IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 1, JANUARY 2010 279 The Rendezvous Problem With Discontinuous Control Policies Giuseppe Conte, Senior Member, IEEE, and Paris Pennesi, Member, IEEE Abstract—In this technical note we consider the Multi-agent Rendezvous Problem and we state new sufficient conditions for characterizing the con- trol policies that assure rendezvous. Our condition are less restrictive than those presented until now in the literature, since, in particular, continuity is substituted by a milder requirement concerning the behavior of the con- trol laws around points of discontinuity. In addition, our results apply to groups of agents moving in , for any finite . Index Terms—Consensus, distributed control, multi-agent system, ren- dezvous. I. INTRODUCTION I N recent years there has been an increasing interest in developing tools and methods to deal with the problem of controlling groups of independent agents ([1], [3], [4], [6], [10]). The fundamental ques- tion to investigate is how to govern the behavior of the global system using distributed control strategies, which are synthesized only on the basis of local information and are implemented locally by the single agents. Here, we consider the Multi-agent Rendezvous Problem introduced in [1], [10]. The Problem concerns a group of agents moving in a spe- cific environment without communicating between them. Assuming that the agents are equipped with limited sensing capability and that they are able to control their motion, the problems focus on the study of distributed control strategies, based on local information only, that guarantees the agents’ rendezvous in a point. A control policy based on the so-called Circumcenter (that is the center of the smallest-radius circle that contains a given set of points) Algorithm was shown in [1] to provide a solution, under suitable hypothesis, to the Multi-agent Ren- dezvous Problem for agents moving in the plane . Maintaining that limitation, a general characterization of the control policies which as- sure the rendezvous was stated in [6] and proved in [7]. Then, the ef- ficacy of control policies based on extensions of the Circumcenter Al- gorithm was proved in more general situations, in which in particular the motion is not confined to the plane and the use of local information depends on suitable proximity relations, in [4]. A different approach to the Problem consists in translating it into a so-called Consensus Problem (see [3], [9], [12]) and representing the behaviour of the overall system of agents by means of a linear dynamic equation. The analysis of conditions for reaching consensus goes back to [11], and has more recently been considered in [5], [8]. Sufficient conditions for the rendezvous are expressed, in this case, in terms of properties of the system’s dynamic matrix. Further results on the gen- eral problem, relaxing the conditions found in [8], are given in [2]. Manuscript received December 03, 2007; revised August 26, 2008 and November 17, 2008. First published December 08, 2009; current version published January 13, 2010. Recommended by Associate Editor F. Bullo. G. Conte is with the Dipartimento di Ingegneria Informatica, Gestionale e dell’Automazione, Università Politecnica delle Marche, Ancona 60100, Italy (e-mail: gconte@univpm.it). P. Pennesi is with the Royal Bank of Scotland, Global Banking & Markets, London EC2M 3UR, U.K. (e-mail: paris.pennesi@gmail.com). Digital Object Identifier 10.1109/TAC.2009.2037249 In [6], the control policies which assure the rendezvous are charac- terized by a number of conditions, one of which is the continuity of the control laws that govern the agents’ motion. The use of the Circum- center Algorithm in defining the control policy assures, as described in [6], Section I.A.2, continuity also in the situations considered in [1] and in [4]. Sufficiency of the conditions stated in [6] for achieving the ren- dezvous is given in [7]. The proof uses a standard Lyapunov argument and it relies heavily, in particular, on continuity. However, the authors says that it remains to be seen “whether or not these requirements (one of which is continuity) can be relaxed by approaching convergence dif- ferently” (compare with [7], end of Section III). In this technical note, we analyze the conditions given in [6], showing, by means of simple illustrative examples, that continuity is not necessary, but also that the other conditions, removing conti- nuity, fail to characterize the class of control policies that guarantee the rendezvous. This suggests that the result could be achieved by substituting the condition about continuity by a milder one. On that basis, we formulate, in the technical note’s main The- orem, new conditions, weaker than those of [6], which characterize a large class of not necessarily continuous, distributed control strate- gies achieving rendezvous and solving the Multi-agent Rendezvous Problem. In particular, we substitute the condition about continuity with a much weaker one, which concerns the behavior of the control laws around the possible points of discontinuity. The main Theorem of this technical note, therefore, provides a pos- itive answer to the question implicitly posed in [7], by showing that actually the continuity condition can be relaxed by approaching con- vergence without Lyapunov theory. In addition, our result holds for motion in , for any arbitrary finite . Therefore, it extends those of [1] and of [6] with respect to the class of solutions and to the dimension of the space in which the agents can move, as well as those of [4] with respect to the class of solutions. The technical note is organized as follows. in Section II we describe the Multi-agent Rendezvous Problem and the class of control policies that are admissible for its solutions. In Section III we analyze the con- ditions given in [6] and we discuss two examples of control policies. In Section IV we state and prove our main result. Conclusions are in Section V. II. PROBLEM FORMULATION In describing the Multi-agent Rendezvous Problem, we follow the general lines of [1] and [6]. Assume we have agents , , that can move in an Euclidean space , whose positions at time are indicated, respectively, by the vectors , . Let us divide the time axis into a sequence of time intervals up to . Each interval is composed by two periods: a sensing period and a maneuvering period . During the sensing period , all the agents rest and each agent senses the environment within a circle centered in and of radius , called sensing region. In particular, any agent measures the number of agents that are actually within its sensing region, namely of its neighbors, and it evaluates the relative distance from any of them (1) During the maneuvering period , each agent decides its own way-point and move toward its next position according to (2) 0018-9286/$26.00 © 2009 IEEE