Observability inequalities by internal observation and their applications (Q1411482)

The systems under study are described by the wave equation \[ {\partial^2 y(t, x) \over \partial t^2} - \Delta y(t, x) = a_1(t, x)y(t, x) + a_2(t, x) {\partial y(t, x) \over \partial t} + \langle a_3(t, x), \nabla y(t, x) \rangle \] in a \(n\)-dimensional domain \(\Omega\) with boundary \(\Gamma,\) with initial conditions \[ y(0, x) = y_0(x),\;\partial y(0, x)/\partial t = y_1(x), \] and the homogeneous Dirichlet boundary condition on \(\Gamma.\) Using Carleman estimates the authors obtain various observability inequalities by internal observation in a subdomain \(G \subseteq \Omega,\) that is, estimates of \(| (y_0, y_1)| _{H_0^1(\Omega) \times L^2(\Omega)}\) by a norm of the restriction \(y| _{(0, T) \times G}.\) The results are applied to a controllability problem and to an inverse wave source problem.