Unbounded perturbation of the controllability for evolution equations (Q1874426)

Consider the system \[ y'(t) = (A + P)y(t) + B(t)u(t), \] where \(A\) is the infinitesimal generator of a strongly continuous semigroup in a Banach space \(E\), \(t\to B(t)\) is continuous, \(B(t)\) is a linear bounded operator from \(U\) to \(E\) \((U\) is a second Banach space). The unperturbed system \((P = 0)\) is exactly controllable, and the author gives conditions on the perturbation \(P\) such that the exact controllability is preserved.