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Let $x$ be a positive scalar variable. Its time derivative satisfies $$|\dot{x}(t)|\le \exp\left\{-\int_{0}^t\frac{1}{x(\tau)}\mathrm{d} \tau\right\}$$ where $|\cdot|$ denotes the absolute value. From the above inequality, can we say $x(t)$ with $t\in[0,+\infty)$ has a finite upper bound? No need to compute an exact upper bound. It is sufficient to show there exists a finite upper bound. I know if $|\dot{x}(t)|\le \exp\left\{-kt\right\}$, then $x(t)=\int_{0}^t \dot{x} \mathrm{d}\tau$ will have an upper bound $1/k$.

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@WillieWong: you could write the answer you wrote at mathoverflow. –  Davide Giraudo Sep 7 '12 at 19:40
    
@DavideGiraudo: Thanks for the comment. I have posted an answer below to refer to the answer given in mathoverflow. –  Shiyu Sep 9 '12 at 13:42
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This problem is solved in Mathoverflow.

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