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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.

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