Strong Duality and Dual Ascent Approach to Continuous-Time Chance-Constrained Stochastic Optimal Control

TL;DR

This paper introduces a duality-based method for solving continuous-time chance-constrained stochastic optimal control problems, transforming probabilistic constraints into expectation-based formulations and solving them via path integral techniques.

Contribution

It develops a strong duality framework for chance-constrained SOC without conservative approximations, enabling efficient numerical solutions through path integral methods.

Findings

Strong duality between primal and dual problems established

Path integral approach effectively solves the dual problem

Simulation results demonstrate the method's applicability to robot motion planning

Abstract

The paper addresses a continuous-time continuous-space chance-constrained stochastic optimal control (SOC) problem where the probability of failure to satisfy given state constraints is explicitly bounded. We leverage the notion of exit time from continuous-time stochastic calculus to formulate a chance-constrained SOC problem. Without any conservative approximation, the chance constraint is transformed into an expectation of an indicator function which can be incorporated into the cost function by considering a dual formulation. We then express the dual function in terms of the solution to a Hamilton-Jacobi-Bellman partial differential equation parameterized by the dual variable. Under a certain assumption on the system dynamics and cost function, it is shown that a strong duality holds between the primal chance-constrained problem and its dual. The Path integral approach is utilized to numerically solve the dual problem via gradient ascent using open-loop samples of system trajectories. We present simulation studies on chance-constrained motion planning for spatial navigation of mobile robots and the solution of the path integral approach is compared with that of the finite difference method.