Optimal Control of PDEs, ODEs, and Variational Inequalities and Applications

小行星9704199 偏微分方程的最优控制，常微分 方程和变分不等式将被研究。 的第一区域 研究偏微分方程混合系统的控制， 常微分方程及其在生物修复中的应用 人口建模 第二个方面是障碍物的最优控制， 变分不等式，椭圆或四阶算子（涉及 双调和算子）。第三个方面是最佳的调查 HIV模型中的化疗策略。 第一，存在一个最优 控制必须显示为模型和目标泛函适当 每个应用领域。 其次，这种最优控制必须是 其特征在于通过必要的条件，从推导一个 最优性系统，它是与伴随系统耦合的状态系统。 由于处理的系统的多样性，存在性，唯一性， 最优控制的表征需要新的和不同的先验 估计和收敛参数。 对于某些应用程序， 将完成数值模拟。 本研究主要涉及局部最优控制的理论和应用 微分方程，常微分方程，变分 不等式 第一个应用领域是生物修复模型的控制 用于鼓励通过细菌代谢来修复污染物。 控制营养物输入的水平以最佳地运行生物反应器。 该方法将在气相生物反应器上进行实验室测试， 降解对二甲苯的恶臭细菌。 其他人口模型将 研究了第二个应用领域是控制变 不等式，其模拟膜的形状的设计， 控制放置在膜下的障碍物。 这些模型产生于流动 通过多孔介质，流体动力润滑，弹塑性分析， 连续铸造和电化学加工。 第三应用 该领域是设计治疗艾滋病毒的最佳化学疗法 感染 可以模拟“各种药物的鸡尾酒”的效果， 一种涉及CD4+T白色血细胞和HIV的免疫学模型 病毒 也是一个模型，涉及自细胞成为 艾滋病毒感染者将被研究。 数值模拟的最佳 控制和产生的控制量（如CD4+T细胞），将是 使用真实的数据完成。

9704199 Lenhart Optimal control of partial differential equations, ordinary differential equations, and variational inequalities will be investigated. The first area of research is the control of hybrid systems of partial differential equations and ordinary differential equations, with applications in bioremediation and population modeling. The second area is optimal control of the obstacle in variational inequalities, with elliptic or fourth order operators (involving biharmonic operators). The third area is the investigation of optimal chemotherapy strategies in HIV models. First, the existence of an optimal control must be shown for the models and objective functionals appropriate to each application area. Secondly, this optimal control must be characterized through necessary conditions from the derivation of an optimality system, which is the state system coupled with an adjoint system. Due to the variety of systems treated, existence, uniqueness, and characterization of the optimal control requires new and different a priori estimates and convergence arguments. For some of the applications, numerical simulations will be completed. This research involves theory and applications of optimal control of partial differential equations, ordinary differential equations, and variational inequalities. The first application area is control of a bioremediation model for encouraging the remediation of contaminants by metabolism of bacteria. The level of the nutrient inputs is controlled to optimally run a bioreactor. The approach will be lab tested on a gas phase bioreactor with pseudomonas putida bacteria degrading paraxylene. Other population models will be investigated. The second application area is the control of variational inequalities, which model the designing of the shape of the membrane by controlling an obstacle put under the membrane. These models arise in flow through porous media, hydrodynamic lubrication, elasto-plastic analysis, continuous casting, and electrochemical machining. The third application area is the design of optimal chemotherapies for the treatment of HIV infection. The effect of a "cocktail of various drugs" can be simulated for an immunology model involving CD4+T white blood cells and the HIV virus. Also a model involving the time elapsed since the cells became infected with HIV will be studied. Numerical simulations for the optimal control and resulting controlled quantities (like CD4+T cells), will be completed using realistic data.