Duality for Evolutionary Equations with Applications to Null Controllability

TL;DR

This paper explores duality in evolutionary equations within weighted L^2 spaces, explicitly characterizing the adjoint system and applying this to analyze null controllability, advancing control theory for such equations.

Contribution

It explicitly describes the ν-adjoint system for evolutionary equations and applies duality to develop new notions of null-controllability.

Findings

Explicit description of the ν-adjoint system

Proof of well-posedness for the adjoint system

Introduction of new null-controllability concepts

Abstract

We study evolutionary equations in exponentially weighted $\mathrm{L}^{2}$-spaces as introduced by Picard in 2009. First, for a given evolutionary equation, we explicitly describe the $\nu$-adjoint system, which turns out to describe a system backwards in time. We prove well-posedness for the $\nu$-adjoint system. We then apply the thus obtained duality to introduce and study notions of null-controllability for evolutionary equations.