Optimal Strategies to Evade Jamming in Heterogeneous Mobile Networks Sourabh Bhattacharya Tamer Bas¸ar Abstract— In this paper, we consider a scenario in which a mobile intruder jams the communication network in a vehicular formation. We formulate the problem as a zero-sum pursuit- evasion game with players possessing heterogeneous dynamics. We use Isaacs’ approach to derive the necessary conditions to arrive at the equations governing the saddle-point strategies of the team of players. I. I NTRODUCTION In the past few years, a lot of research has been done to deploy multiple vehicles in formation to carry out tasks in military as well as civilian scenarios. In general, the mode of communication among vehicles deployed in a team mission is wireless. This renders the communication channel vulnerable to malicious attacks from an intruder present in the vicinity. An example of such an intruder is an aerial or a terrestrial jammer. Jamming is a malicious attack whose objective is to disrupt the communication of the victim network intentionally causing interference or collision at the receiver side. Jamming attack is a well-studied and an active area of research in a wireless networks. Many defense strategies have been proposed by researchers against jamming in wireless networks. In [29], Wu et.al. propose two strategies to evade jamming. The first strategy, channel surfing, is a form of spectral evasion that involves legitimate wireless devices changing the channel that they are operating on. The second strategy, spatial retreats, is a form of special evasion whereby legitimate devices move away from the jammer. In [27], Wood et.al. present a distributed protocol to map jammed region so that the network can avoid routing traffic through it. The solution proposed by Cagalj et.al. [7] uses different worm holes (wired worm holes, frequency- hopping pairs, and uncoordinated channel hopping) that lead out of the jammed region to report the alarm to the network operator. In [28], Wood et.al. investigate how to deliberately avoid jamming in IEEE 802.15.4 based wireless networks. In [8], Lin Chen et.al. propose a strategy to introduce a special node in the network called the anti-jammer to drain the jammer’s energy. To achieve its goal, the anti-jammer configures the probability of transmitting bait packets to attract the jammer to transmit. For a static jammer and mobile nodes, the optimal strategy for the nodes is to retreat away from the jammer after detecting jamming. In case of a mobile jammer intruding Sourabh Bhattacharya and Tamer Bas¸ar are with the Department of Electrical and Computer Engineering, Uni- versity of Illinois at Urbana-Champaign, Urbana, IL 61801 {sbhattac,basar1}@illinois.edu This work was supported in part by a grant from AFOSR, and in part by an ARO MURI grant. the communication network, optimal strategies for retreat are harder to compute due to the mobility of the jammer and constraints in the kinematics of the terrestrial or aerial vehicles in the formation. This attack can be modeled as a zero-sum game [1] between the jammer and the vehicles. Such dynamic games governed by differential equations can be analyzed using tools from differential game theory. There is an extensive amount of research that has been done in the area of two person pursuit-evasion games [15],[11]. In the recent past, techniques from differential game theory have been used to analyze problems that possess potential applications for surveillance systems. In [3],[4], the authors analyze a visibility-based target-tracking problem among obstacles. The problem is formulated as a zero-sum game between the target and the observer and optimal strategies for players are computed using tools from differential game theory. In [12], tools from differential game theory are used to analyze the lion and the man problem in the presence of a circular obstacle in the environment. In contradistinc- tion, there has been limited application of differential game theory to multi-player pursuit-evasion games. Solutions for particular multi-player games were presented by Pashkov and Terekhov [19], Levchenkov and Pashkov [14], Hagedorn and Breakwell [10], Breakwell and Hagedorn [6] and Sriram et.al.[22]. More general treatment of multiplayer differential games was presented by Starr and Ho [5], Bas¸ar and Olsder [11], Vaisbord and Zhukovskiy [26] and Zhukovskiy and Salukvadze [30], and Stipanovi´ c, Hovakimyan and Melikyan [24]. The inherent hardness in obtaining an analytical so- lution to Hamilton-Jacobi-Isaacs equation has led to the development of numerical techniques for the computation of the value function. Recent efforts in this direction to compute an approximation of the reachable sets have been provided by Mitchell and Tomlin [16], Stipanovi´ c, Hwang and Tomlin [23] and Stipanovi´ c, Sriram and Tomlin [9]. In contradistinction with the existing literature, our work analyzes the behavior of multiple vehicles in cooperative as well as non-cooperative scenarios in the presence of a malicious intruder in the communication network. In [2], we envision a scenario in which an aerial jammer intrudes the communication channel in a multiple UAV formation. We model the intrusion as a continuous time pursuit-evasion game between the UAVs and the aerial jammer. In contrast to the previous work in pursuit-evasion games that formulate a payoff based on a geometric quantity in the configuration space of the system, our work formulates a payoff based on the capability of the players in a team to communicate among themselves in the presence of a jammer in the vicinity.