Strong stabilization of controlled vibrating systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 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 : 10.1051/cocv/2010041
Classification : 37L05, 43A60, 47D06, 47H20, 93D15
Mots clés : precompactness, compact resolvent, almost periodic functions, Fourier series, mild solution, integral solution, control theory, stabilization
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     title = {Strong stabilization of controlled vibrating systems},
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Couchouron, Jean-François. Strong stabilization of controlled vibrating systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1144-1157. doi : 10.1051/cocv/2010041. http://www.numdam.org/articles/10.1051/cocv/2010041/

[1] J.M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5 (1979) 169-179. | MR | Zbl

[2] J.M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems. Commun. Pure Appl. Math. 32 (1979) 555-587. | MR | Zbl

[3] J.-M. Coron and B. D'Andréa-Novel, Stabilization of a rotating body-beam without damping. IEEE Trans. Autom. Control. 43 (1998) 608-618. | MR | Zbl

[4] J.-F. Couchouron, Compactness theorems for abstract evolution problems. J. Evol. Equ. 2 (2002) 151-175. | MR | Zbl

[5] J.-F. Couchouron and M. Kamenski, An abstract topological point of view and a general averaging principle in the theory of differential inclusions. Nonlinear Anal. 42 (2000) 1101-1129. | MR | Zbl

[6] R. Courant and D. Hilbert, Methods of Mathematical Physics 1. Interscience, New York (1953). | MR | Zbl

[7] C.M. Dafermos and M. Slemrod, Asymptotic behaviour of nonlinear contraction semigroups. J. Funct. Anal. 13 (1973) 97-106. | MR | Zbl

[8] A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics 377. Berlin-Heidelberg-New York, Springer-Verlag (1974). | MR | Zbl

[9] A. Haraux, Almost-periodic forcing for a wave equation with a nonlinear, local damping term. Proc. R. Soc. Edinb., Sect. A, Math. 94 (1983) 195-212. | MR | Zbl

[10] A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936) 367-379. | MR | Zbl

[11] V. Jurdjevic and J.P. Quinn, Controllability and stability. J. Differ. Equ. 28 (1978) 381-389. | MR | Zbl

[12] A. Pazy, A class of semi-linear equations of evolution. Israël J. Math. 20 (1975) 23-36. | MR | Zbl

[13] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag (1975). | MR | Zbl

[14] J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR | Zbl

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