# Attack Resilient Interconnected Second Order Systems: A Game-Theoretic Approach

## TL;DR

This paper analyzes the resilience of second-order networked systems against strategic attacks using a game-theoretic framework, providing conditions for equilibrium and optimal defense strategies based on network topology.

## Contribution

It introduces a game-theoretic approach to enhance system resilience by strategically placing self-feedback loops, with conditions for Nash and Stackelberg equilibria in attack-defense scenarios.

## Key findings

- Graph-theoretic conditions for Nash equilibria are established.
- Optimal defense node placement is linked to network centrality measures.
- Extension to multiple attack and defense nodes is discussed.

## Abstract

This paper studies the resilience of second-order networked dynamical systems to strategic attacks. We discuss two widely used control laws, which have applications in power networks and formation control of autonomous agents. In the first control law, each agent receives pure velocity feedback from its neighbor. In the second control law, each agent receives its velocity relative to its neighbors. The attacker selects a subset of nodes in which to inject a signal, and its objective is to maximize the $\mathcal{H}_2$ norm of the system from the attack signal to the output. The defender improves the resilience of the system by adding self-feedback loops to certain nodes of the network with the objective of minimizing the system's $\mathcal{H}_2$ norm. Their decisions comprise a strategic game. Graph-theoretic necessary and sufficient conditions for the existence of Nash equilibria are presented. In the case of no Nash equilibrium, a Stackelberg game is discussed, and the optimal solution when the defender acts as the leader is characterized. For the case of a single attacked node and a single defense node, it is shown that the optimal location of the defense node for each of the control laws is determined by a specific network centrality measure. The extension of the game to the case of multiple attacked and defense nodes is also addressed.