On the numerical scheme of a 2D optimal control problem with the hyperbolic system via Bernstein polynomial basis

Abstract

A numerical method for solving a 2D optimal control problem (2DOCP) governed by a linear time-varying constraint is presented in this paper. The method is based upon the Bernstein polynomial basis. The properties of Bernstein polynomial functions are presented. These properties, together with the Ritz method, are then utilized to reduce the given 2DOCP to the solution of an algebraic system of equations. By solving this system, the solution of the proposed problem is achieved. The main advantage of this scheme is that the approximate solutions satisfy all initial and boundary conditions of the problem. We extensively discuss the convergence of the method. Finally, an illustrative example is included to demonstrate the validity and applicability of the new technique.

On two‐dimensional weakly singular fractional partial integro‐differential equations and dual Bernstein polynomials

“…▪ Theorem 2. 16 The operational matrix of integration based of integer order on the two-dimensional BPs vector defined in Equation ( 12) for variables u and v are as follows:…”

“…BPS AND THEIR PROPERTIES Definition 2.1 (4,16,18). The Bernstein basis polynomials of degree 𝑑 in the interval [0, 1] are defined as:…”

Variational problem containing psi‐RL complex‐order fractional derivatives

“…The computational method based on Legendre wavelets (LWs), the Ritz spectral, and the shifted Legendre orthogonal polynomials for fractional optimal control problems (FOCPs) are developed in [9][10][11] respectively. [10,[12][13][14][15][16] present several methods to solve fractional optimal control problems, including the Ritz method, Laguerre functions, and the Bernstein polynomial basis. Complex-order fractional derivatives have also attracted attention.…”