The Maximal Positively Invariant Set: Polynomial Setting

This note considers the maximal positively invariant set for polynomialdiscrete time dynamics subject to constraints specified by a basicsemialgebraic set. The note utilizes a relatively direct, but apparentlyoverlooked, fact stating that the related preimage map preserves basicsemialgebraic structure. In fact, this property propagates to underlyingset--dynamics induced by the associated restricted preimage map in general andto its maximal trajectory in particular. The finite time convergence of thecorresponding maximal trajectory to the maximal positively invariant set isverified under reasonably mild conditions. The analysis is complemented with adiscussion of computational aspects and a prototype implementation based onexisting toolboxes for polynomial optimization.