Global Well-Posedness and Exponential Stability for Heterogeneous Anisotropic Maxwell's Equations under a Nonlinear Boundary Feedback with Delay

TL;DR

This paper proves the global well-posedness and exponential stability of a Maxwell's system with anisotropic materials and nonlinear delayed boundary feedback, advancing understanding of complex electromagnetic boundary control systems.

Contribution

It introduces a novel analysis of Maxwell's equations with nonlinear boundary feedback including delay, establishing well-posedness and stability under new conditions.

Findings

Proved maximal monotonicity of the nonlinear generator.

Established global well-posedness in a Hilbert space.

Demonstrated exponential stability under certain conditions.

Abstract

We consider an initial-boundary value problem for the Maxwell's system in a bounded domain with a linear inhomogeneous anisotropic instantaneous material law subject to a nonlinear Silver-Muller-type boundary feedback mechanism incorporating both an instantaneous damping and a time-localized delay effect. By proving the maximal monotonicity property of the underlying nonlinear generator, we establish the global well-posedness in an appropriate Hilbert space. Further, under suitable assumptions and geometric conditions, we show the system is exponentially stable.