Define a compact set $\mathcal{X}$ such that $x(t) \in \mathcal{X}$ for all $t \geq t_0$

Question

I have state vector $x \in \mathbb{R}^{\text{n}} $, that behaves following this inequality

$ \|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3 $

where $c_1, c_2, c_3$ are positive constants. This inequality shows that $x$ converges to the ball $c_3$ as time passes.

Suppose $x_0$ is in a compact set $\mathcal{X}$, now I want to define a compact set $\mathcal{X}$ such that $x(t) \in \mathcal{X}$ for all $t \geq t_0$

Answer 1

Just set $\mathcal X = \overline{B(0,c_1\|x(t_0)\|+c_3)}$. This is a compact set containing $x(t)$ for all $t\ge t_0$.