Mathematical Sciences: Viscosity Solutions and Control Problems

这个项目涉及几个问题，涉及粘性解和最优控制问题。第一个问题涉及将粘性解的结果推广到具有二次项和非局部耦合项的系统。这些耦合项可以表示竞争和合作系统中的相互作用项。将开发单调迭代格式来求解此类系统。第二个问题涉及与两人博弈有关的二阶椭圆型系统的粘性解。在说明性示例中，一个玩家控制扩散过程的切换，而另一个玩家控制该过程的停止。人们想要证明系统的存在性结果，并证明粘性解给出了对策的鞍点的值。第三个问题是对状态由一阶偏微分方程组控制的最优系统的直接研究。在最小化涉及目标函数的代价泛函时，目标是刻画最优控制并求解最优系统(具有最优控制的状态方程和伴随方程)。

This project concerns several problems involving viscosity solutions and optimal control problems. The first problem involves the extension of viscosity solution results to systems with quadratic and non-local coupling terms. These coupling terms can represent interaction terms in competition and cooperative systems. Monotone iteration schemes will be developed to solve such systems. The second problem involves viscosity solutions for second order elliptic systems associated with two player games. In an illustrative example, one player controls the switching of a diffusion process and the other player controls the stopping of the process. One wants to show existence results for the system and prove that the viscosity solution gives the value of the saddle point of the game. The third problem is a direct study of optimality systems in which the state is governed by first order PDEs. In minimizing a cost functional involving a target function, the goal is to characterize the optimal control and solve the optimality system (state equation with optimal control and the adjoint equation).