Controllability of the Wave Equation

Abstract

Results on controllability of systems with persistent memory have been derived appealing to the corresponding results of the (memoryless) wave equation. For this reason, in this short chapter, we review the key results on the controllability of wave type equations. First, we describe a “hidden regularity” of the normal derivative of the solutions of the uncontrolled system and the observation inequality. A consequence is that the “active part” $\varGamma$ of the boundary must be “large” also in terms of the trace of the eigenvectors of $A$ on $\varGamma$. Finally, we characterize exact controllability in terms of suitable Riesz sequences.