Dynamic stabilization of systems via decoupling techniques

Abstract

Abstract. We give sufficient conditions which allow the study of the exponential stability of systems closely related to the linear thermoelasticity systems by a decoupling technique. Our approach is based on the multipliers technique and our result generalizes (from the exponential stability point of view) the earlier one obtained by Henry et al. Résumé.Nous donnons des conditions suffisantes qui permettent l'étude de la stabilité exponentielle de systèmes similairesà celui de la thermoélasticité en utilisant une technique de découplage. Notre approche est basée sur la technique des multiplicateurs et notre résultat généralise (du point de vue de la stabilité exponentielle) celui obtenu par Henry et al.

Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations

“…Thus, by the boundedness of A 1 2 u n , making use of interpolation, one gets that A 1−α u n = o (1). Moreover, we also have A 1−2α+β u n = o (1).…”

“…Thus, the corresponding (2.18) implies that the semigroup e A α,β t is polynomially stable with order 1 2(β−2α) and 1 2−2(β+2α) , respectively. The above two cases are exactly those found in [11].…”

“…where the operator A α,β : D(A α,β ) ⊆ H → H is defined by where a ∨ b = max{a, b} for any a, b ∈ R. It is known that A α,β (which is identified with its closure) generates a C 0 -semigroup e A α,β t of contractions on H [1]. Then the solution to the evolution equation (1.2) admits the following representation:…”

“…In 1981, Chen-Russell [3] considered the abstract elastic system with direct damping (the so-called linear oscillator) of following form: was shown by Huang [8,9] and Huang-Liu [10] that if B α is equivalent to A α (in a certain sense) with 1 2 ≤ α ≤ 1, the semigroup e A α t is analytic and exponentially stable. Complete regularity results for such a system were obtained by Chen-Triggiani [4,5], which says: When B α is equivalent to A α (in a certain sense), the associated C 0 -semigroup e A α t is analytic for 1 2 ≤ α ≤ 1, is of Gevrey class δ > 1 2α for 0 < α < 1 2 .…”

Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations

“…In particular, the controllability and stability of these thermoelastic systems have been studied in [6,8,10,13,19,20,25].…”