Stabilization of Second Order Evolution Equations with Unbounded Feedback with Delay

Abstract

We now turn to problems with delays, namely in the same Hilbert setting than in the previous chapter we consider the closed loop system (5): $\displaystyle{ \left \{\begin{array}{c} x^{{\prime\prime}}(t) + \mathit{Ax}(t) + B_{1}B_{1}^{{\ast}}x^{{\prime}}(t) + B_{2}B_{2}^{{\ast}}x^{{\prime}}(t-\tau ) = 0,\,t > 0 \\ x(0) = x^{0},\,x^{{\prime}}(0) = x^{1}, \\ B_{2}^{{\ast}}x^{{\prime}}(t-\tau ) = f^{0}(t-\tau ),\,0 < t <\tau.\end{array} \right. }$