Duality for Evolutionary Equations with Applications to Null Controllability

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.