Mathematical Sciences: Methods of Non-Smooth Analysis In Optimal Control

这项研究的目的是利用非光滑分析的概念和方法，在最优随机控制问题和Hamilton-Jacobi-Bellman方程之间建立一种新型的联系，而不考虑潜在过程的可能退化性质。这将对变分问题和某些非线性偏微分方程之间的相互关系提供新的见解，并为确定性和随机最优控制理论提供一种统一的方法。研究的核心是研究最优值函数关于次微分和Clarke导数性质的正则性，以及寻找一类非光滑函数，其中的Hamilton-Jacobi-Bellman方程是唯一可解的。积累关于应用问题的计算和建模经验也是该项目的一部分。

The objective of this research is to use the notions and methods of non-smooth analysis to establish a new type of connection between optimal stochastic control problems and the Hamilton- Jacobi-Bellman equation irrespective of the possible degeneracy properties of the underlying processes. This would provide a new insight into the interrelation between variational problems and certain nonlinear partial differential equations as well as a unified approach to both deterministic and stochastic optimal control theory. The core of the research is the investigation of the regularity of properties of the optimal value function in terms of its subdifferentials, Clarke derivatives and the finding of a class of non-smooth functions in which the Hamilton-Jacobi- Bellman equation is uniquely solvable. Accumulation of computational and modeling experience on application problems are also parts of the project.