Theoretical Guarantees for Satisfaction of Terminal State Constraints for Nonlinear Stochastic Systems

In several engineering applications, it is desired to bring a system from an initial configuration to a specific terminal configuration. A motivational example is the vertical landing of reusable rockets which are required to come to full stop at an exact location on the landing platform in an upright configuration with all linear and angular velocities coming to zero. While in a deterministic setting, one can study these problems and provide theoretical guarantees for the satisfaction of the terminal state requirements, e.g., by employing the Pontryagin Minimum Principle (PMP), no such guarantees can be provided for exact satisfaction of terminal state constraints in a stochastic setting and, inevitably, one needs to seek alternative expressions of the desired requirements and establish guarantees for those alternatives. This article presents two novel approaches, each with an alternative expression of the terminal state requirement, and each providing theoretical guarantees for optimality and the satisfaction of the associated terminal state constraints. The first approach is to impose a constraint on the conditional expectations of the terminal state at all future times in which case the associated optimality conditions are expressed in the form of the Terminally Constrained Stochastic Minimum Principle (TC-SMP). The second approach is to impose a terminal state constraint as the matching of the probability distribution of the terminal state with a desired probability distribution in which case the associated optimality conditions are expressed using Hamilton-Jacobi (HJ) type equations. Numerical examples are provided to illustrate the results.