Exact boundary controllability on \(L_ 2(\Omega)\times H^{-1}(\Omega)\) of the wave equation with Dirichlet boundary control acting on a portion of the boundary \(\partial \Omega\), and related problems (Q1110470)

This paper presents improvements to results on the exact controllability problem for the wave equation obtained by \textit{I. Lasiecka} and the author [J. Differ. Equations 66, 340-390 (1987; Zbl 0629.93047)] via uniform stabilization and by \textit{J.-L. Lions} [Contrôlabilité exacte, perturbations et stabilisation des systèmes distributés. Tome l (1988; Zbl 0653.93002)] via the HUM method. The problem consists in finding a Dirichlet boundary condition control on a part \(\Gamma_ 1\) of the boundary belonging to \(L^ 2(]0,T[\times \Gamma_ 1)\) that steers an arbitrary initial condition in \(L^ 2(\Omega)\times H^{-1}(\Omega)\) to zero. The key technical issue is a lower bound on the \(L^ 2(\Gamma_ 1\times]0,T[)\) norm of the normal derivative of the solution to the corresponding homogeneous problem. This is done by the a priori estimate technique using a vector field which is more general than the radial vector field from an external point \(x_ 0\) which was utilized before. Conditions obtained on \(\Gamma_ 1\) are less restrictive and examples are given. The variable coefficient case is also studied.