Template-based control of relative equilibria in systems with symmetry

We present an approach to the design of feedback control laws that stabilize the relative equilibria of general nonlinear systems with continuous symmetry. Using a templatebased method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. Assuming equivariant actuation, we derive feedback laws for the reduced system that are optimal in the sense that they minimize a quadratic cost function. The method uses standard tools available from finite dimensional linear control theory. We illustrate the method by stabilizing unstable traveling waves of a dissipative PDE possessing translational invariance: the Kuramoto-Sivashinsky equation with periodic boundary conditions.