Several Problems of Stochastic Optimal Controls in Infinite Time Horizon

最优控制理论研究动态系统在一段时间内的控制，从而使目标函数最优化。最优控制问题在工程以及物理、经济和社会科学中经常遇到。最优控制问题的例子包括如何控制火箭推进器的发射，以最小的燃料支出达到选定的目标，或者如何实施货币政策以将失业率降至最低。当所考虑的问题的时间间隔变得无限大，并且需要考虑不确定性，即随机效应时，该理论潜在的数学困难变得更加复杂。这个项目将研究无限范围的随机最优控制问题，并通过引入现有模型的重大扩展来纳入新的影响和新的模型来扩展该领域的知识，这两个模型都需要新的思想和方法来分析它们。该项目还将为本科生和研究生参与这项研究提供机会。本项目将研究无限范围随机最优控制问题的几个重要方面，包括：(I)具有平均场且涉及平均二次成本的线性随机微分方程(SDE)的最优控制，通过不变度量；(Ii)随机最优控制的收费站性质；(Iii)无限范围内随机最优控制的极大值原理；(Iv)无限时间范围内随机最优控制的时间不一致。这些问题需要重新审查旧的方法和开发新的工具，以扩大范围和丰富最优控制理论的领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Optimal control theory deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. Optimal control problems are often encountered in engineering and in physical, economic, and social sciences. Examples of optimal control problems include how to control the firing of rocket thrusters to reach a selected target with minimum fuel expenditure, or how to implement monetary policy to minimize unemployment. The underlying mathematical difficulties of the theory are compounded when the time interval for the problem under consideration becomes infinite, and uncertainties, i.e., stochastic effects, need to be accounted for. This project will study stochastic optimal control problems with infinite horizon and extend the knowledge in the field through the introduction of significant extensions of current models to incorporate new effects, and also new models, both of which necessitate new ideas and approaches for their analysis. The project will also provide opportunities for the involvement of undergraduate and graduate students in this research. This project will investigate several important aspects of stochastic optimal control problems with infinite horizon, including: (i) Optimal control of linear stochastic differential equations (SDEs) having mean-field and involving average quadratic costs, via invariant measures; (ii) Turnpike properties of stochastic optimal controls for SDEs; (iii) maximum principle of stochastic optimal controls in infinite horizon for SDEs and for stochastic Volterra integral equations (SVIEs); (iv) Time-inconsistent optimal controls over infinite time horizon for SDEs. These problems necessitate the re-examination of old approaches and development of new tools to expand the scope and enrich the field of optimal control theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.