Numerical solution of 2D fractional optimal control problems by the spectral method along with Bernstein operational matrix

摘要

This paper presents an approximate method to solve a class of two-dimensional fractional optimal control problems with nonlinear dynamical system. To implement the new method, by considering the initial-boundary conditions, the unknown state and control functions are approximated by the Bernstein polynomials (B-polynomials) basis using spectral Ritz method, then the problem is reduced to an unconstrained nonlinear optimisation problem. Meanwhile, to reduce computational complexity, a new fractional operational Bernstein matrix generalised on an arbitrary interval is constructed and applied. The choice of polynomial basis functions along with the Ritz method provides good flexibility in which all the given initial and boundary conditions are imposed. At last, we extensively argue the convergence of the new method and several illustrative test problems are added to demonstrate the applicability and effectiveness of the new procedure. Moreover, our achievements are compared with the previous results to show the superiority of the proposed method.