Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions (Q1116012)

The paper studies the problem of local exponential stabilizability of abstract systems of the form \[ dy/dt+Ay+B_ 1Fy+B_ 2G(y)=0, \] where F, \(B_ i\) are linear, bounded or not, and G is a nonlinear perturbation. Conditions on the data are given in order to preserve local exponential stability for a large class of perturbations. The major novelty here is that unbounded perturbations can be taken into account. Then the choice of the feedback operator F becomes more critical. General results, both for bounded and unbounded feedback operators F, are given. In the case of bounded perturbation or analytic semigroups, roughly speaking, the principle of linearized stability works. Applications to the wave equation with interior or boundary stabilizing feedback, in the last case acting either in a Dirichlet or Neumann condition, are set up. A high degree of unboundedness is then needed. An application to parabolic problems with boundary stabilization on Dirichlet or Neumann conditions illustrates the ``analytic'' case.