We study evolutionary equations in exponentially weighted (Formula presented.) -spaces as introduced by Picard in 2009. First, for a given evolutionary equation, we explicitly describe the (Formula presented.) -adjoint system, which turns out to describe a system backwards in time. We prove well-posedness for the (Formula presented.) -adjoint system. We then apply the obtained duality to introduce and study notions of null controllability for evolutionary equations.
Buchinger et al. (Sun,) studied this question.