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

Yuan-Hua Ni (Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin, China)

Ji-Feng Zhang (Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)

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.

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