Q-Math Seminar

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T. Chambrion (IMB, U. Bourgogne)

Impulsive control of single-input bilinear quantum systems

Tuesday the 14th of November, 2023, 12:00, Room 2.2.D08

Control systems are said to be bilinear when the dynamics takes the form x’=Ax+uBx, with x the state (in some Hilbert space), A and B two linear (possibly unbounded) operators and u a real valued function of the time (the control law). When the operator B is bounded and the operator A generates a C^0 semi-group, a standard fixed-point argument shows that solutions are well-defined for locally integrable control laws.
In this talk, we investigate the construction of propagators for conservative quantum systems (A and B are skew-symmetric) when the control u is a Radon measure whose atoms can be understood as shocks for the system (hence the name: ”impulsive control”). We present a construction valid for unbounded operators B, under regularity conditions met by most of the quantum systems encountered in the literature.
Under the same regularity conditions, we also present an extension of a celebrated result by Ball, Marsden and Slemrod in 1982 (“when B is bounded, the attainable set with locally integrable controls is meager, hence exact controllability does not hold”), for the case where control laws are Radon measures.

This is a joint work with Nabile Boussaïd from LMB (Besançon) and Marco Caponigro from Università degli Studi di Roma 'Tor Vergata'.

Link for online session (Active on request):

N. Boussaïd (LMB, U. Franche Comté)

Exact controllability in projections of bilinear Schrödinger equations: the mixed spectrum case

Tuesday the 14th of November, 2023, 12:45, Room 2.2.D08

We give sufficient conditions for the exact controllability in projection of bilinear Schrödinger equations with minimal regularity of the control (switching control) in the case where the spectrum of the free Hamiltonian is mixed (a discrete and an essential part).

The idea behind the proof is to use a Galerkin approximation to reduce the problem to the finite dimensional case. The natural Galerkin basis is the one provided by a orthonormal family of eigenvectors. The latter is never complete if the essential spectrum is continuous. When such a situation happens, we use averaging methods and a generalization of the RAGE theorem to decouple the dynamics with respect to the sum of eigenspaces and the one with respect to the continuous spectrum.

This is a joint work with Marco Caponigro from Università degli Studi di Roma 'Tor Vergata' and Thomas Chambrion from the IMB (Dijon).

Link for online session (Active on request):