Strong stabilization of controlled vibrating systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, p. 1144-1157

This paper deals with feedback stabilization of second order equations of the form ytt + A0y + u (t) B0y (t) = 0, t ∈ [0, +∞[, where A0 is a densely defined positive selfadjoint linear operator on a real Hilbert space H, with compact inverse and B0 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 = ⟨yt, B0yH implies the strong stabilization. This result is derived from a general compactness theorem for semigroup with compact resolvent and solves several open problems.

DOI : https://doi.org/10.1051/cocv/2010041
Classification:  37L05,  43A60,  47D06,  47H20,  93D15
Keywords: precompactness, compact resolvent, almost periodic functions, Fourier series, mild solution, integral solution, control theory, stabilization
@article{COCV_2011__17_4_1144_0,
     author = {Couchouron, Jean-Fran\c cois},
     title = {Strong stabilization of controlled vibrating systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {4},
     year = {2011},
     pages = {1144-1157},
     doi = {10.1051/cocv/2010041},
     zbl = {1254.93082},
     mrnumber = {2859869},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_4_1144_0}
}
Couchouron, Jean-François. Strong stabilization of controlled vibrating systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1144-1157. doi : 10.1051/cocv/2010041. http://www.numdam.org/item/COCV_2011__17_4_1144_0/

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