Exponential Stability and Initial Value Problems for Evolutionary Equations

TL;DR

This thesis investigates linear evolutionary equations, establishing well-posedness, exponential stability, and regularity within a Hilbert space framework, and provides a Hille-Yosida type result with illustrative examples.

Contribution

It introduces a unified approach to analyze well-posedness and stability of diverse linear evolutionary equations in a Hilbert space setting, including fractional and delay equations.

Findings

Proved well-posedness of linear evolutionary problems.

Established exponential stability criteria.

Developed a Hille-Yosida type theorem for these equations.

Abstract

In this thesis we consider so-called linear evolutionary problems, a class of linear partial differential equations covering classical elliptic, parabolic and hyperbolic equations from mathematical physics as well as classes of integro-differenital equations, fractional differential equations and delay equations. We address the well-posedness of the problems in a pure Hilbert space setting. Moreover, the exponential stability and the regularity of the problems are studied. In particular, a Hille-Yosida type result is proved, to obtain a strongly continuous semigroup on a suitable state space consisting of admissible initial values and pre-histories. The results are illustrated by various examples.