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Strong stabilization of controlled vibrating systems. (English) Zbl 1254.93082
Summary: This paper deals with feedback stabilization of second order equations of the form \[ y_{tt} + A_0y + u(t) B_0y (t) = 0,\quad t \in [0, +\infty[, \] where \(A_0\) is a densely defined positive selfadjoint linear operator on a real Hilbert space \(H\), with compact inverse and \(B_0\) is a linear map in diagonal form. It is proved here that the classical sufficient ad-condition of Jurdjevic-Quinn and Ball-Slemrod with the feedback control \(u = \langle y_t, B_0y \rangle _H\) implies the strong stabilization. This result is derived from a general compactness theorem for semigroups with compact resolvent and solves several open problems.

MSC:
93C10 Nonlinear systems in control theory
47H20 Semigroups of nonlinear operators
93D15 Stabilization of systems by feedback
93C25 Control/observation systems in abstract spaces
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References:
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