Numerical method for the time optimal control problem governed by the Benjamin-Bona-Mahony equation

Abstract

This paper deals with the numerical approximation for the time optimal control problem governed by the Benjamin-Bona-Mahony BBM equation, which is an unspecified terminal time problem. Firstly, by projecting the original problem with the finite element method FEM, another approximate problem governed by a system of ordinary differential equations will be obtained. Then, the parameterisation method for the optimal time and the control function will be carried out and the unspecified terminal time problem can be reduced to an optimal parameter selection problem with a fixed time horizon [0, 1]. This optimal parameter selection problem is a standard nonlinear mathematical programming problem and can be solved by sequential quadratic programming SQP algorithm. Finally, some numerical simulation studies will be given to illustrate the effectiveness of our numerical approximation method for the time optimal control problem governed by the BBM equation.