Bilinear Control of Evolution Equations

The thesis is devoted to the study of the stabilization and the controllability of the evolution equations $$u'(t) + Au (t) + p (t) Bu (t) = 0$$ by means of a bilinear control $p$. Bilinear controls are coefficients of the equation that multiply the state variable. Multiplicative controls are therefore suitable to describe processes that change their principal parameters in presence of a control. We first present a result of rapid stabilization of the parabolic equations towards the ground state by bilinear control with a doubly exponential rate of convergence. Under stronger hypotheses on the potential $B$, we show results of exact local and global controllability towards the solution of the ground state in arbitrarily small time. We apply these two abstract results to different types of PDE such as the heat equation, or parabolic equations with non-constant coefficients. We then prove local exact controllability of a class of degenerate wave equations relying on a sharp analysis of the spectral properties of the elliptic degenerate operators. We then present a method of constructing multiplicative operators $B$ verifying the sufficient hypotheses required for controllability or stabilization results. This method leads to constructive algorithms of infinite explicit families of such operators $B$. We then prove new controllability results for the Schr{"o}dinger equation with hybrid boundary conditions. We also give applications of our method to parabolic equations leading to results of rapid stabilization, local and global controllability to the ground state which are explicit with respect to the operators $B$.