Mathematical Sciences: Systems of PDEs: Viscosity Solutions and Optimal Control Applications

这是一个多方面的调查，其主要重点是应用偏微分方程及其最优控制问题的力学，人口动力学，数学生态学和数学生理学。 研究的问题包括：Kirchhoff板方程的双线性控制，种群动力学中退化抛物型偏微分方程组的粘性解;模拟免疫系统与人类免疫缺陷病毒（HIV）相互作用的常微分方程系统的最优控制其目的是设计用于化学疗法治疗的最佳调度程序;以及使用营养物的输入泵送水平作为控制变量对用于危险废物的生物修复的模型进行最佳控制。该项目由应用数学计划和经典分析计划共同资助。 这是一个多方面的调查，其主要重点是应用偏微分方程及其最优控制问题的力学，人口动力学，数学生态学和数学生理学。具体研究的问题有：（1）板方程的双线性最优控制，以及在弹性结构控制中的应用;（2）模拟种群动力学的偏微分系统的系统分析，以及在社会、政治、军事和环境情况下的竞争和合作方案的应用;（3）在描述免疫系统与人类免疫缺陷病毒（HIV）的相互作用的模型中，以降低病毒复制率为目标的化疗治疗调度的最优控制;以及（4）通过生物修复过程对危险废物处置的最佳控制，其中控制变量是优化生物修复过程的效率的必需营养物的输送模式。 该项目由应用数学计划和经典分析计划共同资助。

This is a multi-faceted investigation whose primary focus is the application of partial differential equations and their optimal control to problems in mechanics,population dynamics, mathematical ecology and mathematical physiology. Among the problems to be investigated are bilinear control of Kirchhoff plate equations;viscosity solutions of systems of systems of degenerate parabolic partial differential equations that arise in population dynamics; optimal control of a system of ordinary differential equations which models the interaction of the immune system with the human immunodeficiency virus (HIV) with the goal of devising an optimal scheduling procedure for chemotherapy treatments; and optimal control of a model for bioremediation of hazardous waste using the input pumping levels of nutrients as the control variable. This project is jointly funded by the Applied Mathematics Program and the Classical Analysis Program. This is a multi-faceted investigation whose primary focus is the application of partial differential equations and their optimal control to problems in mechanics, population dynamics, mathematical ecology and mathematical physiology. Among the specific problems to be investigated are (1) the bilinear optimal control of plate equations, with applications to control of elastic structures; (2) an analysis of systems of partial differential systems which model population dynamics, with applications to competitive and cooperative scenarios in social, political, military, and environmental situations; (3) optimal control of chemotherapy treatment scheduling in a model which describes the interaction of the immune system with the human immunodeficiency virus (HIV) with the goal of reducing viral replication rates; and (4) optimal control of hazardous waste disposal through the process of bioremediation in which the control variable is the delivery pattern of the essential nutrients that optimizes the efficiency of the bioremediation process. Th is project is jointly funded by the Applied Mathematics Program and the Classical Analysis Program.