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Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system. (English) Zbl 0958.93012

Let \(\Omega\) be an \(m\)-dimensional domain with boundary \(\Gamma\) and denote by \(x = (x_0, x_1, \dots, x_m)\) a generic point in \([0, T] \times \Omega,\) \(x_0\) standing for the time variable \(t.\) The system for the state \((w, \theta)\) is \[ \begin{alignedat}{2} P(x, D)w &= Q_1(x, D) \theta &&\quad (x \in [0, T] \times \Omega),\\ Q(x, D)\theta &= P_1(x, D)w &&\quad (x \in [0, T] \times \Omega),\\ w &= \theta = 0 &&\quad (x \in [0, T] \times \Gamma)\end{alignedat} \] where \(P(x, D)\) is a second order hyperbolic operator and \(Q(x, D)\) is a second order parabolic operator. The coupling terms \(Q_1(x, D)\) and \(P_1(x, D)\) are, respectively, a second order operator (involving the space variables) and a first order operator (involving all variables). The observability question is whether the vanishing of the normal derivative \(\partial_\nu(w, \theta)\) on the boundary causes the vanishing of the solution in \([0, T] \times \Omega\) or, in particular, whether there is an observability estimate of the form \[ \|(w, \theta)\|_{[0, T] \times \Omega} \leq C\|\partial_\nu (w, \theta)\|_{[0, T] \times \Gamma} \] where \(\|\cdot \|_{[0, T] \times \Omega}\) and \(\|\cdot\|_{[0, T] \times \Gamma}\) are suitable Sobolev norms in \([0, T] \times \Omega\) and in \([0, T] \times \Gamma.\) An affirmative answer implies (through duality) an exact null controllability result for the adjoint equation.
Under various assumptions, the authors show that the observability problem has a solution. The method is that of Carleman estimates, which has been previously used for observability-controllability results for hyperbolic as well as parabolic equations (see references in the paper). As the authors point out, the systems under study appear in linear thermoelasticity.

MSC:

93B07 Observability
93C20 Control/observation systems governed by partial differential equations
74F05 Thermal effects in solid mechanics
93B05 Controllability
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