Explicit observability estimate for the wave equation with potential and its application (Q2713419)

The author obtains an observability estimate of the typeNEWLINE\[NEWLINE |w_0|^2_{L^2(\Omega)} + |w_1|^2_{H^{-1}(\Omega)} \leq K \int_0^T \int_{\omega} |w|^2 dx dt, NEWLINE\]NEWLINE where \(w\) denotes the weak solution of the problem \( w'' - \triangle w = q(t,x)w \) in \(Q = (0,T) \times \Omega\), \(w = 0\) on \(\Sigma = (0,T) \times \partial \Omega\), \(w(0) = w_0, w'(0) = w_1\) in \(\Omega\). The potential \(q\) is assumed essentially bounded in \(Q\) and \(\omega\) denotes a subdomain of the bounded domain \(\Omega\) in \(\mathbb{R}^n\) with \(C^{1,1}\) boundary \(\partial \Omega\). The main novelty of the paper is the explicit estimate of the constant \(K\) with respect to \(\ell=|q|_{L^\infty(Q)}\): in fact, it is proved that \(K= O(\exp(\exp(\exp(\ell))))\) as \(\ell \rightarrow \infty\). As usual, this type of estimates can be applied to get the exact internal controllability in \(H_0^1(\Omega) \times L^2(\Omega)\) at (sufficiently large) time \(T\) of the semilinear wave equation \( y'' - \triangle y = f(y) + \chi_\omega(x) u(t,x)\) in \(Q\), where \(f \in C^1(\mathbb{R})\) with \(f' \in L^\infty(\mathbb{R})\), by choosing the control \(u\) in \(L^2(Q)\).