Alessandro Arsie

Distributed Algorithms for Environment Partitioning in Mobile Robotic Networks

A widely applied strategy for workload sharing is to equalize the workload assigned to each resource. In mobile multiagent systems, this principle directly leads to equitable partitioning policies whereby: 1) the environment is equitably divided into subregions of equal measure; 2) one agent is assigned to each subregion; and 3) each agent is responsible for service requests originating within its own subregion. The current lack of distributed algorithms for the computation of equitable partitions limits the applicability of equitable partitioning policies to limited-size multiagent systems operating in known, static environments. In this paper, first we design provably correct and spatially distributed algorithms that allow a team of agents to compute a convex and equitable partition of a convex environment. Second, we discuss how these algorithms can be extended so that a team of agents can compute, in a spatially distributed fashion, convex and equitable partitions with additional features, e.g., equitable and median Voronoi diagrams. Finally, we discuss two application domains for our algorithms, namely dynamic vehicle routing for mobile robotic networks and wireless ad hoc networks. Through these examples, we show how one can couple the algorithms presented in this paper with equitable partitioning policies to make these amenable to distributed implementation. More in general, we illustrate a systematic approach to devise spatially distributed control policies for a large variety of multiagent coordination problems. Our approach is related to the classic Lloyd algorithm and exploits the unique features of power diagrams.

On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

In this paper, we are interested in nontrivial bi-Hamiltonian deformations of the Poisson pencil . Deformations are generated by a sequence of vector fields {X2, X3, X4, ...}, where each Xk is homogeneous of degree k with respect to a grading induced by rescaling. Constructing recursively the vector fields Xk, one obtains two types of relations involving their unknown coefficients: one set of linear relations and an other one which involves quadratic relations. We prove that the set of linear relations has a geometric meaning: using Miura-quasitriviality, the set of linear relations expresses the tangency of the vector fields Xk to the symplectic leaves of ω1 and this tangency condition is equivalent to the exactness of the pencil ωλ. Moreover, extending the results of Lorenzoni P (2002 J. Geom. Phys. 44 331–75), we construct the nontrivial deformations of the Poisson pencil ωλ, up to the eighth order in the deformation parameter, showing therefore that deformations are unobstructed and that both Poisson structures are polynomial in the derivatives of u up to that order.

F-Manifolds with Eventual Identities, Bidifferential Calculus and Twisted Lenard-Magri Chains

Given an

On integrable conservation laws.

F

Complex reflection groups, logarithmic connections and bi-flat F-manifolds

-manifold with eventual identities we examine what this structure entails from the point of view of integrable PDEs of hydrodynamic type. In particular, we show that in the semisimple case the characterization of eventual identities recently given by David and Strachan is equivalent to the requirement that

Poisson bracket on 1-forms and evolutionary partial differential equations

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Purely non-local Hamiltonian formalism, Kohno connections and ∨-systems

has vanishing Nijenhuis torsion. Moreover, after having defined new equivalence relations for connections compatible with respect to the

Inherited structures in deformations of Poisson pencils

F

On an eigenflow equation and its structure preserving properties

-product

Reciprocal F-manifolds

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