On the convergence of discrete-time linear systems: A linear time-varying Mann iteration converges iff the operator is strictly pseudocontractive

TL;DR

This paper investigates the convergence of discrete-time linear systems using an operator-theoretic approach, establishing that a linear Mann iteration converges if and only if the operator is strictly pseudocontractive, with applications to multi-agent systems.

Contribution

It characterizes convergence of linear Mann iterations through operator properties like strict pseudocontractiveness and eigenvalue conditions, linking theory to multi-agent applications.

Findings

Convergence iff the operator is strictly pseudocontractive.

Eigenvalue-based conditions for operator properties.

Application to consensus and game-theoretic dynamics.

Abstract

We adopt an operator-theoretic perspective to study convergence of linear fixed-point iterations and discrete- time linear systems. We mainly focus on the so-called Krasnoselskij-Mann iteration x(k+1) = ( 1 - \alpha(k) ) x(k) + \alpha(k) A x(k), which is relevant for distributed computation in optimization and game theory, when A is not available in a centralized way. We show that convergence to a vector in the kernel of (I-A) is equivalent to strict pseudocontractiveness of the linear operator x -> Ax. We also characterize some relevant operator-theoretic properties of linear operators via eigenvalue location and linear matrix inequalities. We apply the convergence conditions to multi-agent linear systems with vanishing step sizes, in particular, to linear consensus dynamics and equilibrium seeking in monotone linear-quadratic games.