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Belarus Minsk
Section  Mathematics 
Title  Axiomatic representation for smallness classes of coefficient perturbations to linear differential systems 
Author(s)  Makarov E.K.^{a} 
Affiliations  Institute of Mathematics, National Academy of Sciences of Belarus^{a} 
Abstract  A number of problems in the Lyapunov exponent theory of linear differential systems $$\dot x=A(t)x, \quad x\in \mathbb{R}^n, \quad t\geqslant 0,$$ can be reduced to an investigation of the influence of coefficient perturbations on characteristic exponents and other asymptotic invariants of perturbed systems $$\dot y=A(t)y+Q(t)y, \quad y\in \mathbb{R}^n, \quad t\geqslant 0.$$ Here perturbations are assumed to be in some classes of smallness, i.e. certain subsets of the space $\mathrm{KC}_n(\mathbb{R}^+)$ of piecewise continuous and bounded on the positive semiaxis $n\times n$matrices. Commonly used classes of perturbations, such as infinitesimal (vanishing at infinity), exponentially decaying or integrable on the positive semiaxis are defined by specific analytical conditions, but there is no general definition of the smallness class. By analyzing the desirable properties of commonly used classes, we propose an axiomatic definition for this notion, such that most of classes used in the theory of characteristic exponents satisfy this definition. Since the axioms are somewhat cumbersome, for more compact characterization we propose to use the following property of smallness classes: the set of perturbation satisfies the proposed definition if and only if it is a complete matrix algebra over an arbitrary nontrivial ideal of functional ring $\mathrm{KC}_1(\mathbb{R}^+)$ (with the pointwise multiplication) containing at least one strictly positive function. 
Keywords  linear systems, Lyapunov exponents, perturbations 
UDC  517.926.4 
MSC  34D08, 34E10 
DOI  10.20537/vm140104 
Received  20 December 2013 
Language  Russian 
Citation  Makarov E.K. Axiomatic representation for smallness classes of coefficient perturbations to linear differential systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 4657. 
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