Axiomatic representation for smallness classes of coefficient perturbations to linear differential systems

E. K. Makarov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 1 (2014), pp. 46-57

Résumé

A number of problems in the Lyapunov exponent theory of linear differential systems $$ \dot x=A(t)x,\quad x\in\mathbb R^n ,\quad t\geqslant0, $$ can be reduced to an investigation of the influence of coefficient perturbations on characteristic exponents and other asymptotic invariants of perturbed systems $$ \dot y=A(t)y+Q(t)y,\quad y\in\mathbb R^n,\quad t\geqslant0. $$ 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.

Détail

Keywords: linear systems, Lyapunov exponents
Mots-clés : perturbations.

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