area. Existing results on the literature of time scales are restricted to problems of the calculus of variations, which were introduced by Bohner [2] and by Hilscher and Zeidan [3]. There are many opportunities for applications in economics [4, 5]. More general optimal control problems on time scales were studied in [6, 7].

To the best of our knowledge, it seems that there is not too much work about the necessary conditions of optimal control problems on time scales by adapting the method of calculus of variations. That motivates us to investigate new necessary conditions of optimal control problem on time scales. In this paper, based on the Gateaux differential on time scales, we establish necessary conditions for Lagrange optimal control problems on time scales. Moreover, we present an economic model to demonstrate our results.

The paper is organized as follows. We present some necessary preliminary definitions and results about the time scales in Section 2. In Section 3, based on the existence and uniqueness of solutions of a linear dynamic equation on time scales, we derive existence and uniqueness of system solutions for the controlled system. Then, we prove the minimum principle on time scales for the optimal control problem (P) in Section 4. Finally, in Section 5, an example is given to demonstrate our results.

2. Preliminaries

A time scale is a closed nonempty subset of . The two most popular examples are and . The forward and backward jump operators are defined by We put and , where denotes the empty set. If there is the finite , then , and if there exists the finite , then . The graininess function is . A point is called left-dense (left-scattered, right-dense, and right-scattered) if (, , and ) holds. If has a left-scattered maximum value , then we denote . Otherwise, .

Definitions and propositions of Lebesgue -measure and Lebesgue integral can be seen in [8–10].

Definition 2.1. Let denote a proposition with respect to and a subset of . If there exists with such that holds on , then is said to hold , on

Remark 2.2. For each , the single-point set is -measurable, and its -measure is given by Obviously, does not have any right-scattered points. For a set , define the Lebesgue -integral of over by and let (see [8]).

Lemma 2.3 (see [8]). Let . is the extension of to real interval , defined by where , is the index of the set of all right-scattered points of . Then, if and only if . In this case,

Definition 2.4. Suppose that . , if there exists a constant such that

Definition 2.5 (see [10]). A function is said to be absolutely continuous on if for every given constant , there is a constant such that if , with , is a finite pairwise disjoint family of subintervals of satisfying then If , then we denote all absolutely continuous functions on as .

Lemma 2.6. If is Lebesgue -integrable on , then the integral is absolutely continuous on . Moreover,

Proof. where is introduced in and from Lemma 2.3. Now, by the standard Lebesgue integration theory, is an absolutely continuous function on the real interval and Using Definition 2.5, is also absolutely continuous on .
Let be differentiable at for . If is right-scattered, that is, for some , it follows from the continuity of at that If is right dense, Hence, is -differentiable at and That is, The continuity of guarantees that is -differentiable at every right-scattered point . Moreover, implies . We deduce that does not contain any right-scattered points and Hence, is -differentiable -a.e., on and The proof is complete.

It follows from Definition 2.5 and Lemma 2.6 that one can easy to prove the following integration by parts formula on time scales.

Lemma 2.7. If are absolutely continuous functions on , then is absolutely continuous on and the following equality is valid:

Let denote the linear space of all continuous functions on time scale with the maximum norm . The following statement can be understood as a time scale version of the Arzela-Ascoli theorem.

Lemma 2.8 (see [11] (Arzela-Ascoli theorem)). Let be a subset of satisfying the following conditions:(i)is bounded;(ii)for any given , there exists such that , implies for all (i.e., the functions in are equicontinuous).Then, is relatively compact.

3. Existence and Uniqueness of Solutions for a Controlled System Equation

In order to derive necessary conditions, we prove the existence and uniqueness of solutions for controlled system equation .

Definition 3.1. A function is said to be a solution of problem (1.2) if (i) is -differentiable -a.e.on and ;(ii) and We assume the following.[HF] is regressive rd-continuous function and .[HL]The scalar functions along with their partial derivation are continuous and uniformly bounded on for almost all .

Theorem 3.2 (existence and uniqueness of solutions for the controlled system equation). If assumption holds, for any , problem (1.2) has a unique solution in which given by

Proof. For conciseness, we just give a brief proof. Define a function as Then, problem is equivalent to Since , there exists a sequence in such that . Therefore, the Cauchy problem has an unique classical solution given by Now, we define Then, and Lemma 2.6 can be applied to testify that tailors to Definition 3.1.

