Several Problems of Stochastic Optimal Controls in Infinite Time Horizon

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.