Boundary sentinels in cylindrical domains (Q5949010)

The authors continue ideas of J.-L. Lions, who first introduced the sentinel concept, and of their own paper [Port. Math. 51, 421--453 (1994; Zbl 0809.93025)]. These problems discussed here are known as ``incomplete problems''. Two main sources of such incompleteness are: unknown perturbations at the boundary of the domain, here called ``pollution terms'', which are assumed to be ``small'', and some missing information in the initial data. The authors study the vibration of a three-dimensional cylindrical domain \(\Omega^e=\acute\omega\times(-e/2,+e/2)\), \(\acute\omega\) denoting a planar shape. Smoothness of the boundary \(\gamma\) of \(\acute\omega\) is assumed. Cartesian coordinates are normalized by the transformation: \(z_1=x_2\), \(z_2=x_2\), and \(z_3= e^{-1}x_3\), taking \(\Omega^e\) into \(\Omega^e=\acute \omega\times (-1/2,+1/2)\). In these coordinates the wave operator assumes the form \(\square_e=\partial^2/\partial t^2-\Delta_e\), where \(\Delta_e\) denotes \(\partial^2/\partial z_1^2+\partial^2/\partial z_2^2+e^{-2} \partial^2/\partial z_3^2\). Missing terms in the initial data and unknown small perturbation terms on the boundary are introduced. A functional is introduced representing the averages of the data obtained against some given functions. A sentinel functional in the style of \textit{J.-L. Lions} [Sentinelles pour les systèmes distribués. À données incomplètes, Recherches en Mathématiques Apliquées 21 (Mason, Paris) (1992; Zbl 0759.93043)] is introduced, gathering observation independent up to first order of the missing terms. The use of the HUM approach provides information on the exact controllability problem. Finite energy is routinely assumed. Regularity is derived by imposing additional hypotheses on the data. The authors also provide conditions such that the sentinel functional contains no useful information about the pollution terms. Such pollution terms are called stealthy. Necessary and sufficient conditions are derived for the existence of stealthiness.