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In recent years, almost everyone has heard about autonomous vehicles. The well-known DARPA grand challenges of the last two years concerned autonomous vehiclesnavigating challenging courses and terrain. The intent of developing this technology isthat one day vehicles will be able to operate autonomously in a variety of dangeroussituations, from military scenarios to disaster relief. How will these vehicles communi-cate and deploy to cover a space evenly, in an environment where they may only havelocal knowledge of other vehicles? This is an example of the problem that motivatesthis issue’s SIGEST paper.

The paper “NonsmoothCoordination andGeometricOptimization viaDistributedDynamical Systems,” by Jorge Cortes and Francesco Bullo, first appeared in the SIAMJournal on Control and Optimization, 44 (2005), pp. 1543–1574. In this seminal work,the authors design distributed (i.e., reliant on local knowledge) coordination algorithmsfor dynamic networks, and provide formal verification of the correctness of these al-gorithms. One of the important and challenging aspects of the problem they study isthat they assume that the communication topology of the network may change as thesystem evolves, as opposed to remaining fixed.

The paper considers two different scenarios for “even spacing” in the dynamicnetwork, which are easy to describe informally if we continue to consider the exampleof vehicles in a convex polygonal region. The first is to cause each point in the regionto be as close as possible to the nearest vehicle (akin in a static sense to positioningpost offices in a town to minimize the longest distance of any home to the nearestpost office); the second is to require each vehicle to be as distant as possible from allother vehicles, and the boundary of the region. In the case of just one vehicle, the firstscenario corresponds to finding the center of the smallest circle entirely containing thepolygon, whereas the second corresponds to finding the center of the largest circle fullycontained within the polygon. It turns out that these disk-covering and sphere-packingconcepts are related to the solutions in general.

One of the fascinating and beautiful aspects of this paper is that it closely combinesthree rather distinct areas of applied mathematics: geometry and in particular Voronoidiagrams; dynamical systems where relationships can evolve and change; and optimiza-tion, specifically nonsmooth optimization, meaning that the objective functions are notcontinuously differentiable. Indeed, the authors show that dynamical system strategiesbased upon nonsmooth gradients have geometric interpretations related to Voronoidiagrams.

The authors have modified an already very nicely written original paper to make iteven more accessible to a general SIAM audience. We invite readers to enjoy this veryimportant and illuminating view into the world of dynamical systems and optimization.Who knows, one day your carmay incorporate technology stemming from this research!

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