High frequency asymptotic approach for incomplete spectral and local controllability

Abstract

This contribution describes a part of the programm realised in collaboration with G. Lebeau and J. Rauch (cf. [BLR]) and [Le1, Le2].

It has been observed that in the framework of linear hyperbolic problems, the questions of uniqueness of solutions, “Holmgren theorem”, estimation of the error in the observation of a solution, exact controllability and stabilisation are closely related. In particular it has been shown that the “stable observation”, the “exact controllability” and the “uniform stabilization” can be achieved if and only if a geometric condition is satisfied; namely that any bicharacteristic ray intersect the region of “action”. It turns out that this condition is very stringent and generally not satisfied in practical applications. The purpose of this talk is the description of some results which could be obtained when the geometric hypothesis is not satisfied.

Starting from the following observations: most of the properties of hyperbolic problems are local in space-time, the geometric condition is obtained by microlocal analysis i.e. by high frequencies asymptotics, the following question are discussed:

• 1- Composite action (including Neumann, Dirichlet or interior Control).

• 2- Exact controllability in a localised region of space time which can be illuminated from the “action region”.

• 3- Weak controllability involving low frequencies filters and in particular control of the projection of the solution on a finite number of eigenmodes of the propagator.