Shape, Velocity, and Exact Controllability for the Wave Equation on a Graph with Cycle

TL;DR

This paper proves exact controllability for the wave equation on a cyclic graph using a novel approach that combines shape and velocity controllability, highlighting the limitations with a single control.

Contribution

It introduces a method to establish exact controllability on cyclic graphs by combining shape and velocity controllability, addressing moment problems.

Findings

Exact controllability is achieved on graphs with cycles.

Controllability fails with only a single boundary or interior control.

A new proof method links shape and velocity controllability to exact controllability.

Abstract

Exact controllability is proven on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.