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 〈ü(t), ψ〉V ∗,V + d(t; u̇(t), ψ) + a(t;u(t), ψ) = 〈f(t), ψ〉V ∗,V where V ⊂ VD ⊂ H ⊂ V ∗ D ⊂ 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 × V → C and d(t; ·, ·) : VD × VD → 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.