# Well-posedness of time-varying linear systems

## TL;DR

This paper establishes verifiable conditions ensuring the well-posedness of certain time-varying linear PDE systems, including wave and port-Hamiltonian systems, by analyzing their energy properties and evolution families.

## Contribution

It provides new sufficient conditions based on operator smoothness for the well-posedness of perturbed time-varying linear PDE systems, extending existing theory.

## Key findings

- Conditions for well-posedness are easily verifiable.
- Energy inequalities are derived for perturbed systems.
- Application to wave and port-Hamiltonian systems demonstrates practical relevance.

## Abstract

In this paper, we give easily verifiable sufficient conditions for two classes of perturbed linear, passive PDE systems to be well-posed, and we provide an energy inequality for the perturbed systems. Our conditions are in terms of smoothness of the operator functions that describe the multiplicative and additive perturbation, and here well-posedness essentially means that the time-varying systems have strongly continuous Lax-Phillips evolution families. A time-varying wave equation with a bounded multi-dimensional Lipschitz domain is used as illustration, and as a part of the example, we show that the time-invariant wave equation is a "physically motivated" scattering-passive system in the sense of Staffans and Weiss. The theory also applies to time-varying port-Hamiltonian systems.