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Well posedness of inverse problems for systems with time dependent parameters. (English) Zbl 1182.35225

Summary: We investigate the abstract hyperbolic model with time dependent stiffness and damping given by
\[ \big\langle\ddot u(t),\psi\big\rangle_{V^*,V}+ d\big(t;\dot u(t),\psi\big)= \big\langle f(t),\psi\big\rangle_{V^*,V}, \]
where \(V\subset V_D\subset H\subset V_D^*\subset V^*\) are Hilbert spaces with continuous and dense injections, where \(H\) is identified with its dual and \(\langle\cdot,\cdot\rangle\) denotes the associated duality product. We show under reasonable assumptions on the time-dependent sesquilinear forms \(a(t;\cdot,\cdot): V\times V\to C\) and \(d(t;\cdot,\cdot):V_D\times V_D\to C\) that this model allows a unique solution and that the solution depends continuously on the data of the problem. We also consider well-posedness as well as finite element type approximations in associated inverse problems. The problem above is a weak formulation that includes models in abstract differential operator form that include plate, beam and shell equations with several important kinds of damping.

MSC:

35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35L90 Abstract hyperbolic equations
35R30 Inverse problems for PDEs
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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