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Consider a linear ODE system $$\dot x_k=\sum_{j=1}^ma_{kj}(t)x_j,\qquad k=1,\ldots, m,\quad a_{kj}(t)\in C[0,\infty).\quad (1)$$

Proposition. Suppose that $$\sup_{t\ge 0}\Big\{\int_0^t\Big(a_{kk}(s)+\sum_{j\ne k}|a_{kj}(s)|\Big)ds\Big\}<\infty,\quad k=1,\ldots, m.$$ Then system (1) is stable in the sense of Lyapunov.

Please, is there any reference for this proposition or does it perhaps follows from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article.

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  • $\begingroup$ Your system is of the form $x'=(D+R)x$, with $D$ diagonal and $R\in L^1$. With just a little bit of extra information on the $d_j$, Levinson's theorem would give you solutions of the form $x_k=(e_k+o(1))\exp(\int_0^t a_{kk}(s)\, ds)$. $\endgroup$ Commented Oct 3, 2014 at 19:53
  • $\begingroup$ That will not be true here, without extra assumptions, but of course this is also more than you want. You could take a look at the methods used in the proof of Levinson's theorem to check if these can be adapted. $\endgroup$ Commented Oct 3, 2014 at 19:55

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I will assume that the coefficients $a_{kj}(t), t\geq 0$ are all real.

The expression under the integral is what is called the initial growth rate of the matrix $A(t)$ with respect to the $\infty$-norm. Another standard term for this is the matrix measure of $A(t)$ with respect to the $\infty$-norm. Using standard facts about the initial growth rate the desired result follows.

The initial growth rate of a matrix $A \in \mathbb{R}^{m\times m}$ with respect to a norm $\|\cdot\|$ on $\mathbb{R}^m$ and the associated induced norm on the space of matrices may be defined as follows: It is the infimum of the numbers $C$ such that for all $t\geq 0$ we have $$ \|e^{At}\| \leq e^{Ct} .$$

It is well known that the initial growth rate $\mu(A)$ of $A \in \mathbb{R}^{m\times m}$ with respect to $\|\cdot \|_\infty$ is precisely $$ \mu(A) = \max_{k=1,\ldots,n} a_{kk} + \sum_{j\neq k} |a_{kj}| .$$

It follows that for a linear time-varying differential equation $$ \dot{x}(t) = A(t)x(t)$$ with transition matrix $\Phi(t,s)$ we have for all $t \geq t_0$ $$ \| \Phi(t,t_0)\| \leq \exp \left( \int_{t_0}^t \mu(A(s)) ds \right) .$$

So with your assumption we have that for all initial times $t_0$ we have that $\| \Phi(t,t_0)\|$ is bounded in $t$. This is Lapunov stability of the zero position for all initial times. Note that the assumption does not imply uniform stability-

References for facts about the initial growth rate or matrix measure are

Hinrichsen & Pritchard, Mathematical Systems Theory I, Springer, 2005
Vidyasagar, Nonlinear Systems Analysis, SIAM, 2002
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