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.

中文翻译：

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从坐标方向减少一般能量函数的轨迹收敛

我们考虑任意轨迹受到坐标方向能量减少的影响：每个条目的导数符号永远不会与某个能量函数梯度的相应条目的符号相同。我们表明，这个简单的条件可以保证收敛到一个点，收敛到能量函数的最小值，或者收敛到一个集合，其中 Hessian 矩阵具有非常特殊的属性。这扩展并加强了最近仅限于凸二次能量函数的结果。我们通过使用它来证明一类受多重不确定性影响的多代理系统的收敛性来证明我们结果的应用。