# Stability and Well-posedness of a Nonlinear Railway Track Model

## TL;DR

This paper analyzes the stability of a nonlinear PDE model for railway track deflections, establishing conditions for exponential stability and input-to-state stability using Lyapunov functions.

## Contribution

It introduces a Lyapunov-based approach to prove stability and ISS of solutions to a semilinear PDE model of railway tracks with nonlinear foundation behavior.

## Key findings

- Existence of classical solutions under certain inputs
- Exponential stability proven via Lyapunov function
- Input-to-state stability established for mild solutions

## Abstract

Railway tracks rest on a foundation known for exhibiting nonlinear viscoelastic behavior. Railway track deflections are modeled by a semilinear partial differential equation. This paper studies the stability of solutions to this equation in presence of an input. With the aid of a suitable Lyapunov function, existence and exponential stability of classical solutions is established for certain inputs. The Lyapunov function is further used to find an a-priori estimate of the solutions, and also to study the input-to-state stability (ISS) of mild solutions.