Let Define the Hamiltonian as

4. Necessary Conditions for Optimal Control Problem (P)

In this section, we will present the minimum principle on time scales for the optimal control problem (P).

Theorem 4.1 (minimum principle on time scales). Suppose that and hold. If is an optimal solution for problem (P) and is an optimal trajectory corresponding to , then it is necessary that there exists a function satisfying the following conditions:

Proof. This theorem can be proved in the following several steps.
(i) For all and for all , define . Since is a bounded closed convex set, then is also a closed convex subset of and . Because is optimal, (ii) Now, we verify that converges to in as by using Arzela-Ascoli theorem (Lemma 2.8). By boundedness of , we have is uniformly bounded on .
Taking arbitrary points and of the segment and using the absolutely continuity of integral and the boundedness of , we obtain Since Hence, is equicontinuous in .
It follows from that By Arzela-Ascoli theorem (Lemma 2.8), we obtain (iii) Denote Then, satisfies the following initial value problem: with We call and the variational equations.
(iv) We calculate the Gateaux differential of at in the direction . It follows from hypotheses , Lemma 2.3, and (4.4) that Here, the “title’’ is the corresponding extension function in Lemma 2.3. That is, where ,, is the set of all right-scattered points of . Obviously By the variational equations (4.12) and (4.13), we define an operator as Then, is a continuous linear operator. Furthermore, due to the uniform bound of , , given by is also a linear continuous functional. Hence, defined by is a bounded linear functional.
By the Riesz representation theorem (see [12, Theorem  2.34]), there is a such that Using , , and , we obtain Hence we have derived the necessary condition (4.1).
(v) Now, we can claim that and the last part of necessary conditions are true. Using Lemma 2.7, , and as well as , we obtain From the first and the last equalities, we have Hence, similar to Theorem 3.2, one may choose as the solution of the following backward problem: This completes the proof.

Remark 4.2. If the control set , then reduces to

5. Example (A Model in Economics)

In this section, for illustration, we will apply Theorem 4.1 to the following economics model. This model had been discussed by the method of Nabla version calculus of variation on time scales (see [4, 13]). We briefly present it here. A consumer is seeking to maximize his lifetime utility subject to certain constraints. During each period in his life, a consumer has to make a decision regarding how much to consume and how much to spend. Utility is the value function of the consumer that one wants to maximize. It can depend on numerous variables, in this simple example, it depends only on the consumption of some generic production . Utility function abides by the Law of Diminishing Marginal Utility, that is to say, and .

5.1. Discrete Time Model

A representative consumer has to make decisions not just about one period but about the sequence of : . The problem is to find a consumption path that would maximize lifetime utility as follows: where is the consumption during period , is one-period utility, and is the (constant) discount rate. We assume that the future consumption is less than the current consumption, so we discount the future at the rate . The consumer is limited by the budget constraints: where is the amount of assets held at the beginning of period , is the income received in period , and is the constant interest rate. that can be interpreted as “we are not allowed to borrow without limit.”

5.2. Continuous Time Model

The same problem can be solved in a continuous time case, where lifetime utility is the sum of instantaneous utilities: with respect to the path subject to the constraint

5.3. Time Scale Calculus Model

A consumer receives income at one time point, asset holdings are adjusted at a different time point, and consumption takes place at another time point. Consumption and saving decisions can be modeled to occur with arbitrary, time-varying frequency. Hence, the time scale version of this model can de described by subject to the budget constraint where is the Nabla exponential function of , Note that (see [14] for more details) Now, we use Theorem 4.1 to solve this model. The Hamiltonian Optimal consumption satisfies the following necessary conditions: If , then , imply Due to and , it shows that if . Hence, the consumer will wait to consume.If , then , imply It follows from and that if , then . Therefore if the interest rate is higher than the future’s discount rate , the consumer will wait to consume until next periods. Therefore, we obtain the same results as [4].

Acknowledgments

This work is supported by the National Science Foundation of China under Grant 10661004, the Key Project of Chinese Ministry of Education (no. 2071041) and the Talents Foundation of Guizhou University under Grant 2007043, and the technological innovation fund project of Guizhou University under Grant 2007011.