Communications in Information and Systems
Volume 15 (2015)
Stochastic linear-quadratic optimal control without time-consistency requirement
Pages: 521 – 550
In this paper, linear-quadratic optimal control without time-consistency requirement is studied for a class of linear discrete-time systems with multiplicative stochastic disturbances. Both the open-loop and the closed-loop time-consistent solutions are investigated. Necessary and sufficient conditions on the existence of the open-loop time-consistent equilibrium control and the closed-loop time-consistent equilibrium strategy are obtained, respectively. Specifically, the existence of the open-loop time-consistent equilibrium control for all the initial time-state pairs is equivalent to the solvability of two coupled constrained linear difference equations and two coupled constrained generalized difference Riccati equations; the existence of the closed-loop time-consistent equilibrium strategy is equivalent to the solvability of another two coupled constrained generalized difference Riccati equations. It can be found that Riccati equations for the open-loop formulation do not admit symmetry structure, while the ones for the closed-loop formulation do have symmetric solutions.