beginabstract We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decay at the rate 1/t. We also establish observability results, at one or two endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.

We deal with singular perturbations of nonlinear problems depending on a small parameter > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles

By mean of generalized Fourier series and Parseval's equality in weighted L 2 -spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp time, at each of the endpoints of the interval. The observability constants are explicitly given. Using the Hilbert Uniqueness Method we deduce the exact boundary controllability of the wave equation.(1.2)

No abstract

No abstract