Abstract
We investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear O(2) symmetric dynamical systems possessing structurally stable heteroclinic cycles. The leading-order behavior near the equilibria is bilinear and homogeneous in the state variables, while nonlinearities representing the symmetry breaking effect of the controller are crucial. After a series of simplifying transformations, we use ideas from optimal control theory to construct a stabilizing controller. This study is motivated by the desire to stabilize the burst/sweep cycle in low-dimensional models of a turbulent boundary layer. In the last two sections, we apply the techniques to the 10-dimensional system of Aubry et al. (1988). kw]Bilinear systems; dynamical systems; heteroclinic cycles; normal forms; optimal control; symmetry
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