Journal of Control Science and Engineering

Journal of Control Science and Engineering / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 202094 | https://doi.org/10.1155/2009/202094

Jinghao Zhu, Jiani Zhou, "Solution to an Optimal Control Problem via Canonical Dual Method", Journal of Control Science and Engineering, vol. 2009, Article ID 202094, 5 pages, 2009. https://doi.org/10.1155/2009/202094

Solution to an Optimal Control Problem via Canonical Dual Method

Academic Editor: George Gang Yin
Received14 Jun 2009
Accepted01 Sep 2009
Published18 Oct 2009

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 (๐’ซ) in short):

๎€œ(๐’ซ)min๐‘‡0[]๐น(๐‘ฅ)+๐‘ƒ(๐‘ข)๐‘‘๐‘ก,(1)s.t.ฬ‡๐‘ฅ=๐ด(๐‘ก)๐‘ฅ+๐ต(๐‘ก)๐‘ข,๐‘ฅ(0)=๐‘ฅ0โˆˆ๐‘…๐‘›,[],๐‘กโˆˆ0,๐‘‡โ€–๐‘ขโ€–โ‰ค1,(2) where ๐น(โ‹…) is continuous on ๐‘…๐‘›, and ๐‘ƒ(โ‹…) is twice continuously differentiable in ๐‘…๐‘š. An admissible control, taking values on the unit ball ๐ทโˆถ={๐‘ขโˆˆ๐‘…๐‘šโˆฃโ€–๐‘ขโ€–โ‰ค1}, is integrable or piecewise continuous on [0,๐‘‡]. In (2) we assume that ๐ด(๐‘ก),๐ต(๐‘ก) are continuous matrix functions in ๐ถ([0,๐‘‡],๐‘…๐‘›ร—๐‘›) and ๐ถ([0,๐‘‡],๐‘…๐‘›ร—๐‘š), 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:

๐ป(๐‘ก,๐‘ฅ,๐‘ข,๐œ†)=๐œ†โˆ—(๐ด(๐‘ก)๐‘ฅ+๐ต(๐‘ก)๐‘ข)+๐น(๐‘ฅ)+๐‘ƒ(๐‘ข).(3) The state and costate systems are

ฬ‡๐‘ฅ=๐ป๐œ†(๐‘ก,๐‘ฅ,๐‘ข,๐œ†)=๐ด(๐‘ก)๐‘ฅ+๐ต(๐‘ก)๐‘ข,๐‘ฅ(0)=๐‘ฅ0,ฬ‡๐œ†=โˆ’๐ป๐‘ฅ(๐‘ก,๐‘ฅ,๐‘ข,๐œ†)=โˆ’๐ดโˆ—๐œ†โˆ’โˆ‡๐น(๐‘ฅ),๐œ†(๐‘‡)=0.(4)

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 ๐‘ƒ(๐‘ข) is a positive definite quadratic form and ๐น(๐‘ฅ) 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 (๐’ซ).

We know from the Pontryagin principle [1] that if the control ฬ‚๐‘ข is an optimal solution to the problem (๐’ซ), with ฬ‚๐‘ฅ(โ‹…) and ฬ‚๐œ†(โ‹…) denoting the state and costate corresponding to ฬ‚๐‘ข(โ‹…), respectively, then ฬ‚๐‘ข is an extremal control, that is, we have

