Well-posedness of inverse problems for systems with time dependent parameters

In this paper we investigate the abstract hyperbolic model with time dependent stiffness and damping given by V*,V + d(t;(mu)over dot, psi) + a(t;u(t),psi) = <f(t).psi > V*,V where V subset of V-D subset of H subset of V-D* subset of V* are Hilbert spaces with continuous and dense injections, where H is identified with its dual and denotes the associated duality product. We show under reasonable assumptions on the time-dependent sesquilinear forms a (t;.,.) : V x V -> C and d (t;.,.) : V-D x V-D -> 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.