# zbMATH — the first resource for mathematics

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.

##### MSC:
 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:
##### References:
 [1] J.M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim.5 (1979) 169-179. · Zbl 0405.93030 · doi:10.1007/BF01442552 [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. · Zbl 0394.93041 · doi:10.1002/cpa.3160320405 [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. Zbl0908.93055 · Zbl 0908.93055 · doi:10.1109/9.668828 [4] J.-F. Couchouron, Compactness theorems for abstract evolution problems. J. Evol. Equ.2 (2002) 151-175. Zbl1008.47057 · Zbl 1008.47057 · doi:10.1007/s00028-002-8084-z [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. · Zbl 0972.34049 · doi:10.1016/S0362-546X(99)00181-9 [6] R. Courant and D. Hilbert, Methods of Mathematical Physics1. Interscience, New York (1953). · Zbl 0051.28802 [7] C.M. Dafermos and M. Slemrod, Asymptotic behaviour of nonlinear contraction semigroups. J. Funct. Anal.13 (1973) 97-106. Zbl0267.34062 · Zbl 0267.34062 · doi:10.1016/0022-1236(73)90069-4 [8] A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics377. Berlin-Heidelberg-New York, Springer-Verlag (1974). [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. · Zbl 0589.35076 · doi:10.1017/S0308210500015584 [10] A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z.41 (1936) 367-379. Zbl0014.21503 · Zbl 0014.21503 · doi:10.1007/BF01180426 · eudml:168670 [11] V. Jurdjevic and J.P. Quinn, Controllability and stability. J. Differ. Equ.28 (1978) 381-389. · Zbl 0417.93012 · doi:10.1016/0022-0396(78)90135-3 [12] A. Pazy, A class of semi-linear equations of evolution. Israël J. Math.20 (1975) 23-36. · Zbl 0305.47022 · doi:10.1007/BF02756753 [13] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag (1975). · Zbl 0305.47022 [14] J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl.146 (1987) 65-96. · Zbl 0629.46031 · doi:10.1007/BF01762360
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.