# Exponential Stability of Nonlinear Differential Repetitive Processes with Applications to Iterative Learning Control

## TL;DR

This paper establishes conditions for exponential stability of nonlinear differential repetitive processes, linking it to linearized dynamics, with applications to iterative learning control and verified through simulations.

## Contribution

It introduces a novel stability analysis framework for DRPs, showing exponential stability is equivalent to linearized dynamics stability, verified via spectral radius conditions.

## Key findings

- Exponential stability of DRPs is equivalent to that of their linearized systems.
- Spectral radius condition can verify stability efficiently.
- Numerical simulation confirms theoretical results.

## Abstract

This paper studies exponential stability properties of a class of two-dimensional (2D) systems called differential repetitive processes (DRPs). Since a distinguishing feature of DRPs is that the problem domain is bounded in the "time" direction, the notion of stability to be evaluated does not require the nonlinear system defining a DRP to be stable in the typical sense. In particular, we study a notion of exponential stability along the discrete iteration dimension of the 2D dynamics, which requires the boundary data for the differential pass dynamics to converge to zero as the iterations evolve. Our main contribution is to show, under standard regularity assumptions, that exponential stability of a DRP is equivalent to that of its linearized dynamics. In turn, exponential stability of this linearization can be readily verified by a spectral radius condition. The application of this result to Picard iterations and iterative learning control (ILC) is discussed. Theoretical findings are supported by a numerical simulation of an ILC algorithm.