Control Energy of Lattice Graphs

TL;DR

This paper derives an analytical expression for the minimum control energy in infinite lattice graphs with linear dynamics, and shows it accurately predicts control energy in finite lattice graphs, advancing understanding of network controllability.

Contribution

It provides the first analytical formula for control energy in lattice graphs and demonstrates its applicability to finite networks, bridging theory and practical control.

Findings

Analytical expression for control energy using Bessel functions.

Control energy of finite lattices approximates that of infinite lattices.

The derived formula aids in understanding network controllability.

Abstract

The control of complex networks has generated a lot of interest in a variety of fields from traffic management to neural systems. A commonly used metric to compare two particular control strategies that accomplish the same task is the control energy, the integral of the sum of squares of all control inputs. The minimum control energy problem determines the control input that lower bounds all other control inputs with respect to their control energies. Here, we focus on the infinite lattice graph with linear dynamics and analytically derive the expression for the minimum control energy in terms of the modified Bessel function. We then demonstrate that the control energy of the infinite lattice graph accurately predicts the control energy of finite lattice graphs.