Stabilization of unbounded bilinear control systems in Hilbert space

We consider bilinear control systems of the form y′(t)=Ay(t)+u(t)By(t) where A generates a strongly continuous semigroup of contraction (etA)t⩾0 on an infinite-dimensional Hilbert space Y whose scalar product is denoted by 〈.,.〉. We suppose that this system is unbounded in the sense that the linear operator B is unbounded from the state Y into itself. Tacking into account eventual control saturation, we study the problem of stabilization by (possibly nonquadratic) feedback of the form u(t)=−f(〈By(t),y(t)〉). Applications to the heat equation is considered.