Recent results in the study of the Hamilton Jacobi Bellman (HJB) equationhave led to the discovery of a formulation of the value function as a linearPartial Differential Equation (PDE) for stochastic nonlinear systems with amild constraint on their disturbances. This has yielded promising directionsfor research in the planning and control of nonlinear systems. This workproposes a new method obtaining approximate solutions to these linearstochastic optimal control (SOC) problems. A candidate polynomial with variablecoefficients is proposed as the solution to the SOC problem. A Sum of Squares(SOS) relaxation is then taken to the partial differential constraints, leadingto a hierarchy of semidefinite relaxations with improving sub-optimality gap.The resulting approximate solutions are shown to be guaranteed over- andunder-approximations for the optimal value function.
展开▼
