非凸积分-微分扫描过程及其应用
Nonconvex integro-differential sweeping process with applications
摘要：
本文分析和讨论了J. J. Moreau在70年代初的\cite{More71}中以塑性理论为动机提出的所谓扫描过程的一个新变体的适定性。在这个变体中，（轻度非凸的）正则移动集$C(t)$的法锥，假设有一个绝对连续的变化，被Carathéodory映射和积分强迫项的和扰动。强迫项的被积函数依赖于两个时间变量，即我们研究了一般的Volterra型积分微分扫描过程。通过建立适当的半离散化方法，结合一个新的Gronwall-like不等式（微分不等式），证明了积分-微分扫描过程有且仅有一个绝对连续解.我们还建立了解关于初值的连续性。本文的结果被应用于研究具有非线性互补约束的Volterra积分微分方程的组合的非线性积分微分互补系统.另一个应用涉及包含时变电容器和非平滑电子器件（如二极管）的非规则电路。这两个应用程序代表了我们的论文的一个额外的新颖性。
英文摘要：
In this paper, we analyze and discuss the well-posedness of a new variant of the so-called sweeping process, introduced by J.J. Moreau in the early 70's \cite{More71} with motivation in plasticity theory. In this variant, the normal cone to the (mildly non-convex) prox-regular moving set $C(t)$, supposed to have an absolutely continuous variation, is perturbed by a sum of a Carathéodory mapping and an integral forcing term. The integrand of the forcing term depends on two time-variables, that is, we study a general integro-differential sweeping process of Volterra type. By setting up an appropriate semi-discretization method combined with a new Gronwall-like inequality (differential inequality), we show that the integro-differential sweeping process has one and only one absolutely continuous solution. We also establish the continuity of the solution with respect to the initial value. The results of the paper are applied to the study of nonlinear integro-differential complementarity systems which are combination of Volterra integro-differential equations with nonlinear complementarity constraints. Another application is concerned with non-regular electrical circuits containing time-varying capacitors and nonsmooth electronic device like diodes. Both applications represent an additional novelty of our paper.
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