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This paper considers a reach-avoid game on a rectangular domain with two defenders and one attacker. The attacker aims to reach a specified edge of the game domain boundary, while the defenders strive to prevent that by capturing the attacker. First, we are concerned with the barrier, which is the boundary of the reach-avoid set, splitting the state space into two disjoint parts: 1) defender dominance region (DDR) and 2) attacker dominance region (ADR). For the initial states lying in the DDR, there exists a strategy for the defenders to intercept the attacker regardless of the attacker's best effort, while for the initial states lying in the ADR, the attacker can always find a successful attack strategy. We propose an attack region method to construct the barrier analytically by employing Voronoi diagram and Apollonius circle for two kinds of speed ratios. Then, by taking practical payoff functions into considerations, we present optimal strategies for the players when their initial states lie in their winning regions, and show that the ADR is divided into several parts corresponding to different strategies for the players. Numerical approaches, which suffer from inherent inaccuracy, have already been utilized for multiplayer reach-avoid games, but computational complexity complicates solving such games and consequently hinders efficient on-line applications. However, this method can obtain the exact formulation of the barrier and is applicable for real-time updates.

Formation-containment analysis and design problems for high-order linear time-invariant swarm systems with directed interaction topologies are dealt with respectively. Firstly, protocols are presented for leaders and followers respectively to drive the states of leaders to realize the predefined time-varying formation and propel the states of followers to converge to the convex hull formed by the states of leaders. Secondly, formation-containment problems of swarm systems are transformed into asymptotic stability problems, and an explicit expression of the formation reference function is derived. Sufficient conditions for swarm systems to achieve formation containment are proposed. Furthermore, necessary and sufficient conditions for swarm systems to achieve containment and time-varying formation are presented respectively as special cases. An approach to determine the gain matrices in the protocols is given. It is shown that containment problems, formation control problems, consensus problems and consensus tracking problems can all be treated as special cases of formation-containment problems. Finally, numerical simulations are provided to demonstrate theoretical results. 3440 X. DONG ET AL. swarm systems with switching interaction topologies in [19] and [20]. Consensus tracking problems for high-order LTI swarm systems with both fixed and switching interaction topologies were dealt with by Ni and Cheng in [21]. Li et al. [22] proposed sufficient conditions for high-order LTI swarm systems to achieve consensus tracking, where the control input of the leader is nonzero and not available to any follower.In the multiple leaders case, containment problems that require that the states of followers converge to the convex hull formed by the states of leaders arise. Ji et al.[23] investigated containment problems using a hybrid stop-go control strategy. Meng et al.[24] discussed finite-time containment problems for swarm systems with rigid bodies. Containment problems for first-order swarm systems with undirected switching interaction topologies were studied by Notarstefano et al. in [25]. Cao et al. considered containment problems for first-order and second-order swarm systems with both stationary and dynamic leaders in [26] and [27]. Liu et al. [28] presented necessary and sufficient conditions for first-order and second-order swarm systems to achieve containment. Lou and Hong [29] dealt with containment problems for second-order swarm systems with random switching interaction topologies. However, the dynamics of each agent in [23][24][25][26][27][28][29] is restricted to be of first order or second order. In practical applications, the dynamics of agents may be of high order; thus, containment problems for high-order LTI swarm systems make more sense.Li et al.[30] discussed containment problems for high-order LTI swarm systems with directed interaction topologies. Dong et al. [31] investigated containment problems for high-order LTI singular swarm systems with time delays. Sufficient conditions for high-order LTI ...

SUMMARY Containment analysis and design problems for high‐order linear time‐invariant singular swarm systems on directed graphs with time delays are investigated. To eliminate impulse terms in singular swarm systems and ensure that the singular swarm systems can achieve containment, time‐delayed protocols are presented for leaders and followers, respectively. By model transformation, containment problems of singular swarm systems are converted into stability problems of multiple low‐dimensional time‐delayed systems. In terms of linear matrix inequality, sufficient conditions are presented for time‐delayed singular swarm systems to achieve containment, which are independent of the number of agents. By using the method of changing variables, an approach is provided to determine the gain matrices in the protocols. Numerical simulations are shown to demonstrate theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.

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Time-varying formation control problems for high-order linear time-invariant swarm systems with switching interaction topologies are investigated. A general formation control protocol is proposed firstly. Then using a consensus based approach, necessary and sufficient conditions for swarm systems with switching interaction topologies to achieve a given time-varying formation are presented. An explicit expression of the time-varying formation reference function is given. It is revealed that the switching interaction topologies have no effect on the formation reference function and the motion modes of the formation reference can be specified. Furthermore, necessary and sufficient conditions for formation feasibility are presented. An approach to expand the feasible formation set is given and an algorithm to design the protocol for swarm systems with switching interaction topologies to achieve time-varying formations is provided. Finally, numerical simulations are presented to demonstrate theoretical results.