# Convergence in uncertain linear systems

## TL;DR

This paper investigates state convergence in both continuous and discrete uncertain linear systems, providing necessary and sufficient conditions and exploring the differences between weak and strong convergence.

## Contribution

It is the first to analyze state convergence in uncertain linear systems, introducing new conditions and distinctions between convergence types.

## Key findings

- Characterized convergence conditions via Lyapunov and LaSalle methods.
- Established equivalence conditions for weak and strong convergence.
- Showed that state convergence differs from asymptotic stability in duality.

## Abstract

State convergence is essential in several scientific areas, e.g. multi-agent consensus/disagreement, distributed optimization, monotone game theory, multi-agent learning over time-varying networks. This paper is the first on state convergence in both continuous- and discrete-time linear systems affected by polytopic uncertainty. First, we characterize state convergence in linear time invariant systems via equivalent necessary and sufficient conditions. In the presence of uncertainty, we complement the canonical definition of (weak) convergence with a stronger notion of convergence, which requires the existence of a common kernel among the generator matrices of the difference/differential inclusion (strong convergence). We investigate under which conditions the two definitions are equivalent. Then, we characterize weak and strong convergence by means of Lyapunov and LaSalle arguments, (linear) matrix inequalities and separability of the eigenvalues of the generator matrices. We also show that, unlike asymptotic stability, state convergence lacks of duality.