ฬ‡ฬ‚๐‘ฅ=๐ป๐œ†๎€ทฬ‚๐œ†๎€ธ๐‘ก,ฬ‚๐‘ฅ,ฬ‚๐‘ข,=๐ด(๐‘ก)ฬ‚๐‘ฅ+๐ต(๐‘ก)ฬ‚๐‘ข,ฬ‚๐‘ฅ(0)=๐‘ฅ0,(5)ฬ‡ฬ‚๐œ†=โˆ’๐ป๐‘ฅ๎€ทฬ‚๐œ†๎€ธ๐‘ก,ฬ‚๐‘ฅ,ฬ‚๐‘ข,=โˆ’๐ดโˆ—ฬ‚ฬ‚๐œ†๐ป๎€ทฬ‚๎€ธ๐œ†โˆ’โˆ‡๐น(ฬ‚๐‘ฅ),(๐‘‡)=0,(6)๐‘ก,ฬ‚๐‘ฅ(๐‘ก),ฬ‚๐‘ข(๐‘ก),๐œ†(๐‘ก)=minโ€–๐‘ขโ€–โ‰ค1๐ป๎€ทฬ‚๎€ธ,[].๐‘ก,ฬ‚๐‘ฅ(๐‘ก),๐‘ข,๐œ†(๐‘ก)a.e.๐‘กโˆˆ0,๐‘‡(7)

By means of the Pontryagin principle and the dynamic programming theory, many numerical algorithms have been suggested to approximate the solution to the problem (๐’ซ) (see, [3โ€“5]). This is due to the nonlinear integrand in the cost functional. It is even difficult for the case of ๐‘ƒ(๐‘ข) being nonconvex on the unit ball ๐ท in ๐‘…๐‘š. We know that when ๐‘ƒ(๐‘ข) is nonconvex on the unit ball ๐ท, sometimes the optimal control of the problem (๐’ซ) may exist. Let us see the following simple example for ๐‘›=๐‘š=1:

๎€œ(๐’ซ)min10๎‚ƒ1๐‘ฅโˆ’2๐‘ข2๎‚„[]๐‘‘๐‘ก,s.t.ฬ‡๐‘ฅ=๐‘ฅ+๐‘ข,๐‘ฅ(0)=0,๐‘กโˆˆ0,๐‘‡,|๐‘ข|โ‰ค1.(8) In fact, it is easy to see that ฬ‚๐‘ข(๐‘ก)โ‰กโˆ’1;๐‘กโˆˆ[0,๐‘‡] is an optimal control.

In this paper, we consider ๐‘ƒ(๐‘ข) to be nonconvex. If the optimal control of the problem (๐’ซ) exists, we solve the problem (1) to find the optimal control which is an expression of the costate. We see that, with respect to ๐‘ข, the minimization in (7) is equivalent to the following global nonconvex optimization over a sphere:

minโ€–๐‘ขโ€–โ‰ค1๎€บฬ‚๐‘ƒ(๐‘ข)+๐œ†(๐‘ก)โˆ—๎€ป[]๐ต(๐‘ก)๐‘ข,a.e.๐‘กโˆˆ0,๐‘‡,(9) when ๐‘ƒ(๐‘ข) is a nonconvex quadratic function, the problem (9) can be solved completely by the canonical dual transformation [6โ€“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 ๐‘ƒ(๐‘ฅ) to be twice continuously differentiable and nonconvex on the unit ball in ๐‘…๐‘š. Define the set

๎€ฝ๎€บโˆ‡๐บ=๐œŒ>0โˆฃ2๎€ป๐‘ƒ(๐‘ฅ)+๐œŒ๐ผ>0,๐‘ฅโˆ—๎€พ๐‘ฅโ‰ค1.(10) Since ๐‘ƒ(๐‘ฅ) is nonconvex and โˆ‡2๐‘ƒ(๐‘ฅ) is bounded on ๐ทโˆถ={๐‘ฅโˆˆ๐‘…๐‘š๐‘ฅโˆ—๐‘ฅโ‰ค1}, ๐บ is an open interval (๐œŒ,+โˆž) for the nonnegative real number ๐œŒ depending on ๐‘ƒ(๐‘ฅ). Let ๐œŒโˆ—โˆˆ๐บ and ฬƒ๐‘ฅโˆˆ{๐‘ฅโˆ—๐‘ฅโ‰ค1} satisfy the following KKT equation:

โˆ‡๐‘ƒ(ฬƒ๐‘ฅ)+๐œŒโˆ—ฬƒ๐‘ฅ=0.(11) We focus on the flow ฬ‚๐‘ฅ(๐œŒ) defined near ๐œŒโˆ— by the following backward differential equation:

