The Ritz numerical method and hybrid functions (block-pulse functions and Legendre polynomials) for a class of two-dimensional time-delay optimal control problems

Abstract

In this paper, we provide a numerical method for solving a class of two dimensional time-delay optimal control problems (2DTDOCPs) with quadratic cost functional using the Ritz method and orthogonal Legendre Block-Pulse functions. First, the state and control vectors are approximated as a series of hybrid orthogonal Legendre Block-Pulse functions with unknown coefficients. Then, by substituting these approximations in the cost functional, we derive an equation with unknown coefficients. By applying the optimal conditions for this equation, a system of algebraic equations is obtained. Solving this system and substituting the coefficients into the approximation of the guessed functions, the state and control functions are obtained. By increasing the number of blocks as well as the basic functions, we get more accurate solutions. The convergence of the proposed method is discussed and finally, we will present some examples to demonstrate the validity and applicability of the proposed method and evaluate its accuracy and efficiency. Moreover, our results are compared with the previous results to show the superiority of this technique.