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By Wei Ren

Disbursed Coordination of Multi-agent Networks introduces difficulties, versions, and matters comparable to collective periodic movement coordination, collective monitoring with a dynamic chief, and containment regulate with a number of leaders. fixing those difficulties extends the present software domain names of multi-agent networks.

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The main approach used is to define artificial potentials in a proper way such that if two agents are neighbors at a certain time instant, they will always be neighbors afterwards. In [133], consensus with connectivity maintenance is solved when the weights for the edges of the interaction graph are defined properly. In [98], rendezvous of a group of agents with connectivity maintenance is solved based on a perimeter minimizing algorithm. In [282], a controller based on a properly designed potential function is proposed to solve rendezvous of a group of nonholonomic robots with connectivity maintenance.

2 (Rayleigh-Ritz theorem), p. 176]). Let A ∈ Rn×n be symmetric. Then λmin (A)xT x ≤ xT Ax ≤ λmax (A)xT x for T T = minxT x=1 xxTAx , and λmax (A) = all x ∈ Rn , λmin (A) = minx=0n xxTAx x x xT Ax xT Ax maxx=0n xT x = maxxT x=1 xT x . 9]). Let A ∈ Rn×n . If ||| · ||| is any matrix norm, then ρ(A) ≤ |||A|||. 10]). Let A ∈ Rn×n and ε > 0. There is a matrix norm ||| · ||| such that ρ(A) ≤ |||A||| ≤ ρ(A) + ε. 12]). Let A ∈ Rn×n . Then limk→∞ Ak = 0n×n if and only if ρ(A) < 1. 16]). Let A ∈ Rn×n . If ||| · ||| is a matrix norm ∞ and |||A||| < 1.

For a given differentiable x(t), according to Leibniz–Newton formula [114], we have that 0 x(t − τ ) = x(t) − x(t ˙ + s) ds. 10) −τ Suppose that f : R × Cn,τ → Rn is continuous, where Cn,τ denotes the Banach space of continuous vector functions mapping the interval [−τ, 0] into Rn with the topology of uniform convergence. Consider the retarded functional differential equation (RFDE) x(t) ˙ = f (t, xt ). 11) Let φ = xt be defined as xt (θ) = x(t + θ), θ ∈ [−τ, 0]. Suppose that appropriate initial conditions are defined on the delay interval [t0 − τ, t0 ]: xt0 (θ) = φ(θ), ∀θ ∈ [−τ, 0], where t0 ∈ R.

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