**1. Introduction**

Swarms are characterized by large numbers of agents which act individually, yet produce collective, herd-like behaviors. Implementing cooperating swarm strategies for a large-scale swarm is a technical challenge which can be considered to be from the "insider's perspective". It assumes inside control over the swarm's operating algorithms. However, as large-scale 'swarm' systems of autonomous systems become achievable such as those proposed by autonomous driving, UAV package delivery, and military applications—interactions with swarms outside our direct control become another challenge. This generates its own "outsider's perspective" issues.

In this paper, we look at the specific challenge of protecting an asset against an adversarial swarm. Autonomous defensive agents are tasked with protected a high-value unit (HVU) from an incoming swarm attack. The defenders do not fully know the cooperating strategy employed by the adversarial swarm. Nevertheless, the task of the defenders is to maximize the probability of survival of the HVU against an attack by such a swarm. This challenge raises many issues—for instance, how to search for the swarm [1], how to observe and infer swarm operating algorithms [2], and how to best defend against the swarm given algorithm unknowns, and only limited, indirect control through external means. In this paper, we restrict ourselves to the last issue. However, these problems share multiple technical challenges. The preliminary approach we apply in this paper demonstrates some basic methods which we hope will stimulate the development of more sophisticated tools.

For objectives achieved via external control of the swarm, several features of swarm behavior must be characterized: capturing the dynamic nature of the swarm, tracking the collective risk profile created by a swarm, and engaging with a swarm via dynamic inputs, such as autonomous defenders. The many modeling layers create a challenge

**Citation:** Walton, C.; Kaminer, I.; Gong, Q.; Clark, A.H.; Tsatsanifos, T. Defense against Adversarial Swarms with Parameter Uncertainty. *Sensors* **2022**, *22*, 4773. https://doi.org/ 10.3390/s22134773

Academic Editor: Carlo Alberto Avizzano

Received: 11 April 2022 Accepted: 3 June 2022 Published: 24 June 2022

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for generating an effective response to the swarm, as model uncertainty and model error are almost certain. In this paper, we look at several dynamic systems where the network structure is determined by parameters. These parameters set neighborhood relations and interaction rules. Additional parameters establish defender input and swarm risk.

We consider the generation of optimal defense strategies given uncertainty in parameter values. We demonstrate that small deviances in parameter values can have catastrophic effects on defense trajectories optimized without taking error into account. We then demonstrate the contrasting robustness of applying an uncertain parameter optimal control framework instead of optimizing with nominal values. The robustness against these parameter values suggests that refined parameter knowledge may not be necessary given appropriate computational tools. These computational tools—and the modeling of the high-dimensional swarm itself—are expensive. To assist with this issue, we provide dual conditions for this problem in the form of a Pontryagin minimum principle and prove the consistency of these conditions for the numerical algorithm. These dual conditions can, thus, be computed from the numerical solution of the computational method and provide a tool for solution verification and parameter sensitivity analysis.

Although in this paper, optimal strategies against swarms motivate the framework of uncertain parameter optimal control, and the subsequent development of the dual conditions, both the framework and the dual conditions have many applications beyond swarm defense. Optimal control with parameter uncertainty is relevant to robotics—where parts, such as wheels, may have small size and calibration uncertainties; aerospace—where both components and exogeneous factors, such as wind, may be modeled using parameter uncertainty; and search and rescue—where the location of a target object can be considered a parameter uncertainty [3,4]. It is also an instance of mean-field optimal control (which includes this framework, but also more general probability distributions), which is finding application in the training of neural networks [5]. The dual conditions provided in this paper provide both a tool for verification of numerical solutions, as well as another potential route for generating numerical solutions.

The structure of this paper is as follows. Section 2 provides examples of dynamic swarming models and extensions for defensive interactions. Section 3 discusses optimization challenges and describes a general uncertain parameter optimal control framework that this problem could be addressed with. Section 4 provides a proof of the consistency of the dual problem for this control framework, which expands on the results initially presented in the conference paper [6]. Section 5 gives an example of numerical implementation that demonstrates optimal defense against a large-scale swarm of 100 agents. Section 6 discusses the results and future work.
