The reachability cones of essentially nonnegative matrices

Abstract

Let A be an n×n essentially nonnegative matrix and consider the linear differential system . We show that there exists a constant h(A)>0 such that the trajectory emanating from x₀ reaches at a finite time t₀ =t(x ₀)⩾0 if and only if the sequence of points generated by a finite differences approximation from x₀ , with time-step 0<h<h(A), reaches at a finite index k₀ =k(x ₀)⩾0. This generalizes and strengthens earlier results of two of the authors, where some additional spectral restrictions were imposed on A. Our proof makes use of the existence of a basis of nonnegative vectors to the Perron eigenspace.

^*Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860. This author would also like to thank NSERC for making it possible for him to visit Ronald J. Stern in Montreal.

^**Research supported by the Natural Sciences and Engineering Council of Canada, grant No. A4641

^***Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860.

^*Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860. This author would also like to thank NSERC for making it possible for him to visit Ronald J. Stern in Montreal.

^**Research supported by the Natural Sciences and Engineering Council of Canada, grant No. A4641

^***Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860.

Notes

^*Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860. This author would also like to thank NSERC for making it possible for him to visit Ronald J. Stern in Montreal.

^**Research supported by the Natural Sciences and Engineering Council of Canada, grant No. A4641

^***Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860.