Symbolic dynamics for active sets of a class of constrained nonlinear optimal control and MPC problems

Solutions to optimal control problems are usually understood to provide optimal trajectories. In this paper, we show that the optimal state-space system dynamics induce a dynamics of the active sets. More specifically, given the optimal active set at the solution obtained at the current time, its successor optimal active set (which, in turn, defines the successor solution) can be found with index set operations. These operations do not involve any optimal control (or other optimization or integration) problem, but they can be described with simple rules. These rules constitute the symbolic dynamics for active sets. The present paper treats a particular constrained nonlinear problem class, extending earlier results for the constrained linear-quadratic case. Copyright (C) 2024 The Authors.