NULL CONTROLLABILITY OF DEGENERATE AND NON-DEGENERATE SINGULAR PARABOLIC

The present work concerns the study of the controllability problem for the linear heat equation, posed in a bounded domain. Nowadays, the problem of control occupies an important place in the general theory of partial differential equations, in particular because of its many physical applications (fluid mechanics, thermodynamics, propagation phenomena, engineering). Generically, it is a question of intervening on a given evolution system (E) in order to control its solution, i.e. to bring it from an initial (arbitrary) state to a prescribed final state. The system (E) is, depending on the case, hyperbolic (vibratory phenomena), parabolic (heat equation), or of a more complex type. We can also ask the control vector (function) to verify a constraint, such as, to minimize a certain functional. In recent years, the study of this (these) problem (s) has required the implementation of fairly complex theoretical and numerical tools. Control theory took off at the end of the 70s ◦ with the H.U.M (Hilbert Uniqueness Method) method of J.L. Lions. The years 90◦ are marked by two strong points. First, by the arrival of microlocal techniques by C. Bardos, G. Lebeau and J. Rauch. Then, the proof and the use of global Carleman inequalities by Fursikov (and also, for the heat equation in particular, by Lebeau and Robbiano) for the trajectory controllability of second order parabolic equations. A large number of related mathematical problems are equally relevant: stabilization of solutions, problem of uniqueness and unique extension.