Bounded feedback stabilization and global separation principle of distributed parameter systems

We show that the infinite-dimensional system Σ:x˙(t)=Ax(t)+Bu(t), x₀ ε H is globally strongly asymptotically stabilizable by an arbitrarily small smooth feedback. Here, the operator A is the infinitesimal generator of a C₀ semigroup of contractions e^tA on real Hilbert space H and B is a bounded linear operator mapping a Hilbert space of controls U into H. An explicit smooth feedback control law is given. Further, we identify the class of perturbations for which the system is still stabilizable by the same feedback law as for the nominal system. Based on these results and some differential Lyapunov operator equations, we then establish a global separation principle for the system Σ with a Kalman-like observer. Finally, these results are illustrated via an example dealing with the wave equation