On the cost of controlling unstable systems: The case of boundary controls

Abstract

We discuss the cost of controlling parabolic equations of the formy _t + δ² y +kδy = 0 in a bounded smooth domain Ώ ofℝ ^d by means of a boundary control. More precisely, we are interested in the cost of controlling from zero initial state to a given final state (in a suitable approximate sense) at timeT > 0 and in the behavior of this cost ask → ∞. We introduce finite-dimensional Galerkin approximations and prove that they are exactly controllable. Moreover, we also prove that the cost of controlling converges exponentially to zero ask → ∞. This proves, roughly speaking, that when the system becomes more unstable it is easier to control. The system under consideration does not admit a variational formulation. Thus, in order to introduce its Galerkin approximation, we first approximate it by means of a singular perturbation. We also develop a method for the construction of special Galerkin bases well adapted to the control problem.