Volterra扫掠过程带单侧Lipschitz扰动的Filippov定理

中文摘要 AI 辅助

我们在可分Hilbert空间中，对带有外部多值扰动的Volterra型积分微分扫掠过程证明了Filippov型稳定性定理，其假设为：移动集合关于Hausdorff距离一致prox-正则且Lipschitz连续，并且扰动是单侧Lipschitz的。给定系统的一个绝对连续解，该解在多值项的状态参数上受到扰动，并带有外部可积项，我们证明了原始Volterra扫掠过程解的存在性，并利用问题的数据显式估计了两条轨迹之间的距离。当仅存在外部扰动时，该估计变得更精确，并且它恢复了Filippov经典定理中关于初始条件的Lipschitz依赖性。证明结合了将受约束动力学约化为无约束微分包含、可测选择论证以及本文建立的Grönwall不等式的增强版本。作为应用，我们获得了可达集关于初始集在Hausdorff距离下的Lipschitz依赖性，以及一个量化扰动效应的单侧估计。我们还研究了一个具有生态记忆的空间分布渔业模型，其由总生物量触发的捕捞规则是单侧Lipschitz的，但关于Hausdorff距离不是Lipschitz连续的，因此经典框架不适用。对于该模型，我们的估计产生了一个仅由生物数据决定的稳定性常数。

英文摘要

We prove a Filippov-type stability theorem for integro-differential sweeping processes of Volterra type with an outer multivalued perturbation, in a separable Hilbert space, under the assumptions that the moving sets are uniformly prox-regular and Lipschitz continuous with respect to the Hausdorff distance and that the perturbation is one-sided Lipschitz. Given an absolutely continuous solution of the system perturbed both in the state argument of the multivalued term and by an outer integrable term, we show the existence of a solution of the original Volterra sweeping process and estimate the distance between the two trajectories explicitly in terms of the data of the problem. The estimate becomes sharper when only the outer perturbation is present, and it recovers the Lipschitz dependence on the initial condition of Filippov's classical theorem. The proof combines a reduction of the constrained dynamics to an unconstrained differential inclusion, measurable selection arguments, and an enhanced version of Grönwall's inequality established in this work. As an application, we obtain the Lipschitz dependence of the attainable set on the initial set with respect to the Hausdorff distance, together with a one-sided estimate quantifying the effect of the perturbations. We also work out a spatially distributed fishery model with ecological memory whose harvesting rule, triggered by the aggregate biomass, is one-sided Lipschitz but is not Lipschitz continuous with respect to the Hausdorff distance, so that the classical framework does not apply. For this model, our estimates produce a stability constant governed only by the biological data.

发表机构

• Université Bourgogne Europe（勃艮第欧洲大学）
• CNRS（法国国家科学研究中心）
• Universidad de O’Higgins（奥希金斯大学）
• Universidad de Chile（智利大学）

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