Legendre pseudo-spectral method for optimal control problem governed by a time-fractional diffusion equation

ABSTRACT

This paper presents a numerical scheme for optimal control problem governed by a time-fractional diffusion equation based on a Legendre pseudo-spectral method for space discretization and a finite difference method for time discretization. Lagrange interpolating basis polynomials are used to approximate the state, and the differentiation matrix is derived to discrete the spatial derivative. We also discuss the fully discrete scheme for the control problem. A finite difference method developed in Lin and Xu [Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), pp. 1533–1552] is used to discretize the time-fractional derivative. A fully discrete first-order optimality condition is developed based on the ‘first discretize, then optimize’ approach. Furthermore, we design the projected gradient algorithm based on the fully discrete optimality conditions. Numerical examples are given to illustrate the feasibility of the proposed method.

KEYWORDS:

• Legendre pseudo-spectral method
• optimal control problem
• time-fractional diffusion equation
• optimality conditions
• projected gradient algorithm

2010 AMS SUBJECT CLASSIFICATION:

• 65N30

Acknowledgements

The authors are very grateful to the referees for their constructive comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported by National Natural Science Foundation of China [grant numbers 11301311,11471196] and Natural Science Foundation of Shandong Province [grant number ZR2016JL004].