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Year
2014
Issue
1
Pages
46-57
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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 Belarusa
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 non-trivial 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. 46-57.
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