  This paper presents a novel method to synthesize stochastic control Lyapunov
functions for a class of nonlinear, stochastic control systems. In this work,
the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation
is transformed into a linear partial differential equation for a class of
systems with a particular constraint on the stochastic disturbance. It is shown
that this linear partial differential equation can be relaxed to a linear
differential inclusion, allowing for approximating polynomial solutions to be
generated using sum of squares programming. It is shown that the resulting
solutions are stochastic control Lyapunov functions with a number of compelling
properties. In particular, a-priori bounds on trajectory suboptimality are
shown for these approximate value functions. The result is a technique whereby
approximate solutions may be computed with non-increasing error via a hierarchy
of semidefinite optimization problems.
