2019/01/16 by Aleksandar Cvetković, Cvetković, Aleksandar · 1 citation
Mathematics · #15A42 #15B35 #15B48 #90C26 #93C30 #FOS: Mathematics #Optimization and Control (math.OC) #math.OC #msc:15A42 #msc:15B35 #msc:15B48 #msc:90C26 #msc:93C30
paper · pdf · doi:10.48550/arxiv.1901.05522
38 pages
arxiv created 2019/01/16 · arxiv updated 2020/05/20
Metzler matrices play a crucial role in positive linear dynamical systems. Finding the closest stable Metzler matrix to an unstable one (and vice versa) is an important issue with many applications. The stability considered here is in the sense of Hurwitz, and the distance between matrices is measured in l_∞, l1, and in the max norms. We provide either explicit solutions or efficient algorithms for obtaining the closest (un)stable matrix. The procedure for finding the closest stable Metzler matrix is based on the recently introduced selective greedy spectral method for optimizing the Perron eigenvalue. Originally intended for non-negative matrices, here is generalized to Metzler matrices. The efficiency of the new algorithms is demonstrated in examples and by numerical experiments in the dimension of up to 2000. Applications to dynamical systems, linear switching systems, and sign-matrices are considered.