Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities
Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities
复制标题
测度微分包含和演化变分不等式框架下的凸扫描过程
作者：
S. Adly;T. Haddad;L. Thibault
在本文中，我们分析和讨论的两个新的变种的所谓的清扫过程，介绍了Moreau在70年代初（Moreau在Sém Anal Convexe Montpellier，1971）与塑性理论的动机。第一个新的变体是关于移动凸子集$$C（t）$$C（t）的法锥的扰动，假设有一个有界的变化，由Lipschitz映射。在一定的数据假设下，证明了扰动微分测度包含存在且仅有一个有界变差的右连续解。第二个变体，对它进行了大量的分析，涉及移动集合$$C（t）$$C（t）中具有速度的一阶扫描过程。这类问题包括作为一个特殊的情况下，演变变分不等式[广泛用于应用数学和单边力学（Duvaut和狮子在不等式力学和物理。Springer，柏林，1976年]。假设移动子集$$C（t）$$C（t）对每个$$tin [0，T]$$t∈[0，T]有连续变差且$$C（0）$$C（0）有界，我们证明了该问题至少有一个Lipschitz连续解.在算子的非线性假设下，得到了这类扫描过程的适定性。我们还讨论了扫描过程在弹塑性模型中矢量滞后算子的研究（Krejčnik in Eur J Appl Math 2：281-292，1991）、数学经济中的规划过程（亨利in J Math Anal Appl 41：179-186，1973和Cornet in J. Math. Anal. 96：130-147，1983），以及涉及包含非平滑电子器件如二极管的非规则电路（Acary等人，Nonsmooth modeling and simulation for switched circuits.电气工程讲义。Springer，纽约，2011年）。理论结果得到了数值模拟的支持，证明了算法的有效性。我们的方法仅基于凸分析工具。像其他文件在这个集合中，我们在这个演示文稿中显示如何优雅的现代凸分析的影响莫罗的开创性工作。
In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sém Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset $$C(t)$$C(t), supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set $$C(t)$$C(t). This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976]. Assuming that the moving subset $$C(t)$$C(t) has a continuous variation for every $$tin [0,T]$$t∈[0,T] with $$C(0)$$C(0) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Krejčı in Eur J Appl Math 2:281–292, 1991), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179–186, 1973 and Cornet in J. Math. Anal. Appl. 96:130–147, 1983), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau’s seminal work.