Is It Possible to Stabilize Disrete-time Parameterized Uncertain Systems Growing Exponentially Fast?

TL;DR

This paper investigates the stabilizability of discrete-time parameterized uncertain systems, revealing that such systems can be stabilized even if they grow exponentially fast for most states, provided they meet certain growth conditions on a tiny subset.

Contribution

It demonstrates that a system's exponential growth does not preclude stabilizability if the growth is limited to a small subset of states, providing a new perspective on system stabilization.

Findings

Stabilizable systems can grow exponentially fast outside a small subset.

The required growth condition $f(x)=O(|x|^b)$ with $b<4$ holds only on a tiny fraction of states.

The proportion of states satisfying the growth condition can be arbitrarily small in stabilizable systems.

Abstract

This paper derives a somewhat surprising but interesting enough result on the stabilizability of discrete-time parameterized uncertain systems. Contrary to an intuition, it shows that the growth rate of a discrete-time stabilizable system with linear parameterization is not necessarily to be small all the time. More specifically, to achieve the stabilizability, the system function $f(x)=O(|x|^b)$ with $b<4$ is only required for a very tiny fraction of $x$ in $\mathbb{R}$, even if it grows exponentially fast for the other $x$. The proportion of the mentioned set in $\mathbb{R}$, where the system fulfills the growth rate $ O(|x|^b)$ has also been computed, for both the stabilizable and unstabilizable cases. This proportion, as indicated herein, could be arbitrarily small, while the corresponding system is stabilizable.