ASYMPTOTICALLY STABILIZING FEEDBACK CONTROLS AND THE NONLINEAR REGULATOR PROBLEM

Continuous asymptotically stabilizing feedback controls are constructed for two-dimensional, and certain three-dimensional, small time locally controllable affine systems. The Lie products which determine controllability induce a dilation; it suffices to work with the approximating homogeneous system associated with this dilation. A cost functional is then constructed which is such that the associated Hamilton-Jacobi-Bellman equation is homogenous, forcing the solution (which is a Lyapunov function for the optimally controlled system) to be homogeneous and thereby determining its basic form. The process may be viewed as a generalization of the linear regulator construction.