๐‘‘ฬ‚๐‘ฅ+๎€บโˆ‡๐‘‘๐œŒ2๎€ป๐‘ƒ(ฬ‚๐‘ฅ)+๐œŒ๐ผโˆ’1๎€ท๐œŒฬ‚๐‘ฅ=0,๐œŒโˆˆโˆ—โˆ’๐›ฟ,๐œŒโˆ—๎€ป,๎€ท๐œŒ(12)ฬ‚๐‘ฅโˆ—๎€ธ=ฬƒ๐‘ฅ.(13) The flow ฬ‚๐‘ฅ(๐œŒ) can be extended to wherever ๐œŒโˆˆ๐บโˆฉ(0,๐œŒโˆ—] [10]. The dual function [6] with respect to a given flow ฬ‚๐‘ฅ(๐œŒ) is defined as

๐‘ƒ๐‘‘๐œŒ(๐œŒ)=๐‘ƒ(ฬ‚๐‘ฅ(๐œŒ))+2ฬ‚๐‘ฅโˆ—๐œŒ(๐œŒ)ฬ‚๐‘ฅ(๐œŒ)โˆ’2.(14) We have

๎‚ต๐‘‘ฬ‚๐‘ฅ(๐œŒ)๎‚ถ๐‘‘๐œŒโˆ—๎€บโˆ‡2๎€ป๐‘ƒ(ฬ‚๐‘ฅ(๐œŒ))+๐œŒ๐ผ๐‘‘ฬ‚๐‘ฅ(๐œŒ)=๐‘‘๐œŒโˆ’12๐‘‘๎€บฬ‚๐‘ฅโˆ—๎€ป(๐œŒ)ฬ‚๐‘ฅ(๐œŒ)๐‘‘๐‘‘๐œŒ=โˆ’2๐‘ƒ๐‘‘(ฬ‚๐œŒ)๐‘‘๐œŒ2โ‰ฅ0.(15) Consequently

๐‘‘2๐‘ƒ๐‘‘(๐œŒ)๐‘‘๐œŒ2โ‰ค0.(16)

It means that ๐‘‘๐‘ƒ๐‘‘(๐œŒ)/๐‘‘๐œŒ decreases when ๐œŒ increases in ๐บ. If, for a ฬ‚๐œŒโˆˆ๐บ, ๐‘‘๐‘ƒ๐‘‘(ฬ‚๐œŒ)/๐‘‘๐œŒโ‰ค0, then ๐‘‘๐‘ƒ๐‘‘(๐œŒ)/๐‘‘๐œŒโ‰ค0 for ๐œŒโˆˆ๐บโˆฉ[ฬ‚๐œŒ,โˆž). Therefore,

๐‘ƒ๐‘‘(ฬ‚๐œŒ)โ‰ฅ๐‘ƒ๐‘‘(๐œŒ),(17) as long as ๐œŒโ‰ฅฬ‚๐œŒ.

Theorem 1. If the flow ฬ‚๐‘ฅ(๐œŒ),๐œŒโˆˆ๐บโˆฉ(0,๐œŒโˆ—], defined by (11)โ€“(13), passes through a boundary point of the ball ๐ท={๐‘ฅโˆˆ๐‘…๐‘šโ€–๐‘ฅโ€–โ‰ค1} at ฬ‚๐œŒโˆˆ๐บ, that is, []ฬ‚๐‘ฅ(ฬ‚๐œŒ)โˆ—๎€ทฬ‚๐‘ฅ(ฬ‚๐œŒ)=1,ฬ‚๐œŒโˆˆ๐บโˆฉ0,๐œŒโˆ—๎€ป,(18) then ฬ‚๐‘ฅ is a global minimizer of ๐‘ƒ(๐‘ฅ) over the ball ๐ท. Further one has min๐ท๐‘ƒ(๐‘ฅ)=๐‘ƒ(ฬ‚๐‘ฅ)=๐‘ƒ๐‘‘(ฬ‚๐œŒ)=max๐œŒโ‰ฅ๐‘ƒฬ‚๐œŒ๐‘‘(๐œŒ).(19)

