With given ONLY these two pieces of information:
- $z(t) \in \mathcal{L}_{\infty}, \forall t$ and $z(t)$ converges to a ball (or bound) around $0$ for some $t > T$ and stays within that ball (or bound) from thereafter to $t \rightarrow \infty$ ($z(t)$ is generally a vector in $\mathcal{R}^n$, hence the term "ball" is used, although a scalar argument would help initially, hence the term "bound" is used). Note that the exact evolution of $z(t)$ within the ball or bound is not known.
- $\dot{z}(t) \in \mathcal{L}_{\infty}, \forall t$ (even after entering the ball)
Is it possible to argue (that the average) i.e., $\lim_{t \rightarrow \infty} \int_T^{t} z(\tau) d \tau =$ $0$ or some (known or unknown) constant $c$?
If yes, how to rigorously prove it?
If not, what minimal extra condition is needed to make it happen?
Thank you.
Function $z(t)=\sin(t)$ satisfies both assumptions. However, $\int_{T}^{\infty} \sin(\tau)d\tau$ does not converge (for any choice of $T$). Extra assumptions are required, however, I don't known which of them (if any) are necessary (so that they could be described as "minimal").