Stabilization of unbounded bilinear control systems in Hilbert space

We consider bilinear control systems of the form y ′ ( t ) = A y ( t ) + u ( t ) B y ( t ) where A generates a strongly continuous semigroup of contraction ( e t A ) 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 ( 〈 B y ( t ) , y ( t ) 〉 ) . Applications to the heat equation is considered.