  Vortex-dominated flows are ubiquitous in engineering, and the ability to
efficiently manipulate the dynamics of these vortices has broad applications,
from wake shaping to mixing enhancement. However, the strongly nonlinear
behavior of the vortex dynamics makes this a challenging task. In this work, we
investigate the control of vortex dynamics by using a change of coordinates
from the Biot-Savart equations into well-known invariants, such as the
Hamiltonian, linear, and angular impulses, which are Koopman eigenfunctions. We
then combine the resulting model with model predictive control to generate
control laws that force the vortex system using "virtual cylinders". The
invariant model is beneficial as it provides a linear, global description of
the vortex dynamics through a recently developed Koopman control scheme for
conserved quantities and invariants. The use of this model has not been well
studied in the literature in the context of control. In this paper, we seek to
understand the effect of changing each invariant individually or multiple
invariants simultaneously. We use the 4-vortex system as our primary test bed,
as it is the simplest configuration that exhibits chaotic behavior. We show
that by controlling to specific invariant quantities, we can modify the
transition from chaotic to quasiperiodic states. Finally, we computationally
demonstrate the effectiveness of invariant control on a toy example of tracer
mixing in the 4-vortex system.
