Solution to an Optimal Control Problem via Canonical Dual Method

Abstract

The analytic solution to an optimal control problem is investigated using the canonical dual method. By means of the Pontryagin principle and a transformation of the cost functional, the optimal control of a nonconvex problem is obtained. It turns out that the optimal control can be expressed by the costate via canonical dual variables. Some examples are illustrated.

1. Introduction

Consider the following optimal control problem (primal problem in short):

where is continuous on and is twice continuously differentiable in . An admissible control, taking values on the unit ball , is integrable or piecewise continuous on . In (2) we assume that are continuous matrix functions in and , respectively. This problem often comes up as a main objective in general optimal control theory [1].

By the classical control theory [2], we have the following Hamilton-Jacobi-Bellman function:

The state and costate systems are

In general, it is difficult to obtain an analytic form of the optimal feedback control for the problem (1)-(2). It is well known that, in the case of unconstraint, if is a positive definite quadratic form and is a positive semidefinite quadratic form, then a perfect optimal feedback control is obtained by the solution of a Riccati matrix differential equation. The primal goal of this paper is to present an analytic solution to the optimal control problem .

We know from the Pontryagin principle [1] that if the control is an optimal solution to the problem , with and denoting the state and costate corresponding to , respectively, then is an extremal control, that is, we have

By means of the Pontryagin principle and the dynamic programming theory, many numerical algorithms have been suggested to approximate the solution to the problem (see, [3–5]). This is due to the nonlinear integrand in the cost functional. It is even difficult for the case of being nonconvex on the unit ball in . We know that when is nonconvex on the unit ball , sometimes the optimal control of the problem may exist. Let us see the following simple example for :

In fact, it is easy to see that is an optimal control.

In this paper, we consider to be nonconvex. If the optimal control of the problem exists, we solve the problem (1) to find the optimal control which is an expression of the costate. We see that, with respect to , the minimization in (7) is equivalent to the following global nonconvex optimization over a sphere:

when is a nonconvex quadratic function, the problem (9) can be solved completely by the canonical dual transformation [6–8]. In [9], the global concave optimization over a sphere is solved by use of a differential system with the canonical dual function. Because the Pontryagin principle is a necessary condition for a control to be optimal, it is not sufficient for obtaining an optimal control to solve only the optimization (9). In this paper, combing the method given in [6, 9] with the Pontryagin principle, we solve problem (1)-(2) which has nonconvex integrand on the control variable in the cost functional and present the optimal control expressed by the costate via canonical dual variables.

2. Global Optimization over a Sphere

In this section we present a differential flow to deal with the global optimization, which is used to find the optimal control expressed by the costate in the next section. Here we use the method in our another paper (see [9]).

In what follows we consider the function to be twice continuously differentiable and nonconvex on the unit ball in . Define the set

Since is nonconvex and is bounded on , is an open interval for the nonnegative real number depending on . Let and satisfy the following KKT equation:

We focus on the flow defined near by the following backward differential equation:

The flow can be extended to wherever [10]. The dual function [6] with respect to a given flow is defined as

We have

Consequently

It means that decreases when increases in . If, for a , , then for . Therefore,

as long as .

Theorem 1. If the flow , defined by (11)–(13), passes through a boundary point of the ball at , that is, then is a global minimizer of over the ball . Further one has

Proof. Since , . For each and whenever we have By (17), (18), we have Thus This concludes the proof of Theorem 1.

To illustrate the canonical dual method, let us present several examples as follows.

Example 2. Let us consider the following one-dimensional concave minimization problem: We have By choosing , we solve the following equation in : to get a solution . Next we solve the following boundary value problem of the ordinary differential equation: To find a parameter such that we get which satisfies Let be denoted by . To find the value of , we compute the solution of the following algebra equation: and get . It follows from Theorem 1 that is the global minimizer of over .

Example 3. We now consider the nonconvex minimization problem: By choosing , we solve the following equation in : to get a solution . Next we solve the following boundary value problem of the ordinary differential equation: To find a parameter such that we get which satisfies Let be denoted by . To find the value of , we compute the solution of the following algebra equation: and get . It follows from Theorem 1 that is the global minimizer of over .

Example 4. Given a symmetric matrix and a nonzero vector , let be a nonconvex quadratic function. Consider the following global optimization problem over a sphere: Suppose that has distinct eigenvalues . Since is nonconvex, . Let us choose a large such that By solving the boundary value problem of ordinary differential equation we get the unique solution Since is symmetric, there exists an orthogonal matrix such that (a diagonal matrix) and correspondingly (a vector). By (41), we have Since and there exists uniquely such that By Theorem 1, we see that is a global minimizer of the problem.

3. Find an Analytic Solution to the OptimalControl Problem

In this section, we consider in problem (1)-(2) to be constant matrices, and

where and ) is a symmetric matrix. Suppose that has distinct eigenvalues and . Moreover, we need the following basic assumption:

We consider the following optimal control problem:

To solve the above problem, we define the function , where is the solution to the following Cauchy boundary value problem of the ordinary differential equation:

By comparing (48)-(49) with (6) in terms of this special problem (46)-(47), we see that

Noting that and , we have

Thus,

Consequently, we deduce that, for almost every in , the optimal control is

By the relation between and the costate in (50), for given , we need to solve the following nonconvex optimization:

It follows from the basic assumption that for each . By Example 4 and (53), for almost every in , we have

with the dual variable satisfying

We define the function with respect to by the following equation:

and obtain an analytic solution to the optimal control problem via a costate expression

On the other hand, by the solution of the Cauchy boundary value problem of the ordinary differential equation (48)-(49), we have

Example 5. Consider the following optimal control problem: This is a simple case of (46),(47). We have By (59), we have To find an analytic solution of the optimal control problem, we solve the equation to get By (58), we obtain an analytic solution of the optimal control problem which can be expressed as

4. Concluding Remarks

In this paper, a new approach to optimal control problems has been investigated using the canonical dual method. Some nonlinear and nonconvex problems can be solved by global optimizations, and therefore, the differential flow defined by the KKT equation (see (11)) can produce an analytic solution of the optimal control problem. Meanwhile, by means of the canonical dual function, an optimality condition is proved (see Theorem 1). The global optimization problem is solved by a backward differential equation with an equality condition (see (12), (18)). More research needs to be done for the development of applicable canonical dual theory.

Acknowledgment

This research was partly supported by the National Science Foundation of China under grants No. 10671145.

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Copyright

Copyright © 2009 Jinghao Zhu and Jiani Zhou. 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.