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 ct al (1988) [Aubry, N., Holmes, P., Lumley, J. L. and Stone, E. (1988) The dynamics of coherent structures in the wall region of a turbulent boundary layer. Journal of Fluid Mechanics 192, 115-173.].