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Perturbation Theory for Lyapunov Exponents of Discrete Linear Systems

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nauki techniczne » automatyka i robotyka
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This book is devoted to the influence of parametric perturbations and uncertainties on the Lyapunov exponents of discrete linear systems with timevarying coefficients. Different types of perturbations are considered, limited in terms of certain norms, tending to zero with a specified rate. For each of them, we describe their impact on the values of Lyapunov exponents. Results of such considerations, as presented in this book, while being widely known for stationary systems, are new and original when referred to the systems with variable coefficients. The book is addressed primarily to persons interested in the theory of Lyapunov exponents, which well characterize the behavior of the trajectory of time-varying systems. It is also important that the results of this work can be used by graduate students and researchers in new areas of research in dynamics of discrete systems. Analysis of discrete-time systems is also important from cognitive point of view and well established part of control theory and applied computer science.


1 Definitions of characteristic exponents and basic properties 1
2 Bounded perturbations 7
2.1 Stability of Lyapunov exponents  7
2.2 Two-dimensional system  11
2.3 Generalized spectral radius and subradius  18
2.4 Time-invariant system  20
3 Arbitrary small perturbations 23
3.1 Central exponents of families of sequences  24
3.2 Central exponents of linear system. Definitions and basic properties  29
3.3 Central exponents of triangular system  32
3.4 Relations between central and Lyapunov exponents  34
4 Regular systems and regularity coefficients 41
4.1 Coefficients of regularity and basic properties of regular systems  41
4.2 Grobman perturbation  44
4.3 Grobman perturbation for diagonal system  45
4.4 Characterization of regularity in terms of nonhomogeneous system  53
5 Perturbations exponentially tending to zero  61
5.1 Definitions and basic properties of exponential exponent  62
5.2 Formula for the Exponential Exponents  66
5.3 Basic properties of sigma exponents  70
5.4 Relations between sigma exponents and Lyapunov exponents  73
6 Stability by the Linear Approximation  77
6.1 Preliminaries  77
6.2 Perron examples  78
6.3 Conditions for exponential stability  80
7 Perron exponents 85
7.1 Numbers of Perron exponents  86
7.2 Formula for Perron exponent  89
7.3 Bounds for Lyapunov and Perron Exponents  93
8 Conclusion 99
References 103

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