Well-posedness for a general class of differential inclusions

TL;DR

This paper establishes the well-posedness and causality of a broad class of differential inclusions, including those with delays and algebraic constraints, in a Hilbert space framework, with applications to Maxwell's equations.

Contribution

It introduces a unified approach to analyze well-posedness for various differential inclusions under weak assumptions, extending existing theories.

Findings

Proved well-posedness in Hilbert spaces for the class of differential inclusions.

Demonstrated causality of the solution operator.

Applied the theory to a semistatic quasilinear Maxwell's equations variant.

Abstract

We consider an abstract class of differential inclusions, which covers differential-algebraic and non-autonomous problems as well as problems with delay. Under weak assumptions on the operators involved, we prove the well-posedness of those differential inclusions in a pure Hilbert space setting. Moreover, we study the causality of the associated solution operator. The theory is illustrated by an application to a semistatic quasilinear variant of Maxwell's equations.