Controllability of partial differential equations on graphs

TL;DR

This paper investigates the controllability of wave, heat, and Schrödinger equations on finite trees, establishing exact and null controllability results with sharp estimates on control time.

Contribution

It provides new controllability results for PDEs on graphs, including sharp control time estimates for wave equations and controllability for heat and Schrödinger equations.

Findings

Exact controllability in $L_2$-controls for wave equations on trees.

Null controllability for heat equations in arbitrary time.

Exact controllability for Schrödinger equations in arbitrary time.

Abstract

We study the boundary control problems for the wave, heat, and Schr\"odinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. The exact controllability in $L_2$-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. The null controllability for the heat equation and exact controllability for the Schr\"odinger equation in arbitrary time interval are obtained.