Proof. Since ฬ‚๐œŒโˆˆ๐บ, ฬ‚๐œŒ>0. For each ๐‘ฅโˆˆ๐ท and whenever ๐œŒโ‰ฅฬ‚๐œŒ we have ๐‘ƒ๐œŒ(๐‘ฅ)โ‰ฅ๐‘ƒ(๐‘ฅ)+2๎€บ๐‘ฅโˆ—๎€ป๐‘ฅโˆ’1โ‰ฅinf๐ท๎‚ƒ๐œŒ๐‘ƒ(๐‘ฅ)+2๎€บ๐‘ฅโˆ—๎€ป๎‚„๐‘ฅโˆ’1=๐‘ƒ๐‘‘(๐œŒ).(20) By (17), (18), we have ๐‘ƒ(๐‘ฅ)โ‰ฅmax๐œŒโ‰ฅ๐‘ƒฬ‚๐œŒ๐‘‘(๐œŒ)=๐‘ƒ๐‘‘๐œŒ(ฬ‚๐œŒ)=๐‘ƒ(ฬ‚๐‘ฅ(ฬ‚๐œŒ))+2๎€บ(ฬ‚๐‘ฅ(ฬ‚๐œŒ))โˆ—๎€ปฬ‚๐‘ฅ(ฬ‚๐œŒ)โˆ’1=๐‘ƒ(ฬ‚๐‘ฅ(ฬ‚๐œŒ)).(21) Thus min๐ท๐‘ƒ(๐‘ฅ)=max๐œŒโ‰ฅ๐‘ƒฬ‚๐œŒ๐‘‘(๐œŒ).(22) 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: ๐‘โˆ—=min๐‘ƒ(๐‘ฅ)=โˆ’1๐‘ฅ124โˆ’๐‘ฅ2+๐‘ฅ,s.t.๐‘ฅ2โ‰ค1.(23) We have ๐‘ƒ๎…ž(๐‘ฅ)=โˆ’13๐‘ฅ3โˆ’2๐‘ฅ+1,๐‘ƒ๎…ž๎…ž(๐‘ฅ)=โˆ’๐‘ฅ2โˆ’2<0,โˆ€๐‘ฅ2โ‰ค1.(24) By choosing ๐œŒโˆ—=10, we solve the following equation in {๐‘ฅ2<1}: โˆ’13๐‘ฅ3โˆ’2๐‘ฅ+1+10๐‘ฅ=0(25) to get a solution ฬƒ๐‘ฅ=โˆ’0.1251. Next we solve the following boundary value problem of the ordinary differential equation: ๐‘‘๐‘ฅ(๐œŒ)=๐‘‘๐œŒ๐‘ฅ(๐œŒ)๐‘ฅ2๎€ท๐œŒ(๐œŒ)+2โˆ’๐œŒ,๐‘ฅโˆ—๎€ธ=โˆ’0.1251,๐œŒโ‰ค10.(26) To find a parameter such that ๐‘ฅ2(๐œŒ)=1,(27) we get ฬ‚๐œŒ=103,(28) which satisfies ๐‘ƒ๎…ž๎…ž(๐‘ฅ)+ฬ‚๐œŒ=๐‘ƒ๎…ž๎…ž(๐‘ฅ)+103>0,โˆ€๐‘ฅ2โ‰ค1.(29) Let ๐‘ฅ(10/3) be denoted by ฬ‚๐‘ฅ. To find the value of ฬ‚๐‘ฅ, we compute the solution of the following algebra equation: โˆ’13๐‘ฅ3โˆ’2๐‘ฅ+1+103๐‘ฅ=0,๐‘ฅ2=1(30) and get ฬ‚๐‘ฅ=โˆ’1. It follows from Theorem 1 that ฬ‚๐‘ฅ=โˆ’1 is the global minimizer of ๐‘ƒ(๐‘ฅ) over [โˆ’1,1].

