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

TL;DR

The paper investigates controllability, observability, and stabilizability of the heat equation on discrete graphs with a weighted Laplacian $H$, focusing on how the geometry of the control set $D$ affects controllability. The authors establish cost-uniform $\alpha$-controllability for $D$ that is $d_L$-relatively dense by proving a weak observability bound for the adjoint heat flow and translating it via duality, with explicit constants depending on inradius and volume. They also show that $0$-controllability generally fails on discrete graphs, propagate non-$0$-controllability through amenable covering graphs, and derive a necessary condition for controllability in terms of $d_{\mathrm{comb}}$-relative density; stabilization results follow from the controllability framework. The work clarifies the discrete-geometry limitations compared to the continuum, and provides a pathway to stabilization via open-loop and closed-loop feedback when $D$ is sufficiently dense. Overall, the results connect spectral-uncertainty principles at small energies to concrete control and stabilization properties on graphs, with explicit geometric constants.

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 $α$-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.

Theorems & Definitions (30)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6
• Definition 3.7