Distributionally Constrained Convex Duality Optimal Control (DC-CDOC) Subject to Different Forms of Constraining the Terminal State of Nonlinear Stochastic Systems

This article explores different methods of constraining the terminal state of nonlinear stochastic systems and presents the results of the Distributionally Constrained Convex Duality Optimal Control (DC-CDOC) for each case. Specifically, we consider: (a) almost surely equality constraints on the terminal state, (b) constraints on the expected value of the terminal state, and (c) constraints on the probability distributions of the terminal state, both (i) under the total probability measure and (ii) under all conditional probability measures. For each case, the associated optimal control problem is formulated as a convex linear program on the space of Radon measures, and by exploiting the duality relations between the space of measures and that of continuous functions, we derive the optimality conditions in the form of an optimization problem over the space of differentiable functions constrained to a Hamilton-Jacobi (HJ) inequality and, in some of these cases, a terminal value inequality. An iterative algorithm is also proposed for identifying the value function in the studied cases.