Example 3. We now consider the nonconvex minimization problem: ๐‘โˆ—1=min๐‘ƒ(๐‘ฅ)=3๐‘ฅ3+2๐‘ฅ,s.t.๐‘ฅ2โ‰ค1.(31) By choosing ๐œŒโˆ—=โˆš72, we solve the following equation in {๐‘ฅ2<1}: ๐‘ฅ2โˆš+2+72๐‘ฅ=0(32) to get a solution โˆšฬƒ๐‘ฅ=โˆ’2/(4+32). Next we solve the following boundary value problem of the ordinary differential equation: ฬ‡๐‘ฅ=โˆ’๐‘ฅโˆš2๐‘ฅ+๐‘ก,๐‘กโ‰ค๐‘ฅ๎‚€โˆš72,๎‚=72โˆ’2โˆš4+32.(33) To find a parameter such that ๐‘ฅ2(๐œŒ)=1,(34) we get ฬ‚๐œŒ=3,(35) which satisfies ๐‘ƒ๎…ž๎…ž(๐‘ฅ)+ฬ‚๐œŒ=๐‘ƒ๎…ž๎…ž(๐‘ฅ)+3=2๐‘ฅ+3>0,โˆ€๐‘ฅ2โ‰ค1.(36) Let ๐‘ฅ(3) be denoted by ฬ‚๐‘ฅ. To find the value of ฬ‚๐‘ฅ, we compute the solution of the following algebra equation: ๐‘ฅ2+2+3๐‘ฅ=0,๐‘ฅ2=1(37) and get ฬ‚๐‘ฅ=โˆ’1. It follows from Theorem 1 that ฬ‚๐‘ฅ=โˆ’1 is the global minimizer of ๐‘ƒ(๐‘ฅ) over [โˆ’1,1].

Example 4. Given a symmetric matrix ๐บโˆˆ๐‘…๐‘šร—๐‘š and a nonzero vector ๐‘“โˆˆ๐‘…๐‘š, let ๐‘ƒ(๐‘ฅ)=(1/2)๐‘ฅโˆ—๐บ๐‘ฅโˆ’๐‘“โˆ—๐‘ฅ be a nonconvex quadratic function. Consider the following global optimization problem over a sphere: 1min๐‘ƒ(๐‘ฅ)โˆถ=2๐‘ฅโˆ—๐บ๐‘ฅโˆ’๐‘“โˆ—๐‘ฅ,s.t.๐‘ฅโˆ—๐‘ฅโ‰ค1.(38) Suppose that ๐บ has ๐‘โ‰ค๐‘š distinct eigenvalues ๐‘Ž1<๐‘Ž2<โ‹ฏ<๐‘Ž๐‘. Since ๐‘ƒ(๐‘ฅ)=(1/2)๐‘ฅโˆ—๐บ๐‘ฅโˆ’๐‘“โˆ—๐‘ฅ is nonconvex, ๐‘Ž1<0. Let us choose a large ๐œŒโˆ—>โˆ’๐‘Ž1 such that โ€–โ€–๎€ท0<๐บ+๐œŒโˆ—๐ผ๎€ธโˆ’1๐‘“โ€–โ€–<1.(39) By solving the boundary value problem of ordinary differential equation ๐‘‘๐‘ฅ๐‘‘๐œŒ=โˆ’(๐บ+๐œŒ๐ผ)โˆ’1๎€ท๐œŒ๐‘ฅ,โˆ—๎€ธ=๎€ท๐บ+๐œŒโˆ—๐ผ๎€ธโˆ’1๐‘“,๐œŒโ‰ค๐œŒโˆ—,(40) we get the unique solution ๐‘ฅ(๐œŒ)=(๐บ+๐œŒ๐ผ)โˆ’1๐‘“,๐œŒโ‰ค๐œŒโˆ—.(41) Since ๐บ is symmetric, there exists an orthogonal matrix ๐‘… such that ๐‘…๐บ๐‘…โˆ—=๐ทโˆถ=(๐‘Ž๐‘–๐›ฟ๐‘–๐‘—) (a diagonal matrix) and correspondingly ๐‘…๐‘“=๐‘”โˆถ=(๐‘”๐‘–) (a vector). By (41), we have ๐‘ฅโˆ—(๐œŒ)๐‘ฅ(๐œŒ)=๐‘“โˆ—(๐บ+๐œŒ๐ผ)โˆ’2๐‘“=๐‘๎“๐‘–=1๐‘”2๐‘–๎€ท๐‘Ž๐‘–๎€ธ+๐œŒ2.(42) Since ๐‘“โˆ—(๐บ+๐œŒโˆ—๐ผ)โˆ’2๐‘“<1 and lim๐œŒ>โˆ’๐‘Ž1,๐œŒโ†’โˆ’๐‘Ž1๐‘๎“๐‘–=1๐‘”2๐‘–๎€ท๐‘Ž๐‘–๎€ธ+๐œŒ2=+โˆž,(43) there exists ฬ‚๐œŒโˆˆ(โˆ’๐‘Ž1,๐œŒโˆ—) uniquely such that ๐‘ฅโˆ—(ฬ‚๐œŒ)๐‘ฅ(ฬ‚๐œŒ)=๐‘“โˆ—(๐บ+ฬ‚๐œŒ๐ผ)โˆ’2๐‘“=๐‘๎“๐‘–=1๐‘”2๐‘–๎€ท๐‘Ž๐‘–๎€ธ+ฬ‚๐œŒ2=1.(44) By Theorem 1, we see that ๐‘ฅ(ฬ‚๐œŒ)=(๐บ+ฬ‚๐œŒ๐ผ)โˆ’1๐‘“ is a global minimizer of the problem.

