A Spectral Method for Optimal Control Problems Governed by the Time Fractional Diffusion Equation with Control Constraints

Abstract

In this paper, we study the fractional optimal control problem and its spectral approximation. The problem under investigation consists in finding the optimal solution governed by the time fractional diffusion equation with constraints on the control variable. We construct a suitable weak formulation, study its wellposedness, and design a Galerkin spectral method for its numerical solution. The main contribution of the paper includes: (1) a priori error estimates for the spacetime spectral approximation is derived; (2) a projection gradient algorithm is designed to efficiently solve the discrete minimization problem; (3) some numerical experiments are carried out to confirm the efficiency of the proposed method. The obtained numerical results show that the convergence is exponential for smooth exact solutions.

A solution for fractional PDE constrained optimization problems using reduced basis method

“…In Otarola (2017) a piecewise linear FEM was proposed for an optimal control problem involving the fractional powers of a symmetric and uniformly elliptic second order operator. A spectral method for optimal control problems governed by the time fractional diffusion equation with control constraints was presented in Ye and Xu (2014). Authors in Bhrawy et al (2016) provided a space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation.…”

A Space-Time Fractional Optimal Control Problem: Analysis and Discretization

“…Recently, convergence of spectral based techniques has been explored in [22,23] for an optimization problem restricted to fractional order ODEs. Optimal control problems for one dimensional evolution equations with only fractional time derivatives have been recently studied in [37,38]. In these references, the authors derive rigorously first order necessary optimality conditions, propose numerical schemes based on spectral methods and obtain a priori error estimates.…”