Optimal a priori error estimates of parabolic optimal control problems with a moving point control
For researchers in numerical analysis of optimal control problems, this paper corrects an error in prior work and provides rigorous error estimates for a challenging problem involving moving point sources.
This paper provides optimal a priori error estimates for a parabolic optimal control problem with a moving point control, correcting a flaw in a previous analysis. The discretization uses piecewise constant functions in time and continuous piecewise linear finite elements in space, achieving optimal convergence rates modulo logarithmic terms.
In this paper we consider a parabolic optimal control problem with a Dirac type control with moving point source in two space dimensions. We discretize the problem with piecewise constant functions in time and continuous piecewise linear finite elements in space. For this discretization we show optimal order of convergence with respect to the time and the space discretization parameters modulo some logarithmic terms. Error analysis for the same problem was carried out in the recent paper [17], however, the analysis there contains a serious flaw. One of the main goals of this paper is to provide the correct proof. The main ingredients of our analysis are the global and local error estimates on a curve, that have an independent interest.