In a recent Physical Review Letters paper, Vicsek et. al. propose a simple but compelling discrete-time model of n autonomous agents fi.e., points or particlesg all moving in the plane with the same speed but with dierent headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its \neighbors." In their paper, Vicsek et. al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models.

Abstract. We construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls the interaction length scale. The dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion. For the purely incompressible case, the model resembles one for fluid dynamical vortex patches. There exist solutions which have constant population density and compact support for all time. Numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills. The sign of the social interaction term determines the direction of the rotation, and the interaction length scale affects the degree of spiral formation. For the purely potential case, the model resembles a nonlocal (forwards or backwards) porous media equation. The sign of the social interaction term controls whether the population aggregates or disperses, and the interaction length scale controls the balance between transport and smoothing of the density profile. For the aggregative case, the population clumps into regions of high and low density. The characteristic length scale of the density pattern is predicted and confirmed by numerical simulations.

be inserted by the editor)

We investigate a class of continuum models for the motion of a twodimensional biological group under the influence of nonlocal social interactions. The dynamics may be uniquely decomposed into incompressible motion and potential motion. When the motion is purely incompressible, the model possesses solutions which have constant population density and sharp boundaries for all time. Numerical simulations of these "swarm patches" reveal rotating mill-like swarms with circular cores and spiral arms. The sign of the social interaction term determines the direction of the rotation, and the interaction length scale a#ects the degree of spiral formation. When the motion is purely potential, the social interaction term has the meaning of repulsion or attraction depending on its sign. For the repulsive case, the population spreads and the density profile is smoothed. With increasing interaction length scale, the motion becomes more convective and experiences slower di#usive smoothing. For the attractive case, the population self-organizes into regions of high and low density. The characteristic length scale of the density pattern is predicted and confirmed by numerical simulations.

Collaborative/cooperative control of a large group of autonomous vehicles has been received great attentions in recent years. With the rapid advances in sensing, communication, computation, and actuation capabilities, it is extremely appealing to control a large group of unmanned autonomous vehicles (UAVs) to perform dangerous or explorative tasks in various hazardous, unknown or remote environments. Possibilities of a broad range of applications by utilizing UAV swarms have been explored, for example, automated highway systems, mobile sensor networks in ocean resources exploration, space-craft interferometry, satellite formations and robotic border patrol. In such applications, traditional centralized control schemes are always prohibited primarily due to the high communication cost and the high computation cost in a large network of vehicles. In turn, the decentralized/distributed control schemes are preferred to achieve the trade off between the performance and the communication/compuation cost. In past decades, numerous decentralized/distributed control algorithms have been proposed in the literature. Among them, one approach, called bio-inspired approach, is extremely interesting and promising, which ”borrows” algorithms from nature by observ-

coordinated control laws in multiple

In this paper, we study the consensus problem for networked dynamic systems with arbitrary initial states, and present some structural characterization and direct construction of consensus functions. For the consensus problem under similar transformation, we establish some necessary and sufficient conditions by exploiting the structure of consensus functions. Finally, we discuss the consensus problem for dynamic systems under switching by using the common Lyapunov function method.