# 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/221938>.

@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},

month = {11},

number = {4},

pages = {1144-1157},

publisher = {EDP Sciences},

title = {Strong stabilization of controlled vibrating systems},

url = {http://eudml.org/doc/221938},

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

DA - 2011/11//

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/221938

ER -

## References

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