紧性条件下带多值扰动的Volterra扫掠过程
Volterra Sweeping Processes with Multivalued Perturbations under Compactness Conditions
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中文总结 AI 辅助
本文在可分希尔伯特空间中研究带多值扰动的Volterra型积分-微分扫掠过程，在两种紧性假设下证明绝对连续解的存在性，并将结果应用于两类含不确定多值强迫项的实际问题。
中文摘要 AI 辅助
我们在可分希尔伯特空间中研究Volterra型积分-微分扫掠过程，其中速度由 prox-正则移动集的法锥控制，且由外多值扰动项和赋予动力学记忆的依赖历史的积分项共同驱动。假设该扰动可测，具有闭凸值、线性增长，且从强拓扑到弱拓扑上半连续。我们在两种可选紧性假设下研究绝对连续解的存在性：移动集的球紧性，或扰动的非紧性测度条件。在将约束动力学约化为无约束微分包含，并得到状态及其速度的一致先验界后，借助具有可缩值的多值映射的不动点定理即可得到存在性。过程的记忆使可缩性分析变得复杂，我们通过传播动力学历史的延拓论证来获得该性质。作为应用，我们求解了两个问题：具有长记忆的准静态无摩擦黏弹性接触问题，以及具有生态记忆的空间分布生物经济渔业模型，两者均包含不确定的多值强迫项和非球紧的移动约束集。
英文摘要
We study integro-differential sweeping processes of Volterra type in a separable Hilbert space, in which the velocity is governed by the normal cone to a prox-regular moving set and is driven by an outer set-valued perturbation together with a history-dependent integral term that endows the dynamics with memory. The perturbation is assumed measurable, with closed convex values, of linear growth, and upper semicontinuous from the strong to the weak topology. We study the existence of absolutely continuous solutions under either of two alternative compactness hypotheses: ball-compactness of the moving sets, or a measure-of-noncompactness condition on the perturbation. After a reduction of the constrained dynamics to an unconstrained differential inclusion and uniform a priori bounds on the state and its velocity, existence follows from a fixed-point theorem for set-valued maps with contractible values. The memory of the process makes this contractibility delicate, and we obtain it through a continuation argument that propagates the history of the dynamics. As applications, we solve a quasistatic frictionless viscoelastic contact problem with long memory and a spatially distributed bioeconomic fishery model with ecological memory, both featuring uncertain set-valued forcing and a moving constraint set that is not ball-compact.
