Several Problems of Stochastic Optimal Controls in Infinite Time Horizon

最优控制理论研究的是在一段时间内对动态系统进行控制，使目标函数得到优化。最优控制问题经常在工程、物理、经济和社会科学中遇到。最优控制问题的例子包括如何控制火箭推进器的发射，以最小的燃料消耗达到选定的目标，或者如何实施货币政策以最小化失业。当所考虑的问题的时间间隔变得无限时，该理论的潜在数学困难就变得复杂了，并且需要考虑不确定性，即随机效应。本项目将研究具有无限视界的随机最优控制问题，并通过引入现有模型的重大扩展来扩展该领域的知识，以纳入新的效应，以及新的模型，这两者都需要新的思想和方法来分析。该项目还将为本科生和研究生提供参与本研究的机会。本项目将研究具有无限视界的随机最优控制问题的几个重要方面，包括：(i)通过不变测度对具有平均场且涉及平均二次代价的线性随机微分方程（SDEs）的最优控制；（ii） SDEs随机最优控制的收费公路性质；（iii） SDEs和随机Volterra积分方程的无穷视界随机最优控制的极大原理；sde在无限时间范围内的时间不一致最优控制。这些问题需要重新审视旧的方法和开发新的工具，以扩大和丰富最优控制理论的范围和领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.