Optimal control of system governed by nonlinear volterra integral and fractional derivative equations

Abstract

This work presents a novel formulation for the numerical solution of optimal control problems related to nonlinear Volterra fractional integral equations systems. A spectral approach is implemented based on the new polynomials known as Chelyshkov polynomials. First, the properties of these polynomials are studied to solve the aforementioned problems. The operational matrices and the Galerkin method are used to discretize the continuous optimal control problems. Thereafter, some necessary conditions are defined according to which the optimal solutions of discrete problems converge to the optimal solution of the continuous ones. The applicability of the proposed approach has been illustrated through several examples. In addition, a comparison is made with other methods for showing the accuracy of the proposed one, resulting also in an improved efficiency.

Numerical conservation laws of time fractional diffusion PDEs

“…Moreover, for a suitable choice of the function basis, spectral methods are exponentially convergent [62]. Within this class, we recall methods based on Chelyshkov polynomials [46,48] and on Chebyshev polynomials [49,50]. Pseudo-spectral methods based on fractional Lagrange or Müntz-Legendre polynomials have been introduced for the solution of time fractional PDEs, such as Fokker-Plank equation [31], Klein-Gordon equation [47], Black-Scholes equation [54], and a diffusion equation [32].…”