3. Find an Analytic Solution to the OptimalControl Problem

In this section, we consider ๐ด(๐‘ก),๐ต(๐‘ก) in problem (1)-(2) to be constant matrices, ๐น(๐‘ฅ)=๐‘โˆ—๐‘ฅ and

1๐‘ƒ(๐‘ข)=2๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ข,(45)

where ๐‘โˆˆ๐‘…๐‘›ร—1,๐‘โˆˆ๐‘…๐‘šร—1, and ๐บ(โˆˆ๐‘…๐‘šร—๐‘š) is a symmetric matrix. Suppose that ๐บ has ๐‘โ‰ค๐‘š distinct eigenvalues ๐‘Ž1<๐‘Ž2<โ‹ฏ<๐‘Ž๐‘ and ๐‘Ž1<0. Moreover, we need the following basic assumption:

๎€ท๐ตrankโˆ—๎€ธ๎€ท๐ต,๐‘>rankโˆ—๎€ธ.(โˆ—)

We consider the following optimal control problem:

๎€œ(๐’ซ)min๐ฝ(๐‘ข)=๐‘‡0๎‚ƒ๐‘โˆ—1๐‘ฅ+2๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ข๎‚„๐‘‘๐‘ก,(46)s.t.ฬ‡๐‘ฅ=๐ด๐‘ฅ+๐ต๐‘ข,๐‘ฅ(0)=๐‘ฅ0[],๐‘กโˆˆ0,๐‘‡,โ€–๐‘ขโ€–โ‰ค1.(47)

To solve the above problem, we define the function ๐œ™(๐‘ก,๐‘ฅ)=๐œ“โˆ—(๐‘ก)๐‘ฅ, where ๐œ“(๐‘ก) is the solution to the following Cauchy boundary value problem of the ordinary differential equation:

ฬ‡๐œ“(๐‘ก)=โˆ’๐ดโˆ—๐œ“(๐‘ก)+๐‘,(48)๐œ“(๐‘‡)=0.(49) By comparing (48)-(49) with (6) in terms of this special problem (46)-(47), we see that

