1 Department of Mathematics, Technical University of Denmark2 unknown3 Department of Applied Mathematics and Computer Science, Technical University of Denmark
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.
Arabian Journal for Science and Engineering. Section B: Engineering, 2009, Vol 34, Issue 1D, p. 39-58