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In a separable Hilbert space $X$, we study the linear evolution equation\begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A$ is anaccretive self-adjoint linear operator, $B$ is a bounded linear operator on$X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control.  We give sufficient conditions in order for the above control system to belocally controllable to the ground state solution, that is, the solution of thefree equation ($p\equiv0$) starting from the ground state of $A$. We alsoderive global controllability results in large time and discuss applications toparabolic equations in low space dimension.

Exact controllability to the ground state solution for evolution equations of parabolic type via bilinear control