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 $2nd$ 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 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.

Direct Lyapunov method demonstrates almost sure open-loop and asymptotic stability of controlled degenerate diffusion processes, with a new form of Hamilton-Jacobi-Bellman partial differential inequality.

Almost sure stability of controlled degenerate diffusions

Key Takeaway: Direct Lyapunov method demonstrates almost sure open-loop and asymptotic stability of controlled degenerate diffusion processes, with a new form of Hamilton-Jacobi-Bellman partial differential inequality.