The cost of approximate controllability for heat equations: the linear case

Abstract

We consider linear heat equations in a bounded domain of $\mathbb R^{d}$ with Dirichlet boundary conditions. We analyze the problem of controllability when the control acts on a (small) open subset of the domain. It is by now well known that the system is approximately controllable, null-controllable and also finite-approximately controllable. This last property means that there exist controls by means of which we can simultaneously guarantee the approximate controllability and the exact controllability of a projection of the solution over a finite dimensional subspace. In this paper we obtain explicit bounds of the cost of approximate controllability, i.e., of the minimal norm of a control needed to control the system approximately. We also address the problem of simultaneous finite-approximate controllability. The methods we use combine global Carleman estimates, energy estimates for parabolic equations and the variational approach to approximate controllability. In the case of the constant coefficient heat equation, following a different approach, we are able to obtain better bounds. We also show that, in this particular case, the estimates are sharp. As a consequence of our estimates, we can determine the speed of convergence of the limiting process in which the approximate control is obtained through a sequence of penalized optimal control problems.