Stabilization of hyperbolic systems using concentrated sensors and actuators

Certain hyperbolic systems of partial differential equations which are known to be uniformly asymptotically stabilizable using point sensors/actuators (S/A) are considered. The issue to be investigated is the effect on stability when point S/A's are replaced by "concentrated" S/ A's, that is, S/A's which average over small regions of the spatial domain. Although it is known that passing from point to concentrated S/ A's necessarily destroys uniform stability, a necessary and sufficient condition for strong stability is obtained in terms of the S/A weighting functions. In addition, in the special case of a cantilevered beam controlled by a single sensor/actuator pair concentrated at the free end, another, more robust type of stability is shown to hold, even when strong stability does not. The latter result shows that the system energy is bounded by a part which goes uniformly to zero at infinity and a residual which can be explicitly estimated in terms of the support of the weight functions and the initial energy. Furthermore, the residual energy converges to zero as the support reduces to the point at the free end of the beam.