Positively invariant sets for constrained continuous-time systems with cone properties

Abstract

This note deals with some properties of particular bounded sets w.r.t. linear continuous-time systems described by x/spl dot/(t)=A(0)x(t)+c(t), where c(t)/spl isin//spl Omega//spl sub/R/sup n/, /spl Omega/ a compact set, and matrix e/sup tA(0/) has the property of leaving a proper cone K positively invariant, that is e/sup tA(0/)K/spl sub/K. The considered bounded sets /spl Dscr/(K; a, b) are described as the intersection of shifted cones. Necessary and sufficient conditions are given. They guarantee that such sets are positively invariant w.r.t. the considered system. The trajectories starting from x/sub 0//spl isin/R/sup n///spl Dscr/(K; a, b) (respectively to x/sub 0//spl isin/R/sup n/) are studied in terms of attractivity and contractivity of the set /spl Dscr/(K; a, b). The results are applied to the study of the constrained state feedback regulator problem. >