Efficient finite dimensional approximations for the bilinear Schrodinger equation with bounded variation controls

TL;DR

This paper demonstrates that solutions to the bilinear Schrödinger equation can be approximated by finite-dimensional systems under certain spectral and regularity conditions, enabling controllability with bounded variation controls.

Contribution

It introduces a method for finite-dimensional approximation of the bilinear Schrödinger equation that preserves controllability properties for controls of bounded variation.

Findings

Finite-dimensional approximations are uniform in time and control.

Controllability of approximations implies approximate controllability of the original system.

The approach applies to controls with bounded variation and Radon measures.

Abstract

This the text of a proceeding accepted for the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014). We present some results of an ongoing research on the controllability problem of an abstract bilinear Schrodinger equation. We are interested by approximation of this equation by finite dimensional systems. Assuming that the uncontrolled term $A$ has a pure discrete spectrum and the control potential $B$ is in some sense regular with respect to $A$ we show that such an approximation is possible. More precisely the solutions are approximated by their projections on finite dimensional subspaces spanned by the eigenvectors of $A$. This approximation is uniform in time and in the control, if this control has bounded variation with a priori bounded total variation. Hence if these finite dimensional systems are controllable with a fixed bound on the total variation of the control then the system is approximatively controllable. The main outcome of our analysis is that we can build solutions for low regular controls such as bounded variation ones and even Radon measures.