Time-inconsistent optimal control problems

An optimal control problem is time-consistent if for any initial pair of time and state, whenever there exists an optimal control, it will stay optimal thereafter. In real world, however, such kind of time-consistency is hardly true, mainly due to the time-inconsistency of decision maker's time-preference and/or risk-preference. In another word, most optimal control problems, if not all, are not time-consistent, or time-inconsistent. In this paper, some general time-inconsistent optimal control problems are formulated for stochastic differential equations. Recent works of the author concerning the (time-consistent) equilibrium solutions to the time-inconsistent problems are surveyed.