Numerical Solution of a Class of Nonlinear Optimal Control Problems Using Linearization and Discretization

In this paper, a new approach using linear combination property of intervals and discretization is proposed to solve a class of nonlinear optimal control problems, containing a nonlinear system and linear functional, in three phases. In the first phase, using linear combination property of intervals, changes nonlinear system to an equivalent linear system, in the second phase, using discretization method, the attained problem is converted to a linear programming problem, and in the third phase, the latter problem will be solved by linear programming methods. In addition, efficiency of our approach is confirmed by some numerical examples.

The optimal control problems we consider consist of 1) State variables, i.e., variables that describe the system being modeled; 2) Control variables, i.e., variables at our disposal that can be used to affect the state variables; 3) A state system, i.e., ordinary differential equations relating the state and control variables; 4) A functional of the state variables whose minimization is the goal.
Then, the problems we consider consist of finding state and control variables that minimize the given func-tional subject to the state system being satisfied.Here, we restrict attention to nonlinear state systems and to linear functionals.
The approach we have described for finding approximate solutions of optimal control problems for ordinary diffrential equations is of the linearize-then-discretizethen-optimize type.Now, consider the following subclass of nonlinear optimal control problems: where ( ) Here, we use the linear combination property of intervals to convert the nonlinear dy-namical control system (2) to the equivalent linear system.The new optimal control problem with this linear dynamical control system is transformed to a discretetime problem that could be solved by linear programming methods (e.g.simplex method).
There exist some systems containing non-smooth function ( ) .,. h with regard to control variables.In such systems, multiple shooting methods [1][2][3][4] do not dealing with the problem in a correct way.Because, in these methods needing to computation of gradients and hessians of function ( ) .,. h is necessary.However, considering of non-smoothness of function ( ) .,. h could not make any difficulty in our approach.Moreover, in another approaches (see [11,12] ), which discretization methods are the major basis of them, if a complicated function ( ) .,. h is chosen, obtaining an optimal solution seems to be difficult.Here, we show that our strategy acquire better solutions, that attained in fewer time, than one of the abovementioned methods through several simplistic examples, which comparison of the solutions is included in each example.
This paper is organized as follows.Section 2, transforms the nonlinear ( ) .,. h to a corresponding function That is linear with respect to a new control variable.In Section 3, the new problem is converted to a discretetime problem via discretization.In Section 4, numerical examples are presented to illustrate the effectiveness of this proposed method.Finally conclusions are given in Section 5.

Linearization
In this section, problems (1)-( 2) is transformed to an equivalent linear problem.First, we state and prove the following two theorems: Theorem 2.1: be a continuous function where U is a compact and connected subset of m  , then for any arbitrary (but fixed) Since continuous functions preserve compactness and connectedness properties, is compact and connected in  .There- fore  , suppose that the lower and upper bounds of the closed interval ( ) i w t respectively.Thus for 1, 2, , In other words , , .

Discrete-Time Problem
Now, discretization method enables us transforming continuous problem (7) to the corresponding discrete form.Consider equidistance points 0 0 1 2  which defined as where N is a given large number.We use the trapezoidal approximation in numerical integration and the following approximations to change problem (7) to the corresponding discrete form: ( ) ( ) ( ) , where . By solving problem (8), which is a linear programming problem, we are able to obtain optimal solutions kj λ * and kj x * for all Remark 3.1: The most important reason of LCPI (linear combination property of intervals) consideration is that problem ( 8) is an (finite-dimensional) LP problem and has at least a global optimal solution (by the assumptions of the problems (1)-( 2)).However, if problems (1)-(2) be discretized directly then, we reach to an NLP problem which its optimal solution may be a local solution.

Numerical Examples
Here, we use our approach to obtain approximate optimal solutions of the following three nonlinear optimal control problems by solving linear programming (LP) problem (8), via simplex method [20].All the problems are programmed in MATLAB and run on a PC with 1.8 GHz and 1GB RAM.Moreover, comparisons of our solutions with the method that argued in [11] are included in Tables 1, 2 and 3 respectively for each example.
Example 4.1: Consider the following nonlinear optimal control problem: Here, ( ) ( ) Thus by ( 4) and ( 5) for all       The optimal controls * j u , 0,1, 2, ,100 j =  of problem ( 12) is shown in Figure 9. Here, the value of optimal solution of objective function is -0.0435.

Conclusions
In this paper, we proposed a different approach for solving a class of nonlinear optimal control problems which have a linear functional and nonlinear dynamical control system.In our approach, the linear combination property of intervals is used to obtain the new corresponding problem which is a linear optimal control problem.The new problem can be converted to an LP problem by discretezation method.Finally, we obtain an approximate solution for the main problem.By the approach of this paper we may solve a wide class of nonlinear optimal control problems.

2 :
Of problem(10) is obtained by solving problem(8) which is illustrated in Figures1 and 2respectively.Here, the value of optimal solution of objective function is 0.0977.In addition, the corresponding Equation (9) of this example is Consider the following nonlinear optimal control problem: