On the Maximal Parameter Range of Global Stability for a Nonlocal Thermostat Model

TL;DR

This paper investigates the conditions under which a nonlinear nonlocal thermostat model remains globally stable, identifying the maximal parameter range for stability using integral equations and a modified Popov criterion.

Contribution

It introduces a novel analysis of the maximal stability parameter range for a nonlocal thermostat model via integral equations and stability criteria.

Findings

Characterization of the maximal parameter range for global stability.

Application of a modified Popov criterion to a Volterra integral equation.

Identification of conditions ensuring asymptotic decay of solutions.

Abstract

The global asymptotic stability of the unique steady state of a nonlinear scalar parabolic equation with a nonlocal boundary condition is studied. The equation describes the evolution of the temperature profile that is subject to a feedback control loop. It can be viewed as a model of a rudimentary thermostat, where a parameter controls the intensity of the heat flow in response to the magnitude of the deviation from the reference temperature at a boundary point. The system is known to undergo a Hopf bifurcation when the parameter exceeds a critical value. Results on the characterization of the maximal parameter range where the reference steady state is globally asymptotically stable are obtained by analyzing a closely related nonlinear Volterra integral equation. Its kernel is derived from the trace of a fundamental solution of a linear heat equation. A version of the Popov criterion is adapted and applied to the Volterra integral equation to obtain a sufficient condition for the asymptotic decay of its solutions.