Mathematical Sciences: Methods of Non-Smooth Analysis in Optimal Stochastic Control

本研究的目的是继续研究Hamilton-Jacobi Bellman方程的非光滑解及其在最优控制理论中的应用。最近取得的成果建立了HJB方程与最优控制理论之间的一种新型关系。该值函数以其方向导数作为HJB方程的唯一解来表征。结果直观和几何上清晰的性质保证了强大的新一阶和二阶最优性条件，以及对最优策略结构的更深入的了解。对随机问题的推广将提供一种新的统一方法来研究确定性和随机控制理论，而不需要任何退化性或非退化性假设。研究的技术核心将是建立拉格朗日乘子与最优值函数的方向导数之间的必要关系，以及识别和研究合适的非光滑二阶微分算子。

The objective of this research is to continue the investigation of the non-smooth solutions of the Hamilton-Jacobi Bellman equation and their application to optimal control theory. Recently achieved results have established a new type of relationship between the HJB equation and optimal control theory. The value function is characterized in terms of its directional derivatives as the unique solution to HJB equation. The intuitively and geometrically clear-cut nature of the results promises powerful new first and second order optimality conditions as well as a deeper insight into the structure of the optimal policy. An extension to stochastic problem would offer a novel unified approach to deterministic and stochastic control theory free of any kind of degeneracy or non-degeneracy assumptions. The technical core of the research will be to establish the necessary relations between Lagrange multipliers and directional derivatives of the optimal value function as well as to identify and investigate an appropriate non-smooth second order differential operator.