On time-minimal distributed control of vibrations

Abstract

In [1] Fattorini investigates distributed control of vibrations governed by an abstract wave equation and influenced by controls which are Hilbert-space-valuedL ^∞-functions of time and uniformly bounded with respect to the maximum norm. He shows that null-controllability is possible with such controls for sufficiently large times and that there is a time-minimal bounded control which is unique on the minimum time interval and satisfies a strong “bang-bang” principle. We show that null-controllability is possible for every positive time by anL ^∞-control function whose essential supremum norm is as small as possible, which is unique on the given time interval, and satisfies a strong “bang-bang” principle. We further show that the timeminimal bounded control is the minimum norm control on the minimum time interval and is characterized by this property.