Resonance, stabilizing feedback controls, and regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations

It is shown that no resonance and small time local controllability imply many standard necessary conditions for the existence of a continuous, asymptotically stabilizing (state) feedback control (ASFC) for an n-dimensional, real analytic, single input affine control system. Furthermore, no resonance means there exist local coordinates in which the drift vector field is linear, yielding a canonical form for the study of sufficient conditions and construction of an ASFC. The roles of resonance and the Kawski necessary condition are examined in detail, together with their implications on regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations.