Optimal control of a linear time-invariant space–time fractional diffusion process

Abstract

This paper presents a formulation and numerical solutions of an optimal control problem of a linear time-invariant space–time fractional diffusion equation. The main aim of this formulation is minimization of a performance index, which is a functional of both state and control functions of the diffusion system. The dynamics of the system are defined by the space–time fractional diffusion equation in the sense of Caputo and fractional Laplacian operators. The separation of variables technique and a spectral representation of a fractional Laplacian operator are applied to determine the eigenfunctions that represent the space parameters. Therefore, the state and control functions are defined by linear infinite combinations of eigenfunctions. Optimality conditions described by Euler–Lagrange equations are found by using a Lagrange multiplier technique. The Grünwald–Letnikov definition is used to approximate to the time fractional derivative. The applicapability and effectiveness of the numerical scheme are shown by comparison of analytical and numerical solutions for a numerical example. Finally, the variations of problem parameters are analyzed, with some figures obtained using MATLAB.