Nonholonomic regulators

In the context of a nonlinear regulator problem associated with a plant described by ẋ = f(x, u) and a performance measure η = ∫₀^∞ L(x, u)dt, with both L(0, 0) and L(0, 0) vanishing, the form of the optimal control near x₀ = 0 is only well understood if the linearized system at x = 0 is controllable and L(x, u) is suitably approximated by a nondegenerate quadratic. In this case the optimal nonlinear control is approximated near 0 by a linear control law which can be obtained by solving a Riccati equation. However, when the system is controllable, but the linearized system is not, the problem is more difficult and seems not to have been previously explored in any generality. In this paper we investigate a class of such systems which are first bracket controllable and derive the generic form of the optimal regulator in feedback form, mindful of the fact that a certain set of initial conditions will necessarily be excluded because of topological constraints. The explicit formulas for the optimal feedback control found here are remarkable in that they are not analytic at the equilibrium point. The form of the optimal control brings out clearly the nature of the singularity which is necessarily present because of topological obstruction which prevents the existence of a smooth stabilizing control law. The solution can be thought of as providing one answer to the question “what is the closest one can come to a smooth stabilizing feedback when no smooth feedback control exists"?