Exact controllability for wave equation on general quantum graphs with non-smooth controls

TL;DR

This paper establishes the exact controllability of the wave equation on finite quantum graphs with non-smooth controls, using a combination of dynamical and moment methods, and provides a constructive approach for control placement.

Contribution

It introduces a constructive method for exact controllability of wave equations on quantum graphs with non-smooth controls, combining dynamical and moment techniques.

Findings

Exact controllability achieved with Neumann and Dirichlet controls at active vertices and edges.

Control time depends on graph orientation and path decomposition.

Proofs include shape, velocity, and exact controllability results.

Abstract

In this paper we study the exact controllability problem for the wave equation on a finite metric graph with the Kirchhoff-Neumann matching conditions. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if $H^1(0,T)'$ Neumann controllers are placed at the active vertices and $L^2(0,T)$ Dirichlet controllers are placed at the active edges. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the exact controllability utilizes both dynamical and moment method approaches. The control time for this construction is determined by the chosen orientation and path decomposition of the graph.