Realization of joint spectral radius via Ergodic theory
Pages: 22  30,
January
2011
doi:10.3934/era.2011.18.22 Abstract
References
Full text (181.2K)
Related Articles
Xiongping Dai  Department of Mathematics, Nanjing University, Nanjing, 210093, China (email)
Yu Huang  Department of Mathematics, Zhongshan University, Guangzhou 510275, China (email)
Mingqing Xiao  Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, United States (email)
1 
L. Arnold, "Random Dynamical Systems," Springer Monographs in Math., SpringerVerlag, Berlin, Heidelberg, New York, 1998. 

2 
N. Barabanov, Lyapunov indicators of discrete inclusions IIII, Autom. Remote Control, 49 (1988), 152157, 283287, 558565. 

3 
J. P. Bell, A gap result for the norms of semigroups of matrices, Linear Algebra Appl., 402 (2005), 101110. 

4 
M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 2127. 

5 
V. D. Blondel, J. Theys and A. A. Vladimirov, An elementary counterexample to the finiteness conjecture, SIAM J. Matrix Anal. Appl., 24 (2003), 963970. 

6 
V. D. Blondel and Y. Nesterov, Polynomialtime computation of the joint spectral radius for some sets of nonnegative matrices, SIAM J. Matrix Anal. Appl., 31 (2009), 865876. 

7 
V. D. Blondel, R. Jungers and V. Protasov, On the complexity of computing the capacity of codes that avoid forbidden difference patterns, IEEE Trans. Inform. Theory, 52 (2006), 51225127. 

8 
V. D. Blondel and J. N. Tsitsiklis, The boundedness of all products of a pair of matrices is undecidable, Systems Control Lett., 41 (2000), 135140. 

9 
J. Bochi, Inequalities for numerical invariants of sets of matrices, Linear Algebra Appl., 368 (2003), 7181. 

10 
T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77111. 

11 
D. Colella and C. Heil, The characterization of continuous, fourcoefficient scaling functions and wavelets, IEEE Trans. Inform. Theory, 38 (1992), 876881. 

12 
X. Dai, Extremal and Barabanov seminorms of a semigroup generated by a bounded family of matrices, J. Math. Anal. Appl., 379 (2011), 827833. 

13 
X. Dai, Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems, J. Differential Equations, 250 (2011), 35843629. 

14 
X. Dai, Optimal state points of the subadditive ergodic theorem, Nonlinearity, 24 (2011), 15651573. 

15 
X. Dai, Y. Huang and M. Xiao, Almost sure stability of discretetime switched linear systems: A topological point of view, SIAM J. Control Optim., 47 (2008), 21372156. 

16 
X. Dai, Y. Huang, and M. Xiao, Periodically switched stability induces exponential stability of discretetime linear switched systems in the sense of Markovian probabilities, Automatica, 47 (2011), 15121519. 

17 
I. Daubechies and J. C. Lagarias, Two scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22 (1991), 13881410. 

18 
I. Daubechies and J. C. Lagarias, Two scale difference equations. II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal., 23 (1992), 10311079. 

19 
I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), 227263. 

20 
I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), 227263, Corrigendum/addendum, 327 (2001), 6983. 

21 
L. Elsner, The generalized spectralradius theorem: An analyticgeometric proof, Linear Algebra Appl., 220 (1995), 151159. 

22 
L. Gurvits and A. Olshevsky, On the NPhardness of checking matrix polytope stability and continuoustime switching stability, IEEE Trans. Automat. Control, 54 (2009), 337341. 

23 
K. G. Hare, I. D. Morris, N. Sidorov and J. Theys, An explicit counterexample to the LagariasWang finiteness conjecture, Adv. Math., 226 (2011), 46674701. 

24 
C. Heil and G. Strang, Continuity of the joint spectral radius: Applications to wavelets, in "Linear Algebra for Signal Processing," IMA Vol. Math. Appl. 69, SpringerVerlag, New York, (1995), 5161. 

25 
L. Gurvits, Stability of discrete linear inclusions, Linear Algebra Appl., 231 (1995), 4785. 

26 
H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457469. 

27 
R. Jungers, V. Protasov and V. Blondel, Efficient algorithms for deciding the type of growth of products of integer matrices, Linear Algebra Appl., 428 (2008), 22962311. 

28 
V. S. Kozyakin, Structure of extremal trajectories of discrete linear systems and the finiteness conjecture, Autom. Remote Control, 68 (2007), 174209. 

29 
J. C. Lagarias and Y. Wang, The finiteness conjecture for the generalized spectral radius of a set of matrices, Linear Algebra Appl., 214 (1995), 1742. 

30 
B. E. Moision, A. Orlitsky and P. H. Siegel, On codes that avoid specified differences, IEEE Trans. Inform. Theory, 47 (2001), 433442. 

31 
V. V. Nemytskii and V. V. Stepanov, "Qualitative Theory of Differential Equations," Princeton University Press, Princeton, New Jersey, 1960. 

32 
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Lyapunov, exponents of dynamical systems, Trudy Mosk Mat. Obšč., 19 (1968), 179210. 

33 
E. Plischke and F. Wirth, Duality results for the joint spectral radius and transient behavior, Linear Algebra Appl., 428 (2008), 23682384. 

34 
E. S. Pyatnitskiĭ and L. B. Rapoport, Periodic motion and tests for absolute stability of nonlinear timedependent systems, Automat. Remote Control, 52 (1991), 13791387. 

35 
G.C. Rota and G. Strang, A note on the joint spectral radius, Indag. Math., 22 (1960), 379381. 

36 
S. J. Schreiber, On growth rates of subadditive functions for semiflows, J. Differential Equations, 148 (1998), 334350. 

37 
M.H. Shih, J.W. Wu and C.T. Pang, Asymptotic stability and generalized Gelfand spectral radius formula, Linear Algebra Appl., 252 (1997), 6170. 

38 
R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545592. 

39 
R. Sturman and J. Stark, Semiuniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000), 113143. 

40 
J. Theys, "Joint Spectral Radius: Theory and Approximations," Ph.D thesis, Université Catholique de Louvain, 2005. 

41 
J. N. Tsitsiklis and V. D. Blondel, The Lyapunov exponent and joint spectral radius of pairs of matrices are hardwhen not impossibleto compute and to approximate, Math. Control Signals Systems, 10 (1997), 3140. 

42 
P. Walters, "An Introduction to Ergodic Theory," G.T.M., 79, SpringerVerlag, New YorkBerlin, 1982. 

Go to top