[]๐œ“(๐‘ก)=โˆ’๐œ†(๐‘ก),a.e.๐‘กโˆˆ0,๐‘‡.(50)

Noting that ๐œ“(๐‘‡)=0 and ๐‘ฅ(0)=๐‘ฅ0, we have

๎€œ๐ฝ(๐‘ข)=๐‘‡0๎‚ƒ๐‘โˆ—1๐‘ฅ+2๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ข๎‚„=๎€œ๐‘‘๐‘ก๐‘‡0๎‚ƒ๎€ทฬ‡๐œ“(๐‘ก)+๐ดโˆ—๎€ธ๐œ“(๐‘ก)โˆ—1๐‘ฅ+2๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ข๎‚„=๎€œ๐‘‘๐‘ก๐‘‡0๎‚ƒฬ‡๐œ“โˆ—(๐‘ก)๐‘ฅ+๐œ“(๐‘ก)โˆ—1๐ด๐‘ฅ+2๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ข๎‚„=๎€œ๐‘‘๐‘ก๐‘‡0๎‚ƒฬ‡๐œ“โˆ—(๐‘ก)๐‘ฅ+๐œ“(๐‘ก)โˆ—(๐ด๐‘ฅ+๐ต๐‘ข)โˆ’๐œ“(๐‘ก)โˆ—1๐ต๐‘ข+2๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ข๎‚„=๎€œ๐‘‘๐‘ก๐‘‡0๎‚ƒฬ‡๐œ“โˆ—(๐‘ก)๐‘ฅ(๐‘ก)+๐œ“(๐‘ก)โˆ—ฬ‡๐‘ฅ(๐‘ก)โˆ’๐œ“(๐‘ก)โˆ—1๐ต๐‘ข+2๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ข๎‚„=๎€œ๐‘‘๐‘ก๐‘‡0๎‚ƒฬ‡๐œ™(๐‘ก,๐‘ฅ(๐‘ก))โˆ’๐œ“(๐‘ก)โˆ—1๐ต๐‘ข+2๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ข๎‚„+๎€œ๐‘‘๐‘ก=๐œ™(๐‘‡,๐‘ฅ(๐‘‡))โˆ’๐œ™(0,๐‘ฅ(0))๐‘‡0๎‚ƒ12๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ขโˆ’๐œ“(๐‘ก)โˆ—๎‚„๎€œ๐ต๐‘ข๐‘‘๐‘ก=โˆ’๐œ™(0,๐‘ฅ(0))+๐‘‡0๎‚ƒ12๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ขโˆ’๐œ“(๐‘ก)โˆ—๎‚„๐ต๐‘ข๐‘‘๐‘ก.(51) Thus,

๎€œmin๐ฝ(๐‘ข)=โˆ’๐œ™(0,๐‘ฅ(0))+min๐‘‡0๎‚ƒ12๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ขโˆ’๐œ“(๐‘ก)โˆ—๎‚„๐ต๐‘ข๐‘‘๐‘ก.(52)

Consequently, we deduce that, for almost every ๐‘ก in [0,๐‘‡], the optimal control is

ฬ‚๐‘ข(๐‘ก)=argmin๐‘ขโˆ—๐‘ขโ‰ค1๎‚ƒ12๐‘ขโˆ—๐บ๐‘ขโˆ’๐‘โˆ—๐‘ขโˆ’๐œ“(๐‘ก)โˆ—๎‚„๐ต๐‘ข.(53) By the relation between ๐œ“(๐‘ก) and the costate in (50), for given ๐‘กโˆˆ[0,๐‘‡], we need to solve the following nonconvex optimization:

1min2๐‘ขโˆ—๎€ท๐บ๐‘ขโˆ’๐‘โˆ’๐ตโˆ—๎€ธ๐œ†(๐‘ก)โˆ—๐‘ข,s.t.๐‘ขโˆ—๐‘ขโ‰ค1.(54) It follows from the basic assumption (โˆ—) that ๐‘โˆ’๐ต๐‘‡๐œ†(๐‘ก)โ‰ 0 for each ๐‘กโˆˆ[0,๐‘‡]. By Example 4 and (53), for almost every ๐‘ก in [0,๐‘‡], we have

