Solution to Singular Optimal Control by Backward Differential Flow

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.


Introduction
In this paper, we are interested in solving the following optimal control problem: where  ∈  × ,  ∈  × are constant matrices, the initial state  0 is a constant vector in   ,  ∈   is a constant vector, and () is a polynomial of degree 2 having the form where ( 1 ,  2 , . . .,   ) is a polynomial of degree less than 2.We assume that ∇ 2 () ≥ 0. The problem (P) is a singular optimal control problem.The admissible control function (⋅), taking values in   , is integrable and bounded on [0, ].
The set of all admissible controls is denoted by Φ.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  ∈ [0, ] 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.

Global Minimizers of a Polynomial
Let () be a polynomial over   having the same form as () or ℎ  ().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  > 0, all the global minimizers of the polynomial () stay inside the ball ‖  ‖< , where If further let  > max ∈[0,] ‖   () ‖, then there is a bound  > 0 such that for all  the global minimizers of ℎ  () stay inside ‖  ‖< .Therefore, for  ∈ [0, ] 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.
Lemma 6.For every positive parameter ρ, there is solely a point û 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 û() corresponding to the problem (10) does not depend on changing the initial condition.

Solution to Singular Optimal Control
To solve the singular optimal control problem (P), by Lemma 6 and Theorem 8, we present the following algorithm.
Since () is continuous and the solution of the canonical backward differential equation relies on the parameter  ∈ [0, ] for the initial value continuously, we see that  *  will be dependent of the parameter  ∈ [0, ] continuously.It follows from Krotov extension method that the optimal control of (P) is û() :=  *  ,  ∈ [0, ].

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:

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.

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