Sufficient conditions for the controllability of wave equations with a transmission condition at the interface

Abstract

We consider waves travelling in two different mediums each endowed with a different constant speed of propagation. At the interface between the two mediums, the refraction of the rays of the optic geometry obeys the Snell’s law. We provide sufficient conditions on the geometry’ of the mediums and on the speed of propagation for the exact boundary controllability.

An inverse problem for a transmission wave equation with a flat interface in Rn

“…See also [8] for the study of Carleman estimate concerning the observability of wave equations with singular potentials, a problem where the geometry of the domain also plays a key role in the construction of the weight function. Finally, me mention the works [9,10], concerning hyperbolic equations with discontinuous main coefficients, and also [3,13], where other Equations with discontinuities are studied.…”

“…Despite the fact that (47) is a better minimal time than (10) (mainly for a close to 0), we stated the main result with the later estimate in order to simplify the presentation.…”

“…The definition of µ for x n > 0 is such that it satisfies the transmission conditions (5) on x n = 0. In section 5 we explain this choice for µ and also we prove the minimality of T 0 given by (10).…”