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arXiv 2203.033162022-03math.OCcs.SYeess.SYmath.DS

一般能量函数坐标方向下降下的轨迹收敛性

Trajectory Convergence from Coordinate-wise Decrease of General Energy Functions

Julien M. Hendrickx, Balázs Gerencsér

备注：: 9 pages, 1 figure

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AI总结：

本文研究一般能量函数在坐标方向下降条件下的轨迹收敛性，证明该条件可保证收敛到点、最小值或具有特定Hessian性质的集合，并将其应用于多智能体系统收敛性分析。

摘要：

我们考虑满足坐标方向能量下降的任意轨迹：每个分量的导数符号与某个能量函数梯度对应分量的符号从不相同。我们证明这一简单条件保证轨迹收敛到一个点、能量函数的最小值，或一个其Hessian具有非常特定性质的集合。这扩展并加强了最近仅限于凸二次能量函数的结果。我们通过利用该结果证明一类受多种不确定性影响的多智能体系统的收敛性，展示了其应用。

英文摘要：

We consider arbitrary trajectories subject to a coordinate-wise energy decrease: the sign of the derivative of each entry is never the same as that of the corresponding entry of the gradient of some energy function. We show that this simple condition guarantees convergence to a point, to the minimum of the energy functions, or to a set where its Hessian has very specific properties. This extends and strengthens recent results that were restricted to convex quadratic energy functions. We demonstrate the application of our result by using it to prove the convergence of a class of multi-agent systems subject to multiple uncertainties.

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