The Maximal Positively Invariant Set: Polynomial Setting

This note considers the maximal positively invariant set for polynomial discrete time dynamics subject to constraints specified by a basic semialgebraic set. The note utilizes a relatively direct, but apparently overlooked, fact stating that the related preimage map preserves basic semialgebraic structure. In fact, this property propagates to underlying set--dynamics induced by the associated restricted preimage map in general and to its maximal trajectory in particular. The finite time convergence of the corresponding maximal trajectory to the maximal positively invariant set is verified under reasonably mild conditions. The analysis is complemented with a discussion of computational aspects and a prototype implementation based on existing toolboxes for polynomial optimization.