Control of integrable Hamiltonian systems and degenerate bifurcations

Abstract

We discuss control of low-dimensional systems which, when uncontrolled, are integrable in the Hamiltonian sense. The controller targets an exact solution of the system in a region where the uncontrolled dynamics has invariant tori. Both dissipative and conservative controllers are considered. We show that the shear flow structure of the undriven system causes a Takens-Bogdanov bifurcation to occur when control is applied. This implies extreme noise sensitivity. We then consider an example of these results using the driven nonlinear Schrödinger equation.