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Stability of linearly perturbed linear nonautonomous system

  1. Sep 3, 2014 #1
    I have a linear time-varying linearly perturbed ODE of the form:

    [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 is whether 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] is exponentially stable. 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 is exponentially stable, 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?
     
  2. jcsd
  3. Sep 3, 2014 #2
    Found the answer. According to the theorem 1.1 from this paper, there indeed exists sufficiently small [itex]\beta[/itex] such that [itex]\dot{x}=\left[A(t)+B(t)\right]x[/itex] has negative maximum lyapunov exponent.
     
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