Optimal control for a variable-order diffusion-wave equation with a reaction term; A numerical study

Abstract

In this paper, optimal control for a variable-order diffusion-wave equation with a reaction term is numerically studied, where the variable-order operator is defined in the sense of Caputo proportional constant. Necessary optimality conditions for the control problem are derived. Existence and uniqueness for the solutions of fractional optimal control problem are derived. The nonstandard weighted average finite difference method and the nonstandard leap-frog method are developed to study numerically the proposed problem. Moreover, the stability analysis of the methods is proved. Finally, in order to characterise the memory property of the proposed model, three test examples are given. It is found that the nonstandard weighted average finite difference method can be applied to study such variable-order fractional optimal control problems simply and effectively.

Keywords

• Lagrange method for the optimal control of partial differential equations
• Optimality system
• Diffusion-wave equation with a reaction term
• Variable-order fractional derivatives
• Constant-proportional-Caputo hybrid fractional operator
• Nonstandard leap-frog method