Exact controllability for the wave equation with variable coefficients (Q5951508)

The author studies the exact controllability for the wave equation \(y'' -Ay=0\) in \(\Omega\times \mathbb{R}^{+}\), where \(A=\partial_i (a_{ij}\partial_j)\) and \(a_{ij}\) are functions of \(t\) and \(x\) with some regularity. The control acts on the boundary as \(y=v\) on \(\Gamma\times \mathbb{R}^{+}\). The author proves that, for \(T\) sufficiently large and for every initial data in \(L^2(\Omega)\times H^{-1}(\Omega)\), the system can be transferred to rest at time \(T\). The approach is based on the well-known Hilbert uniqueness method of J. L. Lions. Previous results refer to the case when \(a_{ij}\) are independent of \(t\) or when \(a_{ij}=\delta_{ij} a(t)\).