Quantum and ensemble control in small times – QUEST

A variety of physically relevant quantum systems, whose evolution is governed by bilinear Schrödinger PDEs, are known to be controllable in large time. Nevertheless, there are examples which are controllable in large time, but not in small times. This obstruction happens e.g. in the presence of sub-quadratic potentials, due to the conservation of Gaussian states, for small times. Time-optimality represents a challenge in quantum control, both from a fundamental viewpoint and for practical implementation of control theory to quantum technology. Recently, using geometric control methods such as controllability of groups of diffeomorphisms, we were able to furnish the first examples of bilinear Schrödinger and wave PDEs that are controllable in arbitrarily small times. Such examples are posed on euclidean spaces (e.g. the quantum harmonic oscillator), and tori (e.g. rotating rigid molecules, or cold atoms trapped in periodic optical lattices). The general scope of this project is to push forward the understanding of such geometric control methods in order to obtain new results of controllability for bilinear PDEs. The nature of the project is thus two-sided: from the methodology point of view, we aim at developing new methods of geometric control theory in infinite dimensions; from the point of view of the results, we expect to prove new small-time global approximate controllability properties for several physically relevant PDE models. More precisely, the project is divided in three parts:

1) Genericity and higher regularity properties of small-time approximate controllability of bilinear PDEs; such study is mostly devoted to Schrödinger equations posed on compact manifolds, describing physically relevant quantum systems, although it has direct consequences also for other PDEs such as Liouville and wave equations.

2) Relation between approximate controllability of PDEs and controllability of groups of diffeomorphisms; diffeomorphisms, simplectomorphisms, and Hamiltonian flows, arise naturally in the study of Schrödinger and Liouville equations, and describe the dynamics of continuous ensembles of particles through transportation of densities.

3) Interaction between small-time approximate controllability and finite speed of propagation in wave equations; necessary and sufficient conditions on the initial states (for small-time approximate controllability) can capture such feature which is typical of waves and Maxwell equations.

The main innovative aspect of the scientific program is thus in the development of new geometric control methods, such as low mode forcing and controllability of groups of diffeomorphisms, and their application to nonlinear infinite-dimensional control-to-state problems.
The significance of the expected results is in the capability of obtaining new small-time approximate controllability properties of several models of bilinear PDEs: such results were out-of-reach with previous techniques, and are highly demanded in applications.