Symmetry Reduction and Control of the Dynamics of a 2D Rigid Circular Cylinder and a Point Vortex: Vortex Capture and Scattering

Symmetry reduction and control of the Hamiltonian system of a 2D rigid circular cylinder dynamically interacting with a point vortex external to it is presented. This dynamic model is an idealized example in an inviscid framework of fully coupled solid–fluid systems interacting in the presence of vorticity and has potential applications to problems in engineering and in nature involving the interaction of coherent vortices with bodies moving (primarily) under their influence. The dynamics of the system generically gives rise to two types of vortex orbits relative to the moving cylinder: bound and scattering orbits. The control input of a bounded external force acting through the center of mass of the cylinder is then added. Exploiting the S1-symmetry in the system, symplectic reduction is employed to formulate an S1-invariant control system, that preserves the momentum map, on the 2D symplectic reduced space. On this reduced space, both nonoptimal and optimal controllers, the latter using Pontryagin's maximum principle, are investigated with the control objective of changing the vortex orbit from a bound to a scattering type and vice versa.