- #1

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- Homework Statement
- Assume that the homogenous system ##x'=Ax## is asymptotically stable. Show that if ##b(t)## is bounded for ##t\geq t_0## then every solution of the system ##x'=Ax+b(t)## is bounded for ##t\geq t_0##.

- Relevant Equations
- A system ##x'=Ax## is said to be asymptotically stable iff all eigenvalues of ##A## have a negative real part. Moreover, the general solution to ##x'=Ax+b(t)## and ##x(t_0)=x_0## is given by ##x(t)=e^{tA}x_0+\int_{t_0}^te^{(t-\tau)A}b(\tau)d\tau##.

We need to show ##\lVert x(t)\rVert## is bounded. It is given that ##\lVert b(t)\rVert\leq c_1## for ##t\geq t_0##. A TA has claimed that ##\lVert e^{tA}\rVert\leq ce^{-\epsilon t}## holds for some ##\epsilon>0## and a constant ##c##, when ##t\geq0##. I have a hard time confirming this claim and I'd be grateful if anyone could comment on this. If this bound is true, then the statement in the exercise follows from the following estimates

First, for ##t\geq t_0##, \begin{align}\left\lVert \int_{t_0}^te^{(t-\tau)A}b(\tau)d\tau\right\rVert&\leq \int_{t_0}^t \left\lVert e^{(t-\tau)A}\right\rVert \left\lVert b(\tau) \right\rVert d\tau \nonumber \\

&\leq c\cdot c_1 \int_{t_0}^te^{-\epsilon(t-\tau)}d\tau \nonumber \\

& =c\cdot c_1\left(\frac1{\epsilon}-\frac{e^{-\epsilon(t-t_0)}}{\epsilon}\right) \nonumber \\

&\leq \frac{c\cdot c_1}{\epsilon} \nonumber\\

& =\frac{C}{\epsilon}

\nonumber\end{align}

Second, for ##t\geq t_0##, $$\lVert e^{tA}x_0\rVert\leq ce^{-\epsilon t}\lVert x_0\rVert \leq ce^{-\epsilon t_0} \lVert x_0\rVert=D$$

Finally then, ##\lVert x(t)\rVert## must be bounded by ##\frac{C}{\epsilon}+D## when ##t\geq t_0##.

But why does ##\lVert e^{tA}\rVert\leq ce^{-\epsilon t}## hold? I know that every element in ##e^{tA}## is a linear combination of terms of the form ##t^je^{\lambda t}##, where ##\lambda## is an eigenvalue of ##A## and ##j## is less than the multiplicity of that eigenvalue. Moreover, I know of ##\lVert e^{A}\rVert\leq e^{\lVert A\rVert}##, but I don't know if this is helpful.

Also, I have assumed in my computations that ##t_0\geq 0##. I guess it makes no sense for ##t_0<0##, right?

First, for ##t\geq t_0##, \begin{align}\left\lVert \int_{t_0}^te^{(t-\tau)A}b(\tau)d\tau\right\rVert&\leq \int_{t_0}^t \left\lVert e^{(t-\tau)A}\right\rVert \left\lVert b(\tau) \right\rVert d\tau \nonumber \\

&\leq c\cdot c_1 \int_{t_0}^te^{-\epsilon(t-\tau)}d\tau \nonumber \\

& =c\cdot c_1\left(\frac1{\epsilon}-\frac{e^{-\epsilon(t-t_0)}}{\epsilon}\right) \nonumber \\

&\leq \frac{c\cdot c_1}{\epsilon} \nonumber\\

& =\frac{C}{\epsilon}

\nonumber\end{align}

Second, for ##t\geq t_0##, $$\lVert e^{tA}x_0\rVert\leq ce^{-\epsilon t}\lVert x_0\rVert \leq ce^{-\epsilon t_0} \lVert x_0\rVert=D$$

Finally then, ##\lVert x(t)\rVert## must be bounded by ##\frac{C}{\epsilon}+D## when ##t\geq t_0##.

But why does ##\lVert e^{tA}\rVert\leq ce^{-\epsilon t}## hold? I know that every element in ##e^{tA}## is a linear combination of terms of the form ##t^je^{\lambda t}##, where ##\lambda## is an eigenvalue of ##A## and ##j## is less than the multiplicity of that eigenvalue. Moreover, I know of ##\lVert e^{A}\rVert\leq e^{\lVert A\rVert}##, but I don't know if this is helpful.

Also, I have assumed in my computations that ##t_0\geq 0##. I guess it makes no sense for ##t_0<0##, right?