Consider the linear dynamical system $$\dot{x}(t)=Ax(t)$$ $$y(t)=Cx(t)+d(t),$$ for all $t\geq 0$. The common observer design to estimate the state $x$ while observing the output $y$ is given by the Luenberger observer: $$\dot{\hat{x}}(t)=A\hat{x}(t)+L(y(t)-C\hat{x}(t)).$$ Indeed, if $A-LC$ is Hurwitz (all eigenvalues have negative real part) and the disturbance $d=0$ then the estimation error $$e(t):=x(t)-\hat{x}(t)\longrightarrow 0,$$ as $t$ goes to infinity.
On the other hand, denote the non-negative real line by $\mathcal{R}^+$. We say a function $\gamma:\mathcal{R}^+\longrightarrow\mathcal{R}^+$ is of class $K$ if $\gamma(0)=0$, it is strictly increasing and continuous. A function $\beta:\mathcal{R}^+\times\mathcal{R}^+\longrightarrow\mathcal{R}^+$ is of class $\mathcal{KL}$ if $\beta(\cdot,r)$ is of class $\mathcal{K}$ and $\beta(s,\cdot)$ is decreasing such that for each $s$: $$\beta(s,t)\longrightarrow 0,$$ as $t$ goes to infinity.
The Luenberger observer has the property that there exist such functions satisfying: $$|e(t)|\leq\beta(|e(0)|,t)+\gamma(|d|_\infty),$$ for all $t\geq 0$ where the disturbance $d$ is bounded with supremum norm $|d|_\infty$.
For general non-linear systems with specific structures, the bounded/convergent observers respectively corresponding to noisy/perfect outputs that I know satisfy this property also. Is it equivalent to the boundedness/convergence of the observer or is it a much stronger assumption?