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This paper addresses a continuous-time continuous-space chance-constrainedstochastic optimal control (SOC) problem via a Hamilton-Jacobi-Bellman (HJB)partial differential equation (PDE). Through Lagrangian relaxation, we convertthe chance-constrained (risk-constrained) SOC problem to a risk-minimizing SOCproblem, the cost function of which possesses the time-additive Bellmanstructure. We show that the risk-minimizing control synthesis is equivalent tosolving an HJB PDE whose boundary condition can be tuned appropriately toachieve a desired level of safety. Furthermore, it is shown that the proposedrisk-minimizing control problem can be viewed as a generalization of theproblem of estimating the risk associated with a given control policy. Twonumerical techniques are explored, namely the path integral and the finitedifference method (FDM), to solve a class of risk-minimizing SOC problems whoseassociated HJB equation is linearizable via the Cole-Hopf transformation. Usinga 2D robot navigation example, we validate the proposed control synthesisframework and compare the solutions obtained using path integral and FDM.

Chance-Constrained Stochastic Optimal Control via Path Integral and Finite Difference Methods