Find a Nonzero solution $x(t)$ satsifying $\|x(t)\|\le 1$ for every $t\in \mathbb{R}$ or explain why no such solutions exists

Question

Find a Nonzero solution $x(t)$ satsifying $\|x(t)\|\le 1$ for every $t\in \mathbb{R}$ or explain why no such solutions exists

I have found the solution using eigenvalues and eigenvector

but how to solve question (b).

Answer 1

If $c_2$ or $c_3$ is non zero, then the exponentials in $x(t)$ will mean that $\|x(t)\| \to \infty$.

The solution with $c_1= 1$, $c_2 = c_3 = 0$ is constant and of norm $1$.

Answer 2

The solutions with $|c1| \leq 1$ and $c_2 = c_3 = 0$ are constant and of norm less or equal to $1$, i.e.:

$$x(t) = \begin{bmatrix}c_1 \\ 0 \\ 0 \end{bmatrix} ~\forall t \geq 0,$$

and

$$\|x(t)\| = |c_1| \leq 1 ~\forall t \geq 0.$$

In all other cases, the exponentials will blow up to infinity.