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

In this paper, we provided a numerical method to solve a class of two dimensional time-delay optimal control problems (2DTDOCPs) with quadratic cost functional using Ritz method and orthogonal Legendre Block-Pulse functions. First, the state and control vectors are approximated as a series of hybrid functions(block-pulse functions and Legendre polynomials) with unknown coefficients. Then, we derive an equation with unknown coefficients by substituting these approximations in the cost functional. A system of algebraic equations is obtained by applying the optimal conditions for this equation. Solving this system and substituting the coefficients into approximating 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 proposed method is discussed, and finally, we will present some examples to show the validity and applicability of proposed method, and evaluate its accuracy and efficiency. Moreover, our results are compared to previous results to show the superiority of this technique.