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I have a problem:

Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$

and $$\begin{cases} & \mathrm{ } \lim \inf_{t \to \infty}\int_{t_0}^{t}Tr(A(t_1)dt_1> - \infty;(1') \\ \\ & \mathrm{ } \int_{t_0}^{\infty}\left \|f(t_1) \right \|dt_1<+\infty; (2'). \end{cases}$$

Prove that all the solutions of (2): $$\frac{dy}{dt}=A(t)y+f(t)$$ are bounded in $ \left[t_0,+\infty \right )$.

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Ok, Here's my solution: (I'm not sure, but I still post it).

  • Assume that $X(t)$ be a fundamental matrix of (1), $X(t_0)=E$. Then $$y(t)=X(t)X^{-1}(t)x(t_0)+X(t)\int_{t_0}^{t}X^{-1}(s)f(s)\mathrm{ds}$$
  • Because system (1) is stable, so $X(t)$ is bounded

  • Applying Ostrogradski - Liouville, we have:$$\det X(t)=\det E \cdot \exp \left[\int_{t_0}^{t} TrA(s)\mathrm{ds}\right]=\exp \left[\int_{t_0}^{t} TrA(s)\mathrm{ds}\right]$$ whence $$\lim \inf_{t \to \infty}\int_{t_0}^{t}Tr(A(s)\mathrm{d}s=\lim \inf_{t \to \infty}\ln\det X(t)> - \infty$$ We see that, provided (1') is satisfied, $X^{-1}(t)$ is bounded as $t \to \infty$.

  • Let $c_1=\max\{\sup_{t \ge t_0}\|X(t)\|;\sup_{t \ge t_0}\|X^{-1}(t)\|;\sup\|y(t_0)\|\}$.

Therefore, $\|y(t)\| \le c_{1}^{3}+c_{1}^{2}\int_{t_0}^{t}\|f(s)\|\mathrm{d}s$.

Since (2'): $\int_{t_0}^{\infty}\left \|f(t_1) \right \|dt_1<+\infty$, we're done!

I've made several errors with my solution, may be! :(. Anyone can check it help me? Thanks!

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Just a notational comment: this would be a lot clearer if you indicated that $x$ and $y$ were vectors, or at least mentioned $y\in \mathbb R^n$ somewhere. –  Alqatrkapa Nov 7 at 13:44

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