Advances in the theory and applications of projected dynamical systems and double-layered dynamics

该提案旨在推进无限维投影动力系统（PDS）的理论和应用，以及多时间尺度动力学的理论和应用。 PDS 在 70 年代以微分包含的形式引入数学文献中（在凸变分分析及其经济应用的背景下），然后通过将其视为希尔伯特空间上的一类不连续微分方程而演变为现在的形式。一般来说，PDS 描述了某些约束问题从不平衡到平衡的时间演化。 PDS 与变分不等式问题有着密切的关系，并构成了一种跨学科兴趣的数学理论。它们已被应用于工程、交通科学、运筹学、经济学、金融和博弈论等领域。我们建议通过继续研究周期循环的存在性和稳定性问题以及新结果对应用问题的影响来推进 PDS 理论。该提案的作者已获得（并发布）了该方向的第一个结果。目前这个方向的研究也涉及一名研究生。我们建议扩展现有的 PDS 理论，作为干预双层动力学 DLD 的理论（通过将 PDS 扩展到 Lp 空间），并将 DLD 的理论和领域（参见当前出版物）扩展到网络交互和互联网流量问题。我们首次开始在流行病学中使用 PDS 理论，为具有不同感染/疫苗接种风险评估的人群建模疫苗接种策略游戏。我们计划进一步研究此类游戏的 DLD 模型。     最后但并非最不重要的一点是，鉴于本提案的作者（作为 PI）收到了 CFI 高性能计算设施资助，我们建议为 PDS/DLD 应用开发数值模拟。

This proposal is meant to advance both the theory and the applications of the infinite-dimensional projected dynamical systems (PDS), as well as to advance the theory and applications of multiple time-scale dynamics. PDS were introduced in the mathematical literature in the 70's in the form of differential inclusions (in the context of convex variational analysis and its economic applications) and then evolved to their present form via their treatment as a class of discontinuous differential equations on Hilbert spaces. In general, a PDS describes the time evolution of certain constraint problems from disequilibrium to equilibrium. PDS share an intimate relation to variational inequality problems and constitute a mathematical theory of interdisciplinary interest. They have been used in areas such as engineering, transportation science, operations research, economics, finance and game theory. We propose to advance the theory of PDS by continuing to study the question of existence and stability of periodic cycles and the implications of the new results for applied problems. The first results in this direction have been obtained (and published) by the author of this proposal. The current study in this direction also involves a graduate student. We propose to extend the present theory of PDS, as intervenes in that of double layered dynamics-DLD (by extending PDS to Lp-spaces), and to extend the theory and areas of DLD (see current publications) to networks interaction and internet traffic problems. We started using, for the first time, PDS theory in epidemiology, by modelling vaccination strategies games for groups of populations with distinct infection/ vaccination risk assessments. We plan to further this investigation towards a DLD model for such games.     Last but not least, given the CFI High Performance Computing Facility grant the author of this proposal received (as PI), we propose the development of numerical simulations for the applications of PDS/DLD.