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Explicit observability estimate for the wave equation with potential and its application. (English) Zbl 0976.93038
The author obtains an observability estimate of the type$$ |w_0|^2_{L^2(\Omega)} + |w_1|^2_{H^{-1}(\Omega)} \leq K \int_0^T \int_{\omega} |w|^2 dx dt, $$ 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 $\bbfR^n$ 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 $f \in C^1(\bbfR)$ with $f' \in L^\infty(\bbfR)$, by choosing the control $u$ in $L^2(Q)$.

93C20Control systems governed by PDE
35L05Wave equation (hyperbolic PDE)
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