๎€ทฬ‚๐‘ข(๐‘ก)=๐บ+๐œŒ๐‘ก๐ผ๎€ธโˆ’1๎€บ๐‘โˆ’๐ตโˆ—๎€ป๐œ†(๐‘ก),(55) with the dual variable ๐œŒ๐‘ก>โˆ’๐‘Ž1 satisfying

๎€ท๐‘โˆ’๐ตโˆ—๎€ธ๐œ†(๐‘ก)โˆ—๎€ท๐บ+๐œŒ๐‘ก๐ผ๎€ธโˆ’2๎€ท๐‘โˆ’๐ตโˆ—๎€ธ๐œ†(๐‘ก)=1.(56) We define the function ๐œŒ(๐œ†) with respect to ๐œ† by the following equation:

๎€ท๐‘โˆ’๐ตโˆ—๐œ†๎€ธโˆ—(๐บ+๐œŒ(๐œ†)๐ผ)โˆ’2๎€ท๐‘โˆ’๐ตโˆ—๐œ†๎€ธ=1,๐œŒ(๐œ†)>โˆ’๐‘Ž1(57) and obtain an analytic solution to the optimal control problem via a costate expression

ฬ‚๐‘ข=(๐บ+๐œŒ(๐œ†)๐ผ)โˆ’1๎€ท๐‘โˆ’๐ตโˆ—๐œ†๎€ธ.(58)

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

๐œ†(๐‘ก)=โˆ’๐œ“(๐‘ก)=๐‘’๐ดโˆ—๐‘‡๎€œ0๐‘‡โˆ’๐‘ก๐‘’โˆ’๐ดโˆ—๐‘ก๐‘’โˆ’๐ดโˆ—๐‘ ๐‘‘๐‘ ๐‘=๐‘’๐ดโˆ—(๐‘‡โˆ’๐‘ก)๎‚ธ๎€œ0๐‘‡โˆ’๐‘ก๐‘’โˆ’๐ดโˆ—๐‘ ๎‚น๐‘‘๐‘ ๐‘.(59)

Example 5. Consider the following optimal control problem: ๎€œ(๐’ซ)min10๎‚ƒ1๐‘ฅโˆ’2๐‘ข2๎‚„๐‘ฅ[],๐‘‘๐‘กs.t.ฬ‡๐‘ฅ=๐‘ฅ+๐‘ข,(0)=0,๐‘กโˆˆ0,1|๐‘ข|โ‰ค1.(60) This is a simple case of (46),(47). We have ๐บ=โˆ’1,๐‘=1,๐‘=0,๐ด=1,๐ต=1,๐‘‡=1. By (59), we have ๐œ†(๐‘ก)=๐‘’1โˆ’๐‘ก๎€œ01โˆ’๐‘ก๐‘’โˆ’๐‘ ๐‘‘๐‘ =๐‘’1โˆ’๐‘กโˆ’1โ‰ 0(๐‘กโ‰ 1).(61) To find an analytic solution of the optimal control problem, we solve the equation (๐œŒโˆ’1)โˆ’2๐œ†2(๐‘ก)=1,๐œŒ>1(62) to get ||||๎€บ๐‘’๐œŒ=1+๐œ†(๐‘ก)=1+1โˆ’๐‘ก๎€ปโˆ’1.(63) By (58), we obtain an analytic solution of the optimal control problem which can be expressed as ฬ‚๐‘ข(๐‘ก)=(๐œŒโˆ’1)โˆ’1[]=๎€ท๐‘’โˆ’๐œ†(๐‘ก)1โˆ’๐‘ก๎€ธโˆ’1โˆ’1๎€บ1โˆ’๐‘’1โˆ’๐‘ก๎€ป=โˆ’1,(๐‘กโ‰ 1).(64)

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 © 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.

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