Global Optimization for Solving Linear Non-quadratic Optimal Control Problems

This paper presents a global optimization approach to solving linear non-quadratic optimal control problems. The main work is to construct a differential flow for finding a global minimizer of the Hamiltonian function over a Euclid space. With the Pontryagin principle, the optimal control is characterized by a function of the adjoint variable and is obtained by solving a Hamiltonian differential boundary value problem. For computing an optimal control, an algorithm for numerical practice is given with the description of an example. In this paper, the notation. represents a norm for the specified space concerned. The primal goal of this paper is to present a solution to the following optimal control problem (primal problem () in short). () () 0 min d T F x P u t +     ∫ (1.1) () [ ] s.t. x Ax Bu x a t T x R u R = + = ∈ ∈ ∈ (1.2) where ()


Primal Problem.
In this paper, the notation .represents a norm for the specified space concerned.The primal goal of this paper is to present a solution to the following optimal control problem (primal problem (  ) in short).where ( ) . In the control system, , A B are given matrices in n n R × and n m R × respectively and α stands for a given vector in n R .We assume that ( ) If ( ) P u is a positive definite quadratic form with respect to u and ( ) F x is a posi- tive semi-definite quadratic form with respect to x, then the problem (  ) is a classical linear-quadratic optimal control problem [1].
The rest of the paper is organized as follows.In Section 2, we focus on Pontryagin principle to yield a family of global optimizations on the adjoint variable.In Section 3, we deal with the global optimization for the Hamiltonian function.In Section 4, we show that there exists an optimal control to the primal (  ) and present a mathe- matical programming.In Section 5 and 6, we discuss how to compute the global minimizer by a differential flow and present an algorithm for the numerical practice with the description of an example.

Pontryagin Principle
Associated with the optimal control problem (  ), let's introduce the Hamiltonian fun- ction with the state and adjoint systems We know from Pontryagin principle [2] that if ( ) u is an optimal control to the problem (  ), then it is an extremal control.Associated with the state variable ( )
Since in (2.6) the global optimization is processed on the variable u over m R for a given t, it is equivalent to deal with the optimization (for obtaining a global minimizer): Therefore we turn to consider the following optimization with respect to a given

Global Optimization
In this section, for a given parameter vector ( ) to create a function It follows from (1.3) that there exist positive numbers β and r such that ( ) It follows from (1.3) that there exist positive numbers β and r, such that, when u r > , ( ) Without loss of generalization, we assume that ( ) Since we have shown above that, for all The lemma has been proved.
Consequently, by Lemma 3.1 we conclude the following lemma.
Lemma 3.2.Let u λ be a minimizer of ( ) ( ) Then, it follows by Lemma 3.2 that u λ is also the unique minimizer of ( ) Therefor u λ is uniquely determined by the equ- ation By elementary calculus [5], the above equation defines an implicit function of the variable λ , denoted by which is continuously dependent of the parameter λ .

Hamiltonian Boundary Value Problem
In this section we solve the following Hamiltonian boundary value problem: Equation ( 4.2) can be rewritten by the integral form Substituting it into Equation (4.1), we have In the following we show that Equation (4.4) has a solution, then together with (4.3) we obtain a solution to Hamiltonian boundary value problem (4.1), (4.2).

Since ( )
F x is twice continuously differentiable on n R , we may define ( ) Consider the ball centered at a in X  (regarding a as a function constantly equal to the vector a): , define an operator : G X Ω → , which acts on each element x ∈ Ω to produce an image Gx satisfying (noting that the integral in (4.4) needs the information of ( ) By an elementary estimation we have, for which implies that Gx ∈ Ω .It is also clear that G is a continuous and compact mapping.
Then by Schauder fixed-point theorem, there is an element x ∈ Ω such that ˆĜx x = .
It follows that ( ) By a traditional approach in the classical theory of ordinary differential equation, we see that the solution is also an optimal control to the optimal control problem (  ).
In other words, in the practice for solving (  ), we only need to compute a solution of the following differential boundary value problem: ( ), We present a numerical method to deal with the differential boundary value Equation (4.9), Equation (4.10) as follows.Define a mesh by dividing the time interval [ ] Consider solving for 0 1 , , , N x x x , with i x the intended approximation of ( ) For the requirement on the adjoint variable ( ) 0 N t λ = (due to the boundary condition of the differential boundary value Equation (4.9), Equation (4.10)), we consider the following difference equation: Solving the differece equation above we can get the valyue 0 λ .According to classical numerical analysis theory, the solution of above difference equation will converge to the solution of differential boundary value problem (4.8) -(4.10).Apparently, we need to compute ( ) h λ numerically.It will be given in next section.

Computing h(λ) by a Differential Flow
In this section we study how to compute ( ) ρ ∈ ∞ .Thus, combining (5.3), ( ) λ ξ ρ is the unique solution of the equation ( ) In what follows we choose a real number * 1 ρ > such that, on Moreover, we have where the positive constant C is only dependent of the parameters λ .In the following there are several times of appearing the character C which may denote different positive constants only dependent of the parameters λ .
It follows from (5.4) that ( ) ( ) Although it is hard to get u λ and * u exactly, we can com- pute numerically another vector instead of * u by the following result.
Theorem 5.1.Let the flow ( ) ξ ρ be got by the following backward differential equation ( ) where the positive constant C is only dependent of the parameters λ and * ρ is selected to be sufficiently large satisfying (5.4), (5.5).
Proof: When * 1 ρ > is sufficiently large, * u is near the origin.In a neighborhood of the origin, by (5.4), we have In the following we need to keep in mind that  In what follows, we give an algorithm to compute ( ) , stop; Otherwise, go to 5); We deal with the following difference equation.

A Description of an Example
Let's consider to solve the following optimal control problem numericaly: this paper, for a given adjoint variable, we solve the optimization (2.8) to create a function ( ) u h λ = .Then in Hamiltonian boundary problem (2.2), (2.3) we replace the variable u with the function ( ) h λ and solve the following equation

(
we deal with the following global optimization problem  [3][4] global problem  is equivalent to the the following global problem *  :

.=
For a given parameter vector n R λ ∈ , we solve the following global optimization problem  In the following we will determine the value of ( ) h λ by a differential flow.Since the Hessen matrix function of ( ) P u is positive definite, by the classical theory of ordinary differential equation, for given since u λ is the minimizer of ( ) T P u Bu λ + over m R (Lemma 3.2).To explain the uniqueness of ( ) λ ξ ρ , we refer to the fact that ) and the uniqueness of the flow ≥ can also be got by the following Cauchy initial value problem the inequality process the value of the constant C has been changed several times but only dependent of given information like , , P B λ .
noting that in the inequality process the value of the constant C takes different positive values which are only dependent of given information like , , the computation practice, we can solve the Cauchy initial problem (5.9) (5.10), instead of (5.8) to get ( ) solving the following Cauchy initial problem

2 .
For the step 3) of above algorithm, we present a numerical method to deal with the Cauchy initial problem as follows.Define a mesh by dividing the time interval [ ]

2 ,
We need to solve the following differential boundary value equation: given positive integers , N M (properly large) and positive real numbers  (properly small), * 1 ρ > , we consider the following difference equation: