"Nonsmooth dynamics associated to variational inequalities, generalized Nash games and applications"

这一建议研究了三个重要的数学结构之间的关系：非光滑动力系统、拟变分不等式和广义Nash对策，以及它们的适当应用。变分不等式(VI)是在60年代初在偏微分方程组的框架内引入的。后来的研究表明，它们等价地重新表述了一大类均衡问题(如纳什、瓦洛普、沃尔拉斯、古诺、交通均衡等)，从而在运筹学中变得突出起来。由于这种等价性，VI的解通常被称为平衡。在我的工作中，我特别感兴趣的是一类特殊的VI，称为“准变分”(QVI)。本质上，QVI模拟均衡问题，约束集依赖于均衡解。 从本质上讲，VI/QVI问题是静态的。为了描述这类问题的动态，方法范围从进化的不平等到最近的进化和微分变分不等概念。我的方法依赖于将VI/QVI与由微分方程式或包含的解给出的动力系统相关联，从而满足两个条件：VI/QVI的解与方程或包含的驻点重合；方程/包含的解始终保持在约束集中。这些系统属于非光滑动力学领域。QVI理论的结果表明，相关动力学方程存在定常解。然而，非定常解的存在唯一性和稳定性分析在很大程度上是一个未解决的问题。调查这些问题是我的第一个目标。这个目标是我之前将VI与非光滑系统联系起来的一些工作的自然扩展。我的第二个目标是利用上面的结果提供解决广义Nash(GN)博弈的新方法，它可以等价地重新表述为QVI问题。一些GN游戏的解决方案是已知的，但总的来说，这是一个悬而未决的问题。我的第三个目标涉及上述结果在决策的动态人口模型中的应用，在网络拓扑变化的时间依赖的交通网络问题中的应用，以及在流行病学中的应用。

This proposal investigates the relationship between three important mathematical constructs: nonsmooth dynamical systems, quasivariational inequalities, and generalized Nash games, and their appropriate applications. Variational inequalities (VI) were introduced in the early 60s within the framework of partial differential equations. Later it was shown they equivalently reformulate large classes of equilibrium problems (such as Nash, Wardrop, Walras, Cournot, traffic equilibrium etc.), thus becoming prominent in operations research. Due to this equivalence, solutions of VI are often called equilibria. Of specific interest in my work is a particular class of VI, called "quasivariational" (QVI). Essentially a QVI models equilibrium problems with constraint sets dependent on the equilibrium solution. At their core VI/QVI problems are static. To describe the dynamics of such problems, approaches range from inequalities of evolution to recent concepts of evolutionary and differential variational inequalities. My approach relies on associating VI/QVI to dynamical systems given by solutions to a differential equation or inclusion so that two conditions are met: the solutions of the VI/QVI coincide with the equation's or inclusion's stationary points; the solutions of the equation/inclusion remain inside the constraint set at all times. These systems belong to the area of nonsmooth dynamics. Results from QVI theory imply existence of stationary solutions for the associated dynamics. However, existence, uniqueness and stability analysis of non-stationary solutions is largely an open problem. Investigating these questions is my first objective. This objective is a natural extension of some of my previous work relating VI with nonsmooth systems. My second objective is to use results above to provide novel ways to solve generalized Nash (GN) games which can be equivalently reformulated as QVI problems. Solutions for some GN games are known, but in general this is an open question. My third objective concerns applications of results above in dynamic population models of decision making, in time-dependent traffic network problems with changes in network topology, and in epidemiology.