Stabilisation and polynomial decay estimate for distributed semilinear systems

Abstract

In this article, we propose a family of feedback controls that guarantee weak and strong stabilisability for distributed semilinear systems. In the case of strong stabilisation, an optimal decay of the state is estimated. Applications to wave equations are provided.

Strong stabilisation and decay estimate for unbounded bilinear systems

“…In the case where B is bounded so that BH & H, the problem of stabilisation by quadratic feedback control has been studied by many authors (Ball and Slemrod 1979;Berrahmoune 1999;Bounit and Hammouri 1999;Ouzahra 2008). In Ball and Slemrod (1979) a result of weak stabilisation was obtained under the following condition: …”

“…then (Berrahmoune 1999;Ouzahra 2008) the control (4) is a strongly stabilising one with the decay estimate…”

Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

“…Then using the techniques as in [15], we can obtain the result of Theorem 3.1 if (4) is changed to (37). 6.…”

Constrained Feedback Stabilization for Bilinear Parabolic Systems

“…However, in this way the convergence of the resulting closed loop state is not better than (8). In [7] the rational decay rates are established i.e. using the following feedback control:…”

“…in H , then the system (1) is strongly stable with the explicit decay estimate (8), using the control (9), provided that the estimate (7) holds. Here, we will establish an explicit decay estimate of the stabilized state and the robustness of the control (9) for a large class of bilinear systems as considered in [3] [8] [9].…”