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# Communications in Information and Systems

## Volume 15 (2015)

### Number 4

### Stochastic linear-quadratic optimal control without time-consistency requirement

Pages: 521 – 550

DOI: http://dx.doi.org/10.4310/CIS.2015.v15.n4.a5

#### Authors

#### Abstract

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.