Stabilizability conditions in terms of critical Hamiltonians and symbols

Symmetric functions of critical Hamiltonians, called symbols, were used in such problems of nonlinear control as the characterization of symmetries and feedback invariants. We derive here a stabilizability condition in the class of almost continuous feedback controls based on symbols. The methodology proposed consists of defining a selector of the multivalued covector field satisfying an extra degree condition on a sphere close to the origin. Then the above selector is extended to some neighborhood in order to get an exact differential form. Integration of that form gives us a control Lyapunov function candidate (Theorem 3.1). The condition proposed is applied to the analysis of a planar control-affine system with polynomial homogeneous vector fields. Unlike known results in the literature, we consider the case when each vector field vanishes at the origin and the control is bounded. We give stabilizability conditions explicitly in terms of the control system parameters (Theorem 4.1).