On exponential stabilizability with arbitrary decay rate for linear systems in Hilbert spaces

In this paper, we deal with linear infinite dimensional systems in Hilbert spaces. In the beginning, systems are continuous. In the second part, we use a sampling to transform our first system in a discrete-time system. For these systems, exact controllability implies that the extended controllability gramian is uniformly positive definite. This operator allows us to define a feedback control law which stabilizes exponentially the closed-loop system. In the first case, we want to push eigenvalues in the negative part of the complex plane and to have an arbitrary decay rate. In the discrete-time system, we want to push eigenvalues of the closed loop system in the unit ball that is to say to minimize the norm of the closed loop operator. Results of continuous-time system are applied to a system described by a wave equation.