On controllability, observability and stabilizability of the heat equation on discrete graphs

TL;DR

This paper investigates the controllability, observability, and stabilizability of the heat equation on discrete graphs, providing cost-uniform controllability results using weak observability estimates and discussing their optimality.

Contribution

It introduces cost-uniform controllability for the heat equation on discrete graphs using weak observability estimates, and analyzes stabilizability properties.

Findings

Cost-uniform $oldsymbol{ extit{ extalpha}}$-controllability established.

Weak observability estimates proved for the dual problem.

Discussion on the optimality and stabilizability implications.

Abstract

We consider linear control problems for the heat equation of the form $\dot f (t) = -Hf (t) + \mathbf{1}_D u (t)$, $f (0) \in \ell_2 (X,m)$, where $H$ is the weighted Laplacian on a discrete graph $(X,b,m)$, and where $D \subseteq X$ is relatively dense. We show cost-uniform $\alpha$-controllability by means of a weak observability estimate for the corresponding dual observation problem. We discuss optimality of our result as well as consequences on stabilizability properties.