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

We consider linear control problems for the heat equation of the form $\dot{f}(t)=-Hf(t)+{\mathbf{1}}_{D}u(t)$, $f(0)\in {\mathrm{\ell }}_2(X,m)$, 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.