stability of linearly perturbed linear nonautonomous system

I have a linear time-varying linearly perturbed ODE of the form:

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

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

The question is whether there exists a sufficiently small $\beta$ such that the origin is asymptotically stable (not necessarily exponentially! I don't care about that).

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

Found the answer. According to the theorem 1.1 from this paper, there indeed exists sufficiently small $\beta$ such that $\dot{x}=\left[A(t)+B(t)\right]x$ has negative maximum lyapunov exponent.
• Cool. Would be great to hear your opinion. I think condition C2 is satisfied cuz all $A_{kk}(t)$ are negative by definition. The question however is still open if $A$ is block-triangular with negative definite blocks on the main diagonal for all $t$ – dmytro Sep 4 '14 at 8:26