Mathematical Control and Related Fields (MCRF)

Indirect stabilization of weakly coupled systems with hybrid boundary conditions

Pages: 413 - 436, Volume 1, Issue 4, December 2011      doi:10.3934/mcrf.2011.1.413

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Fatiha Alabau-Boussouira - Present position Délégation CNRS at MAPMO, UMR 6628, Current position Université Paul Verlaine-Metz, Ile du Saulcy, 57045 Metz Cedex 1, France (email)
Piermarco Cannarsa - Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy (email)
Roberto Guglielmi - Dipartimento di Matematica - Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1 - 00133 Roma, Italy (email)

Abstract: We investigate stability properties of indirectly damped systems of evolution equations in Hilbert spaces, under new compatibility assumptions. We prove polynomial decay for the energy of solutions and optimize our results by interpolation techniques, obtaining a full range of power-like decay rates. In particular, we give explicit estimates with respect to the initial data. We discuss several applications to hyperbolic systems with hybrid boundary conditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively.

Keywords:  Indirect stabilization, energy estimates, interpolation spaces, evolution equations, hyperbolic systems.
Mathematics Subject Classification:  93D15, 35L53, 47D06, 46B70.

Received: March 2011;      Revised: July 2011;      Available Online: November 2011.