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Decay rates for delayed abstract thermoelastic systems with Cattaneo law. (English) Zbl 1373.93266

Summary: In this paper, we consider a thermoelastic type system with Cattaneo’s law and internal time delay. Under a suitable assumption on the weight of delay, we prove that the exponential stability of this system is reduced to an observability estimate for the corresponding uncontrolled system. The proof of the main results uses the methodology introduced by A. Haraux [Port. Math. 46, No. 3, 245–258 (1989; Zbl 0679.93063)] and generalized by K. Ammari and M. Tucsnak [ESAIM, Control Optim. Calc. Var. 6, 361–386 (2001; Zbl 0992.93039)]. An illustrating example is given.

MSC:

93D20 Asymptotic stability in control theory
93C15 Control/observation systems governed by ordinary differential equations
74A15 Thermodynamics in solid mechanics
93C25 Control/observation systems in abstract spaces
93B07 Observability
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