Regular propagators of bilinear quantum systems

The present analysis deals with the regularity of solutions of bilinear control systems of the type x′ = (A+ u(t)B)x where the state x belongs to some complex in nite dimensional Hilbert space, the (possibly unbounded) linear operators A and B are skew-adjoint and the control u is a real valued function. Such systems arise, for instance, in quantum control with the bilinear Schrödinger equation. For the sake of the regularity analysis, we consider a more general framework where A and B are generators of contraction semi-groups. Under some hypotheses on the commutator of the operators A and B, it is possible to extend the de nition of solution for controls in the set of Radon measures to obtain precise a priori energy estimates on the solutions, leading to a natural extension of the celebrated noncontrollability result of Ball, Marsden, and Slemrod in 1982.