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Stabilization of Second Order Evolution Equations with Unbounded Feedback with Delay

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2124)

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): \(\displaystyle{ \left \{\begin{array}{c} x^{{\prime\prime}}(t) + \mathit{Ax}(t) + B_{1}B_{1}^{{\ast}}x^{{\prime}}(t) + B_{2}B_{2}^{{\ast}}x^{{\prime}}(t-\tau ) = 0,\,t > 0 \\ x(0) = x^{0},\,x^{{\prime}}(0) = x^{1}, \\ B_{2}^{{\ast}}x^{{\prime}}(t-\tau ) = f^{0}(t-\tau ),\,0 < t <\tau.\end{array} \right. }\)

Keywords

  • Unbounded Feedbacks
  • Order Evolution Equations
  • Hilbert Setting
  • Closed-loop System
  • Explicit Decay Rate

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Ammari, K., Nicaise, S. (2015). Stabilization of Second Order Evolution Equations with Unbounded Feedback with Delay. In: Stabilization of Elastic Systems by Collocated Feedback. Lecture Notes in Mathematics, vol 2124. Springer, Cham. https://doi.org/10.1007/978-3-319-10900-8_3

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