We study a class of third order hyperbolic operators in with triple characteristics on . We consider the case when the fundamental matrix of the principal symbol for has a couple of non vanishing real eigenvalues and is strictly hyperbolic for We prove that is strongly hyperbolic, that is the Cauchy problem for is well posed in for any lower order terms .
We study existence, uniqueness, continuous dependence, general decay of solutions of an initial boundary value problem for a viscoelastic wave equation with strong damping and nonlinear memory term. At first, we state and prove a theorem involving local existence and uniqueness of a weak solution. Next, we establish a sufficient condition to get an estimate of the continuous dependence of the solution with respect to the kernel function and the nonlinear terms. Finally, under suitable conditions...
We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior...
We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of...
We study controllability for a nonhomogeneous string and ring under an axial stretching tension that varies with time. We consider the boundary control for a string and distributed control for a ring. For a string, we are looking for a control f(t) ∈ L2(0, T) that drives the state solution to rest. We show that for a ring, two forces are required to achieve controllability. The controllability problem is reduced to a moment problem...
We analyze the problem of boundary observability of the finite-difference space semidiscretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the meshsize) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski [9]. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm...
In this paper we consider a smooth and bounded domain of dimension with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains...