Numerical Solution for A Special Class of optimal Control Problem by using Hermite polynomial

- In this paper, a numerical solution for solving a special class of optimal control problems is considered. The main idea of the solution is to parameterize the state space by approximating the state function using a linear combination of Hermite polynomial with unknown coefficients an iterative method is proposed in order to facilitate the computation of unknown coefficients. Some illustrated examples are included to test the efficiency of algorithm.


Introduction
Optimal control has many applications in every area of science and engineering. And has been studied by many researches [1][2][3][4].
Since the analytic solution is not always available for optimal control problems, therefor a numerical solution must be found. Numerical methods for solving optimal control problem are vary in their approach and complexity. in [5] ,the authors suggested a new algorithm for solving optimal control problems and controlled duffing oscillator using Chebyshev polynomial as a basis function. While numerical solution for solving optimal control problems based on state parameterization technique were consider in [6] and [7]. Furthermore the fundamental of control parameterization method and solving its various applications were introduced in [8]. In addition, control parameterization technique for discrete value control problems was considered in [9].
In recent year different approximate methods and many algorithms has been introduced to solve the optimal control problems [10][11][12][13]. The organization of this paper is presented into the following sections. In section 2 the Hermite polynomial which are used as a basis function are reviewed briefly. Section 3, is about mathematical formulation of optimal control problem.in section4, the proposed algorithm is derived. While section 5 includes numerical example and results. Finally, the paper is concluded in section 6.

Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arises in probability, such as the Edgeworth series, in combinatorics, as an example of an Appell sequence, obeying the umbral calculus, and in physics, where they give rise to the Eigenstates of the quantum harmonic oscillator. They are named in honor of Charles Hermite." " in a sense to be described below, of the form ( ) ( ) For n=1,2,3,….. The first four Hermite polynomials are ( ) ( ) ( ) ( ) ( ) " 1-1 Definition: "For n ∈ N, we define Hermite polynomials To find ( ) expand the right hand side of (1) as a Maclaurin series in r and equate coefficients. From Equation (1) we derive the closed expression Where ⌊ ⌋ denoted the largest integer less than or equal to t .checking with n=0,1,2,…. We find that (2) yields the expected Hermite polynomials."

Mathematical formulate
The process illustrated by the following system of nonlinear differential equation on the final time is the control variable, and f is a real valued continuously differential function. Along with the controlled process (3-4) a cost functional of the form There are admissible control are always assume that pass through (0,x 0 )and (1,x 1 ) and in the set of controls, the control variable is searched which minimizes J and call it optimal control.

The proposed algorithm
The following approximate for ( ) Is first consider which is in terms the Hermite polynomials ( ) ( ) ( ) ( ) ( ) (6) Using the boundary condition (4), yields: (7) By substitution of (7) into (6), we obtain The control variable u(t) are then obtained using eq.(3). Then, substituting x 1 (t) and u(t), we obtain J as a function of a 2 . The solution of the optimal control problem (3-4) is J(a * ) (a * is the value which minimizes J(a 2 )). The state and control variables are also found from a * approximately. In the second step, the following approximated is use From (10-11) we have (12) In this case the solution of optimal control problem (3-4)is J(a * ) where a * is the value which minimizes J(a 3 ). In general, the approximate solution in the n th step will be Form the second condition of (4) we obtained ( ) ( ) ( ) (15) We solve the equation (14) and (15) simultaneously to obtain and as a function of as follows: Multiply eq. (14) and (15)by ( ) ( ) respectively ,yields: From the above equations ,one can get The denominator in Eq. (16) and (17) Therefore ( ) ( ) ( ) ( ) The proposed algorithm can be summarized by the following steps: Step 1: Choose an Step 2: For n=1 , calculate : ( ) ( ) ( ) ( ) ( ) ( )And then calculate a 2.

Numerical Examples
The efficiency of the proposed algorithms is the illustrated by same examples which have analytical solutions, so that the validation of the method can be allowed by comparing with the results of the exact solution.

Conclusion
The proposed algorithm for treating optimal control problem depending on Hermite polynomial and their propertied provided a simple way to obtain an optimal control with fast convergence. [9] L. H. and T. K., "Control parameterization enhancing technique for optimal discrete -valued control problems", Automatica Vol.35, pp.1401-1407, 1999.