Almost sure stabilizability of controlled degenerate diffusions

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton–Jacobi–Bellman partial differential inequality of second order. We give local and global versions of the first and second Lyapunov theorems, assuming the existence of a lower semicontinuous Lyapunov function satisfying such an inequality in the viscosity sense. An explicit formula for a stabilizing feedback is provided for affine systems with smooth Lyapunov function. Several examples illustrate the theory.