Finite dimensional compensators for some hyperbolic systems with boundary control

The posslblility of finite dimensional compensators for infinite dimensional systems was first realized by Schumacher in [11],[12], who designed finite dimensional stabilizing schemes via dynamic output ~eedbacK for a large class of systems, including parabolic and delay systems. The main restriction was that the control action and the observation be implemented by bounded operators B and C. For parabolic systems this restriction was eliminated by Curtain in [4], who used a different compensator scheme, but still by dynamic output ~ecdbacK° Hyperbolic systems such as the wave equation cannot be made exponentially stable by finite dimensional state feedback [8], essentially because of the way the eigenvalues cluster along vertical asymptotes. The two schemes [J2],[4] work by shifting finitely many eigenvalues to stabilize the system and it is essential that the original system has finitely many eigenvalues to the right of Re[s) = -a. This is not satisfied by the wave equation, but it is satisfied by some hyperbolic systems used in modelling, for example [6] and [9]. For such hyperbolic systems the approach cf Schumacher is applicable and as noted in [5] the construction of Curtain in [4] can also be applied to stabilize these systems, provided that B and C are bounded. For more background on this, the reader is referred to [5].