Bilinear control of evolution equations of parabolic type

Fatiha Alabau Boussouira, Piermarco Cannarsa, Cristina Urbani

Published 2018-11-21, updated 2018-11-22Version 2

We prove a result of bilinear controllability for a class of abstract parabolic equations of the form \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\in [0,T] \end{equation*} where the operator $-A$ is the infinitesimal generator of an analytic semigroup of bounded linear operators on a Hilbert space and $p(\cdot)$ is the control function. The proof is based on maximal regularity properties associated with analytic semigroups and on a linearization argument. To study the linearized system we use the moment method. Finally, the controllability of the nonlinear system is deduced through the classical inverse mapping theorem. We give several applications of our result to different kinds of parabolic equations.