Stochastic optimal control using semidefinite programming for moment dynamics

TL;DR

This paper introduces a semidefinite programming approach to approximate solutions for stochastic optimal control problems with polynomial dynamics, leveraging moment dynamics to derive control strategies.

Contribution

It presents a novel method to solve stochastic control problems by formulating them as deterministic problems in moment space using semidefinite programming.

Findings

Exact solution for linear quadratic cases.

Provides a systematic way to improve approximation accuracy.

Offers a lower bound on the optimal control solution.

Abstract

This paper presents a method to approximately solve stochastic optimal control problems in which the cost function and the system dynamics are polynomial. For stochastic systems with polynomial dynamics, the moments of the state can be expressed as a, possibly infinite, system of deterministic linear ordinary differential equations. By casting the problem as a deterministic control problem in moment space, semidefinite programming is used to find a lower bound on the optimal solution. The constraints in the semidefinite program are imposed by the ordinary differential equations for moment dynamics and semidefiniteness of the outer product of moments. From the solution to the semidefinite program, an approximate optimal control strategy can be constructed using a least squares method. In the linear quadratic case, the method gives an exact solution to the optimal control problem. In more complex problems, an infinite number of moment differential equations would be required to compute the optimal control law. In this case, we give a procedure to increase the size of the semidefinite program, leading to increasingly accurate approximations to the true optimal control strategy.