On the influence of the exponentially decaying perturbations of matrices coefficients on the Lyapunov exponents of the discrete time-varying linear system

In the paper we examine the influence of parametric perturbations on the Lyapunov exponents of the linear discrete time-varying system. We show on an example that even the perturbations tend to zero exponentially they may change the values of Lyapunov exponents. However, the main result of the paper shows that if the perturbations tends to zero exponentially with sufficiently large exponent, then the Lyapunov exponents of the perturbed and unperturbed systems are the same.

[1]  Michal Niezabitowski,et al.  The complete description of Lyapunov coefficients of irregularity of mutually adjoint linear discrete time-varying systems , 2015, 2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR).

[2]  A. Czornik,et al.  On the Lyapunov exponents of a class of second-order discrete time linear systems with bounded perturbations , 2013 .

[3]  A. Czornik,et al.  On the perturbations of the Lyapunov exponents of discrete linear system , 2010 .

[4]  V. M. Millionshchikov.  A proof of the attainability of the central exponents of linear systems , 1969 .

[5]  N. A. Izobov.  Lyapunov exponents and stability , 2012 .

[6]  Adam Czornik,et al.  Exact boundaries of mobility of the Lyapunov, Perron and Bohl exponents of linear difference systems under small perturbations of the coefficient matrices , 2016 .

[7]  A. Czornik,et al.  Lyapunov exponents for systems with unbounded coefficients , 2013 .

[8]  Aleksander Nawrat,et al.  Generalized spectral radius and Lyapunov exponents of linear time varying systems , 2007, 2007 IEEE International Conference on Control Applications.

[9]  A. Czornik,et al.  On the spectrum of discrete time-varying linear systems , 2013 .

[10]  A. Czornik,et al.  On the perturbations preserving spectrum of discrete linear systems , 2011 .

[11]  A. Czornik,et al.  Corrigendum Lyapunov exponents for systems with unbounded coefficients , 2013 .

[12]  O. Perron.  Die Ordnungszahlen linearer Differentialgleichungssysteme , 1930 .

[13]  Aleksander Nawrat,et al.  On the stability of Lyapunov exponents of discrete linear systems , 2013, 2013 European Control Conference (ECC).

[14]  M. G. Kreïn.  Stability of solutions of differential equations in Banach space , 2007 .

[15]  E. A. Barabanov,et al.  Necessary and sufficient conditions for the stability of Lyapunov exponents of linear differential systems with exponentially decaying perturbations , 2007 .