Boundary results for positively invariant cones and their reachability cones

Abstract

We say that a closed, convex, and solid cone is positively invariant with respect to the linear system of differential equations . In a previous paper, [2], the authors obtained formulas for the closure of the reachability cone

under certain additional assumptions on A. The purpose of the present paper is to characterize the set X_A (K) ⋂ ∂ X_A (K) which consists of those points on the boundary of X_A (K) which actually reach A, in the case when K is a simplicial cone. We use this characterization to construct an algorithm for the computation of X_A (K) from its closure. Furthermore, to obtain our characterization we develop a result which is of interest in itself, namely, that the maximal positively invariant subset of the boundary of a positively invariant simplicial cone k is, precisely, the union of its positively invariant faces.