On the approximation of time optimal control problem with the impulse PDE constraint

This paper is concerned with the finite element approximation of the time optimal control problem with the impulse heat PDE constraint. The discretization is applied both for the state space and the control space. Without any additional assumptions, the error estimate of optimal controls$ \begin{equation*} \|u^*-u_l^*\|_{{\omega}}\le C(h^2+l^2) \end{equation*} $is established, where $ h $ and $ l $ represent mesh sizes of discretizations for the state space and the control space, respectively. Moreover, error estimates of states and adjoint states are also obtained.