Journal of Control Science and Engineering

Volume 2013 (2013), Article ID 678679, 5 pages

http://dx.doi.org/10.1155/2013/678679

Research Article

Solution to Singular Optimal Control by Backward Differential Flow

Jinghao Zhu,¹ Shangrui Zhao,¹ and Guohua Liu1,2

¹Department of Mathematics, Tongji University, Shanghai 200092, China

²College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 28 October 2012; Revised 14 March 2013; Accepted 21 March 2013

Academic Editor: Mohamed Zribi

Copyright © 2013 Jinghao Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper presents a backward differential flow for solving singular optimal control problems. By using Krotov equivalent transformation, the cost functional is converted to a class of global optimization problems. Some properties of the flow are given to reveal the significant relationship between the dynamic of the flow and the geometry of the feasible set. The proposed method is also used in solving a class of variational problems. Some examples are illustrated.

1. Introduction

In this paper, we are interested in solving the following optimal control problem:

where , are constant matrices, the initial state is a constant vector in , is a constant vector, and is a polynomial of degree having the form

where is a polynomial of degree less than . We assume that . The problem (P) is a singular optimal control problem. The admissible control function , taking values in , is integrable and bounded on . The set of all admissible controls is denoted by . This problem often comes up as a main objective in general optimal control theory [1, 2].

With Krotov [3] equivalent transformation, the optimal control problem (P) can be converted to some auxiliary optimization problems with constraints. A class of backward differential equations [4] relying continuously on time point is introduced for solving these optimization problems.

The rest of the paper is organized as follows. In Sections 2 and 3, with Krotov extension method [3], we convert the optimal control problem (P) to a class of constrained global optimization problems. In Sections 4 and 5, we introduce the theory of backward differential flow [4] for global optimizations. We illustrate the solution to the problem (P) by an example in Section 6. An application of the proposed method in a kind of variational problem is included in Section 7. The last section is the conclusion of the paper.

2. Krotov Equivalent Extension

Let be the solution to the following ordinary equation:

The cost functional of the problem (P) can be rewritten as follows:

noting that . So we have

It suffices for minimizing over to do the following global optimization for each :

For each , let . For given , is a polynomial having the same form as in (2) except for the linear part. In particular, , ().

Remark 1. Krotov extension [3] has been used in (5) for finding an optimal control.

3. Global Minimizers of a Polynomial

Let be a polynomial over having the same form as or . Here we briefly introduce the main result in [5, 6] for estimating a bound of global minimizers for the polynomial over . For the polynomial , we have the constrained optimization with the minimum denoted by as follows:

The following result gives a bound of all global minimizers for the polynomial over .

Theorem A (see [6]). If , all the global minimizers of the polynomial stay inside the ball , where

If further let , then there is a bound such that for all the global minimizers of stay inside . Therefore, for the optimization problem (6) is equivalent to the following constrained optimization problem:

Remark 2. The unconstrained global optimization (6) has been converted to a constrained one here for constructing a backward flow in next section.

4. Canonical Backward Differential Flow

In what follows the polynomial still stands for or as in the previous section. We keep the assumption as in Section 1. We introduce the theory of canonical backward differential flow [2] for solving the following optimization problem

where with the radius being got in Section 2 such that all minimizers of over stay in .

Since , we have, for

For the pair with satisfying the following K-T equation:

we define the flow near by the following ordinary differential equation:

The canonical dual function [4, 7] with respect to the flow is defined as follows:

Lemma 3 (see [4]). For a given flow defined by (13), one has

Lemma 4 (see [4]). Let be the flow defined by (13) and be the corresponding canonical dual function defined by (14), one has

(i)for every , ;(ii)if , then monotonously decreases in ;(iii)if and , then monotonously decreases in .

Definition 5 (see [4]). Let be the flow defined by (13). When being restricted in is called the backward differential flow.

5. Global Optimization

In this section, we use the backward differential flow to find a global minimizer to the optimization problem (10). In what follows we assume that .

Since is closed and bounded, there is a such that and , for all . By Brown fixed-point theorem [8], we have a nonzero point such that

Since in , in provided that . It follows from the differential equation,

that a differential flow corresponding to the problem (10) exists in . By (14), (15), we see that on the flow for every

Since , the flow as by Lemma 4. We claim that the backward flow also keeps staying inside . If this is not true, then there is a such that . Since is convex, by (19) we deduce that, for ,

This contradicts the fact that the minimizer of over will stay in . By (15), this implies that, for every , is the unique point over such that

Consequently, by the classical ordinary differential equation theory, the backward differential flow got by (18) can also be got by the following initial value problem:

In conclusion we have reached the following result.

Lemma 6. For every positive parameter , there is solely a point located inside , such that

Further, the differential flow corresponding to the problem (10) is unique and can be got by the equation

Remark 7. In other words, the differential flow corresponding to the problem (10) does not depend on changing the initial condition.

Theorem 8. Let be the backward flow corresponding to the problem (10). Then exists and is a global minimizer of over .

Proof. By Lemma 3, the backward flow is well defined in and , for all . Since is bounded, there is a point and a decreasing sequence of positives such that and . By Lemma 4, noting that is increasing monotonously as the positive variable goes to zero monotonously, we have

On the other hand, along the flow , for each we have

and for all

noting that on . Thus, for all ,

Let , and we have, by (28), (25), for all ,

We conclude that is a global minimizer of over the ball .

6. Solution to Singular Optimal Control

To solve the singular optimal control problem (P), by Lemma 6 and Theorem 8, we present the following algorithm.

Algorithm 9. (i) Choose .

(ii) For , solve .

(iii) Solve

(iv) For , . Go back to (ii).

Since is continuous and the solution of the canonical backward differential equation relies on the parameter for the initial value continuously, we see that will be dependent of the parameter continuously. It follows from Krotov extension method that the optimal control of (P) is .

Example 10. Given the polynomial on :

consider the following singular optimal control problem:

where

For , we have . Noting that , we have . By Theorem A, for , we have the global optimization problem

By Algorithm 9, for simplicity, choose and solve the following algebraic differential equation

to get the corresponding canonical backward differential flow . For each define . We get an optimal control for the problem (32): . Table 1 contains some data of approximate value on the optimal control.

Table 1: Some data on the optimal control for Example 10.

0
(0.4816–1.0559)

0.1
(0.4697–0.9972)

0.2
(0.4516–0.9387)

0.3
(0.4284–0.8801)

0.4
(0.3996–0.8207)

0.5
(0.3646–0.7595)

0.6
(0.3219–0.6949)

0.7
(0.2686–0.6240)

0.8
(0.1985–0.5409)

0.9
(0.0927–0.4274)

0.95
(0.0040–0.3376)

1.0
(−0.1652–0.1652)

7. Application in a Variational Problem

In this section, we use backward differential flows developed in Sections 4 and 5 to deal with the following variational problem:

This kind of problem appears very often in many complex systems from nonconvex analysis of phase transitions to discrete optimization in network design and communication [9, 10].

The functional of the variational problem (P) can be rewritten as follows:

For , let . Noting that , with Algorithm 9, for simplicity, choose . To get the corresponding canonical backward differential flow , we have the following differential equation:

We have

For each defining , we get the solution to the variational problem (36):

8. Concluding Remarks

In this paper, a new approach to solve singular optimal control is presented. As the first step of this approach, we convert the original optimal control problem to a ball constrained optimization problem. Then a differential equation is established by the K-T equation with the ball constrained nonlinear programming. The main contribution is the development of constructive backward differential flow which can be effectively used for finding a global minimizer. A kind of variational problems can also be solved by this method.

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