# Strong stabilization of controlled vibrating systems

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 4, page 1144-1157
- ISSN: 1292-8119

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topCouchouron, Jean-François. "Strong stabilization of controlled vibrating systems." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1144-1157. <http://eudml.org/doc/272894>.

@article{Couchouron2011,

abstract = {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, B0y⟩H implies the strong stabilization. This result is derived from a general compactness theorem for semigroup with compact resolvent and solves several open problems.},

author = {Couchouron, Jean-François},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {precompactness; compact resolvent; almost periodic functions; Fourier series; mild solution; integral solution; control theory; stabilization; bilinear systems in operator form; feedback stabilization; Jurdjevic-Quinn and Ball-Slemrod ad-condition},

language = {eng},

number = {4},

pages = {1144-1157},

publisher = {EDP-Sciences},

title = {Strong stabilization of controlled vibrating systems},

url = {http://eudml.org/doc/272894},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Couchouron, Jean-François

TI - Strong stabilization of controlled vibrating systems

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2011

PB - EDP-Sciences

VL - 17

IS - 4

SP - 1144

EP - 1157

AB - 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, B0y⟩H implies the strong stabilization. This result is derived from a general compactness theorem for semigroup with compact resolvent and solves several open problems.

LA - eng

KW - precompactness; compact resolvent; almost periodic functions; Fourier series; mild solution; integral solution; control theory; stabilization; bilinear systems in operator form; feedback stabilization; Jurdjevic-Quinn and Ball-Slemrod ad-condition

UR - http://eudml.org/doc/272894

ER -

## References

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