Numerical approximation for a time optimal control problems governed by semi-linear heat equations

In this paper, we study the optimal time for a time optimal control problem ( P ) , governed by an internally controlled semi-linear heat equation. By projecting the original problem via the finite element method, we obtain another time optimal control problem ( P h ) governed by a semi-linear system of ordinary differential equations. Here, h is the mesh sizes of the finite element spaces. The purpose of this study is to approach the optimal time for the problem ( P ) through the optimal time for the problem ( P h ) . We obtain error estimates between the optimal times in terms of h.