Ogura, M.; Martin, C. F. A limit formula for joint spectral radius with \(p\)-radius of probability distributions. (English) Zbl 1294.15015 Linear Algebra Appl. 458, 605-625 (2014). Summary: In this paper we show a characterization of the joint spectral radius of a set of matrices as the limit of the \(p\)-radius of an associated probability distribution when \(p\) tends to \(\infty\). Allowing the set to have infinitely many matrices, the obtained formula extends the results in the literature. Based on the formula, we then present a novel characterization of the stability of switched linear systems for an arbitrary switching signal via the existence of stochastic Lyapunov functions of any higher degrees. Numerical examples are presented to illustrate the results. Cited in 1 Document MSC: 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A18 Eigenvalues, singular values, and eigenvectors 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory 93E15 Stochastic stability in control theory 93C05 Linear systems in control theory Keywords:joint spectral radius; \(p\)-radius; Lyapunov functions; absolute exponential stability; switched linear system; numerical example Software:JSR PDFBibTeX XMLCite \textit{M. Ogura} and \textit{C. F. Martin}, Linear Algebra Appl. 458, 605--625 (2014; Zbl 1294.15015) Full Text: DOI arXiv References: [1] Rota, G.-C.; Strang, W. G., A note on the joint spectral radius, Indag. Math., 22, 379-381 (1960) · Zbl 0095.09701 [2] Jungers, R. M., The Joint Spectral Radius, Lecture Notes in Control and Inform. Sci., vol. 385 (2009), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg [3] Tsitsiklis, J. N.; Blondel, V. D., The Lyapunov exponent and joint spectral radius of pairs of matrices are hard - when not impossible - to compute and to approximate, Math. Control Signals Systems, 10, 31-40 (1997) · Zbl 0888.65044 [4] Blondel, V. D.; Nesterov, Y., Computationally efficient approximations of the joint spectral radius, SIAM J. Matrix Anal. Appl., 27, 256-272 (2005) · Zbl 1089.65031 [5] Parrilo, P. A.; Jadbabaie, A., Approximation of the joint spectral radius using sum of squares, Linear Algebra Appl., 428, 2385-2402 (2008) · Zbl 1151.65032 [6] Protasov, V. Y.; Jungers, R. M.; Blondel, V. D., Joint spectral characteristics of matrices: a conic programming approach, SIAM J. Matrix Anal. Appl., 31, 2146-2162 (2010) · Zbl 1203.65093 [8] Jia, R.-Q., Subdivision schemes in \(L_p\) spaces, Adv. Comput. Math., 3, 309-341 (1995) · Zbl 0833.65148 [9] Wang, Y., Two-scale dilation equations and the mean spectral radius, Random Comput. Dyn., 4, 49-72 (1996) · Zbl 0872.39007 [10] Protasov, V. Y., The generalized joint spectral radius. A geometric approach, Izv. Math., 61, 995-1030 (1997) · Zbl 0893.15002 [11] Jungers, R. M.; Protasov, V. Y., Fast methods for computing the \(p\)-radius of matrices, SIAM J. Sci. Comput., 33, 1246-1266 (2011) · Zbl 1236.65036 [13] Ogura, M.; Martin, C. F., Generalized joint spectral radius and stability of switching systems, Linear Algebra Appl., 439, 2222-2239 (2013) · Zbl 1280.93090 [15] Lin, H.; Antsaklis, P., Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Automat. Control, 54, 308-322 (2009) · Zbl 1367.93440 [16] Shorten, R.; Wirth, F.; Mason, O.; Wulff, K.; King, C., Stability criteria for switched and hybrid systems, SIAM Rev., 49, 545-592 (2007) · Zbl 1127.93005 [17] Seidman, T. I.; Schneider, H.; Arav, M., Comparison theorems using general cones for norms of iteration matrices, Linear Algebra Appl., 399, 169-186 (2005) · Zbl 1071.15025 [18] Bertram, J.; Sarachik, P., Stability of circuits with randomly time-varying parameters, IRE Trans. Circuit Theory, 6, 260-270 (1959) [19] Ahmadi, G., On the mean square stability of linear difference equations, Appl. Math. Comput., 241, 233-241 (1979) · Zbl 0412.39003 [20] Feng, X.; Loparo, K. A.; Ji, Y.; Chizeck, H. J., Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control, 31, 38-53 (1992) · Zbl 0747.93079 [21] Dayawansa, W.; Martin, C., A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Trans. Automat. Control, 44, 751-760 (1999) · Zbl 0960.93046 [22] Molchanov, A.; Pyatnitskiy, Y., Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Systems Control Lett., 13, 59-64 (1989) · Zbl 0684.93065 [23] Lang, R., A note on the measurability of convex sets, Arch. Math., 47, 90-92 (1986) · Zbl 0607.28003 [24] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), SIAM: SIAM Philadelphia · Zbl 0484.15016 [25] Brewer, J., Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits Syst., 25, 772-781 (1978) · Zbl 0397.93009 [26] Bogachev, V. I., Measure Theory (2007), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg · Zbl 1120.28001 [27] Gurvits, L., Stability of discrete linear inclusion, Linear Algebra Appl., 231, 47-85 (1995) · Zbl 0845.68067 [28] Wirth, F., The generalized spectral radius and extremal norms, Linear Algebra Appl., 342, 17-40 (2002) · Zbl 0996.15020 [29] Xu, J.; Xiao, M., A characterization of the generalized spectral radius with Kronecker powers, Automatica, 47, 1530-1533 (2011) · Zbl 1227.15013 [30] Ando, T.; Shih, M.-H., Simultaneous contractibility, SIAM J. Matrix Anal. Appl., 19, 487-498 (1998) · Zbl 0912.15033 [31] Protasov, V. Y., Extremal \(L_p\)-norms of linear operators and self-similar functions, Linear Algebra Appl., 428, 2339-2356 (2008) · Zbl 1147.15023 [32] Vandergraft, J. S., Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math., 16, 1208-1222 (1968) · Zbl 0186.05701 [33] Fang, Y.; Loparo, K. A., On the relationship between the sample path and moment Lyapunov exponents for jump linear systems, IEEE Trans. Automat. Control, 47, 1556-1560 (2002) · Zbl 1364.93562 [34] Arnold, L., A formula connecting sample and moment stability of linear stochastic systems, SIAM J. Appl. Math., 44, 793-802 (1984) · Zbl 0561.93063 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.