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Published June 2014 | public
Book Section - Chapter

Semidefinite Relaxations for Stochastic Optimal Control Policies

Abstract

Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation have led to the discovery of a formulation of the value function as a linear Partial Differential Equation (PDE) for stochastic nonlinear systems with a mild constraint on their disturbances. This has yielded promising directions for research in the planning and control of nonlinear systems. This work proposes a new method obtaining approximate solutions to these linear stochastic optimal control (SOC) problems. A candidate polynomial with variable coefficients is proposed as the solution to the SOC problem. A Sum of Squares (SOS) relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function.

Additional Information

© 2014 AACC. The authors would like to thank Venkat Chandrasakaran for guidance and suggestions. This work was partially supported by DARPA, through the ARM-S and DRC programs, as well as the Robotics Technology Consortium Alliance (RCTA).

Additional details

Created:
August 20, 2023
Modified:
October 20, 2023