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.

In this paper, we consider optimal control problem (P). Find

Time scale calculus was initiated by Hilger in his Ph.D. thesis in 1988 [

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

A time scale

Definitions and propositions of Lebesgue

Let

For each

Let

Suppose that

A function

If

Let

It follows from Definition

If

Let

Let

for any given

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

A function

We assume the following.

The scalar functions

If assumption

For conciseness, we just give a brief proof. Define a function

Let

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

Suppose that

This theorem can be proved in the following several steps.

(i) For all

Taking arbitrary points

It follows from (

(iv) We calculate the Gateaux differential of

By the Riesz representation theorem (see [

(v) Now, we can claim that

If the control set

In this section, for illustration, we will apply Theorem

A representative consumer has to make decisions not just about one period but about the sequence of

The same problem can be solved in a continuous time case, where lifetime utility is the sum of instantaneous utilities:

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

If

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.