Exact Controllability for the Wave Equation on a Graph with Cycle and Delta-Prime Vertex Conditions

TL;DR

This paper proves exact controllability of the wave equation on a specific metric graph with cycles and delta-prime conditions, using boundary and internal controls, with implications for general graph controllability.

Contribution

It introduces new controllability results for wave equations on graphs with cycles and delta-prime conditions, expanding understanding of control in complex network structures.

Findings

Proves exact controllability on a cycle with delta-prime conditions.

Analyzes controllability in a tripod graph with fixed leaves.

Demonstrates the effectiveness of boundary and internal controls.

Abstract

Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.