Series of Lectures:

Title:  Global Carleman Estimates for wave type and heat type equations and applications

Speaker:  Jean-Pierre Puel, University of Versailles

Time: 1:30-4:30 pm, Wednesday, January 9

1:30-4:30 pm, Thursday, January 10

1:30-4:30 pm, Friday, January 11

1:30-4:30 pm, Monday, January 14

1:30-4:30 pm, Wednesday, January 16

Venue:  Room 2001, Guanghua Est Main-Building, Handan Campus

Abstract:  Concerning the wave equations with a potential term we are motivated by two problems: exact controllability (with control acting on a part of the boundary) on one hand and some inverse problems for retrieving the potential from boundary measurements on the other hand. We will establish global Carleman estimates with a suitable weight function in order to obtain an observability inequality. This, applied to an adjoint problem will enable us to obtain exact controllability even in presence of a bounded potential. As a second application, we will obtain Lipschitz stability for the potential with respect to boundary measurements of the flux in suitable norms. These two results require some geometrical conditions which will be explicited. For the heat equation, the motivation comes again from exact controllability to trajectories and also from data assimilation problems. We will give a global Carleman estimate in the standard case (with a quasi complete proof) and also present the corresponding results in the case of weak right hand sides and non-homogeneous boundary conditions. The adapted weight functions are here different and have to present singularities in the time variable. We will apply these results to problems of exact controllability to trajectories and also to a new version of data assimilation problems which consist in retrieving the state value of the solution at some time from boundary measurements of the flux of the solution. If time permits, we will also apply these results to Stokes equations and linearized Navier-Stokes equations with the same kind of applications.