Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary

Abstract

For the observation or control of solutions of second-order hyperbolic equation in $\mathbb{R}_t \times \Omega $, Ralston’s construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. 807–823] showed that it is necessary that the region of control meet every ray of geometric optics that has, at worst, transverse reflection at the boundary. For problems in one space dimension, the method of characteristics shows that this condition is essentially sufficient. For problems on manifolds without boundary, the sufficiency was proved in [J. Rauch and M. Taylor, Indiana Univ. Math. J., 24 (1974)]. The theorems regarding propagation of singularities [M. Taylor, Comm. Pure Appl. Math., 28 (1975), pp. 457–478], [R. Melrose, Acta Math., 147 (1981), pp. 149–236], [J. Sjostrand, Communications in Partial Differential Equations, 1980, pp. 41–94] allows the extension of the latter argument to the problem of interior control [C. Bardos, G. Lebeau, and J. Rauch, Rendiconti del Seminario Mathematico, Universita e Politecnico di Torino, 1988, pp. 11–32]. In this paper, the sufficiency is proved for problems of control and observation from the boundary. For multidimensional problems, the region of control must meet each ray in a nondiffractive point, and a new microlocal lower bound on the trade of solutions at the boundary at gliding points is required. This paper treats linear problems with variable coefficients and solutions of all Sobolev regularities. The regularity of the controls is precisely linked to the regularity of the solutions.