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

We consider linear control problems for the heat equation of the form $\dot f (t) = -Hf (t) + \mathbf{1}_D u (t)$, $f (0) \in \ell_2 (X,m)$, 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 $α$-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.

💡 Research Summary

The paper investigates linear control, observation, and stabilization problems for the heat equation on weighted discrete graphs. The dynamics are given by (\dot f(t) = -H f(t) + \mathbf 1_D u(t)), where (H) is the weighted graph Laplacian acting on (\ell^2(X,m)) and (\mathbf 1_D) embeds a control function (u) supported on a subset (D\subset X). The authors aim to determine under which geometric conditions on (D) the system is controllable with a uniform cost, to establish weak observability estimates for the dual system, and to derive consequences for both open‑loop and closed‑loop stabilization.

First, the paper sets up the graph framework. A weighted graph ((X,b,m)) satisfies connectivity, bounded degree, and bounded vertex measure. Two metrics are introduced: the combinatorial distance (d_{\text{comb}}) and the length distance (d_L). Relative density of a set (D) with respect to a metric means that a finite covering radius exists. The Laplacian (H) is defined by \