An Iterative Method for Solving Quadratic Optimal Control Problem Using Scaling Boubaker Polynomials

  Abstract     In this paper, an iteration method was used for solving a quadratic optimal control problem (QOCP) by the aid of state parameterization technique and scaling Boubaker polynomials. Some numerical examples were added to show the applicability of the method, also a comparison with other method was presented. The process steps were illustrated by some numerical examples with graphs done by Matlab.


Introduction
Optimal control represents a large field in which many researchers gave different methods of various aspects. The crucial aim for solving an optimal control problem (OCP) is to find the control variable which minimizes a given performance index while all the given constraints are satisfied. State parameterization method is one of the mostly used direct methods in solving (OCPs) by the researchers. Kafash B., Delavarkhalafi A. and Karbassi S. M., used state parameterization for solving (NOCPs) and the controlled Duffing Oscillator [1]. Ouda E. H., utilized generalized Laguerre polynomials as a basis function with the aid of state-control parameterization to find an approximate solution for (OCPs) [2].
Scaling functions are a functional dilation equation has the general following form In this paper, an iterative method was used for solving a quadratic optimal control problem (QOCP) by the aid of state parameterization technique and scaling Boubaker polynomials.
Some numerical examples were added to show the applicability of the method, also a comparison with other method was presented. The process steps were illustrated by some numerical examples with graphs done by Matlab.
With a nonzero solution, this kind of equations has been used in many fields (e.g., interpolating subdivision schemes and wavelet theory) [3]. Yousfi S. A., presented a numerical solution of Emden-Fowler equations using Legendre scaling function approximation [4]. Ouda E. and Ahmed I. utilized direct methods (state and control Parameterization) with the scaling Boubaker function for solving OCP [5].
Iterative technique was also largely used for solving OCPs in last decades, Keyanpour M. and Azizsefat M., had an iterative approach with a hybrid of perturbation and parameterized methods for this purpose [6]. Ramezanpour H. et al., presented a new procedure depending on homotopy perturbation method with iterative technique to solve an optimal control problem of a bilinear systems [7]. Elaydi H., Jaddu H. and Wadi M., utilized Legendre scaling function with iterative technique for solving NOCP [8]. Jaddu H. and Majdalawi A., presented an iterative technique in two proceedings, firstly by parameterizing the state variables by finite length Chebyshev series [9], secondly by a finite length Legendre polynomials [10].
Eskandri M. et al., introduced a method for solving a class of nonlinear quadratic optimal control problem (NQOCP) based on variational iteration method [11]. Ramezani M., proposed a new iterative method utilizing 2nd kind Chebyshev wavelet [12]. Shihab S. and Delphi M., used the iterative technique on B-spline Bernestein polynomials [13].
Boubaker polynomials are proved to be a good tool for solving (OCPs), Samaa F. et al., used indirect method based on Boubaker polynomial [14]. Ouda E. H., deduced the operational matrices of derivative and integration and using them with the indirect method [15]. Many other researchers deal with this kind of polynomials in different proceedings. The novelty of our approach is using scaling Boubaker polynomials for solving (QOCP's), this method was proved to be efficient and accurate. A comparison was introduced to show the capability of this method with some other methods.
This paper is arranged as follows, in section2, Boubaker polynomials have been presented. In section3, scaling Boubaker polynomials were introduced. In section 4, the proposed method was presented in steps. In section 5, some numerical examples with comparison for the first example and illustrative figures were added to show the capability of this method, at the end conclusion and the references.

Boubaker polynomials
The Boubaker polynomials were established for the first by Boubaker et al. as a tool for solving heat equation inside physical model, and then it was used for solving different equations in many applications. [16] Boubaker polynomial is introduced by the following equation [17] () ( 4 ) 2 ( ) Scaling boubaker polynomials (SBP) The Scaling Boubaker polynomials (SBP), can be defined as follows [18] The arguments of scaling (k, n, m, t), k is positive integer, n = 0, 1, 2, 3,..., 2k, m is degree of Boubaker polynomials and t is the time.
Choosing k=1 and m=5. The first five terms Scaling Boubaker SBm(t) were found by using (3) to be: , The convergence of this method with state parameterization technique has been treated in [2]. From table1, we noticed that our proposed method has less absolute error with respect to Delphi and Mehne in which power and Bernestein polynomials have been used.

Conclusion
An iterative method with state parameterization technique using scaling Boubaker polynomials was proved to be a good tool for evaluating the optimal solution of (QOCP) by its rapid convergence and simplicity. The numerical examples show the applicability and accuracy of this method, also comparison with other methods proves its efficiency.