Nonlinear Stochastic Model Predictive Control: Existence, Measurability, and Stochastic Asymptotic Stability

In this article, we establish a collection of new theoretical properties for nonlinear stochastic model predictive control (SMPC). Based on the concept of stochastic input-to-state stability (SISS), we define robust asymptotic stability in expectation (RASiE) and establish that nonlinear SMPC renders the origin of the closed-loop system RASiE. Moreover, we establish several new foundational results that have not been addressed in previous research. Specifically, we verify that, under basic regularity assumptions, a solution to the SMPC optimization problem exists and the closed-loop trajectory is Borel measurable thereby guaranteeing that all relevant stochastic properties of the closed-loop system are indeed well-defined. We present a numerical example to demonstrate the nonintuitive behavior that can arise from nonlinear SMPC.