On the convergence rate of the fraction of simple algebras

This paper considers flocking to the virtual leader in network of agents with double-integrator. A locally linear algorithm is employed which guarantees exponential flocking to the virtual leader. A lower bound for flocking rate is calculated which is independent of the initial conditions. Simulations are provided to validate the result and it is shown that the calculated rate is not over bound the actual convergence rate. The effect of coefficients of algorithm is investigated and it is shown that the similar results can be inferred from the calculated formula for the convergence rate.

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On a Lower Bound for the Convergence Rate in a Local Limit Theorem

Theory of Probability and Its Applications ◽

10.1137/1130100 ◽

1986 ◽

Vol 30 (4) ◽

pp. 810-814 ◽

Cited By ~ 1

Author(s):

V. K. Matskyavichyus

Keyword(s):

Limit Theorem ◽

Lower Bound ◽

Convergence Rate ◽

Local Limit Theorem ◽

Local Limit

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A Lower Bound for the Convergence Rate in the Central Limit Theorem in Hilbert Space

Theory of Probability and Its Applications ◽

10.1137/1131104 ◽

1987 ◽

Vol 31 (4) ◽

pp. 730-733 ◽

Cited By ~ 2

Author(s):

F. A. Aliev

Keyword(s):

Hilbert Space ◽

Central Limit Theorem ◽

Limit Theorem ◽

Lower Bound ◽

Convergence Rate ◽

Central Limit ◽

The Central Limit Theorem

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Weighted L 2-contractivity of Langevin dynamics with singular potentials

Nonlinearity ◽

10.1088/1361-6544/ac4152 ◽

2021 ◽

Vol 35 (2) ◽

pp. 998-1035

Author(s):

Evan Camrud ◽

David P Herzog ◽

Gabriel Stoltz ◽

Maria Gordina

Keyword(s):

Lower Bound ◽

Probability Measure ◽

Convergence Rate ◽

Exponential Convergence ◽

Invariant Probability Measure ◽

Langevin Dynamics ◽

Convergence To Equilibrium ◽

Singular Potentials ◽

Lennard Jones ◽

Polynomial Type

Abstract Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by Hérau and developed by Dolbeault et al, we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2(dμ) and L 2(W* dμ), where μ denotes the invariant probability measure and W* is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min(γ, γ −1). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.

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