I have a linear time-varying linearly perturbed ODE of the form:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\dot{x} = [A(t)+B(t)]x

[/tex]

where [itex]A(t)[/itex] is a bounded lower-triangular matrix with negative functions on the main diagonal, i.e. [itex]0>a^0\ge a_{ii}(t)[/itex]. The matrix [itex]B(t)[/itex] is bounded, so that [itex]||B(t)|| \le \beta[/itex].

The question iswhether there exists a sufficiently small bound [itex]\beta[/itex] such that the origin is asymptotically stable (not necessarily exponentially! I don't care about that).

This paper gives a positive answer to the question, BUT the assumption is that [itex]\dot{x}=A(t)x[/itex] isexponentiallystable. In my case, while it is possible to show inductively stability of [itex]\dot{x}=A(t)x[/itex], it is not clear to me that it isexponentiallystable, since, for example [itex]\dot{x}_2 = a_{22}(t)x_2 + a_{21}(t)x_1(t)[/itex] such that [itex]x_1(t)\to0[/itex] is exponentially bounded. Is [itex]x_2(t)[/itex] also exponentially bounded?

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# Stability of linearly perturbed linear nonautonomous system

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