Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 3

Solution of two-dimensional optimal control problem using Legendre Block-Pulse polynomial basis

Pages: 1075 – 1094

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n3.a7


S. M. Hosseini (Department of Mathematics, Payame Noor University, Tehran, Iran)

F. Soltanian (Department of Mathematics, Payame Noor University, Tehran, Iran)

K. Mamehrashi (Department of Mathematics, Payame Noor University, Tehran, Iran; and Mathematics Unit, School of Science and Engineering, University of Kurdistan Hewler, Erbil, Kurdistan Region, Iraq)


In this paper, a numerical method is presented for solving a class of two-dimensional optimal control problems using the Ritz method and orthogonal Legendre block-pulse functions.

The most important reason for using the Ritz method is its high flexibility in boundary and initial conditions. 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 into cost functional, we derive an unconstrained optimization problem. By applying optimality conditions for this problem, a system of algebraic equations is obtained. Solving this system, the unknown coefficients and consequently the state and control functions are obtained. At last the convergence of the proposed method is discussed and the accuracy and efficiency of the proposed method is demonstrated in comparison with other methods by providing several examples.


two-dimensional optimal control, Ritz method, Legendre block-pulse, numerical method

2010 Mathematics Subject Classification

Primary 49M37. Secondary 65K10.

Received 20 December 2021

Received revised 13 February 2022

Accepted 23 February 2022

Published 24 July 2022