Control and Stabilization for the Wave Equation in a Bounded Domain

Abstract

Exact controllability problem of the wave equation is studied by the stabilizability method of D. L. Russell. It is shown that if we use the linear “velocity feedback,” then the energy of the system will decay uniformly exponentially and exact controllability can be achieved. In case the velocity feedback contains a certain nonlinear dissipative term, then we prove that this system evolves as a nonlinear semigroup with respect to time. Certain estimates for the nonlinear equation are also obtained.