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Time-inconsistent stochastic linear quadratic control for discrete-time systems

Abstract

This paper is mainly concerned with the time-inconsistent stochastic linear quadratic (LQ) control problem in a more general formulation for discrete-time systems. The time-inconsistency arises from three aspects: the coefficient matrices depending on the initial pair, the terminal of the cost function involving the initial pair together with the nonlinear terms of the conditional expectation. The main contributions are: firstly, the maximum principle is derived by using variational methods, which forms a flow of forward and backward stochastic difference equations (FBSDE); secondly, in the case of the system state being one-dimensional, the equilibrium control is obtained by solving the FBSDE with feedback gain based on several nonsymmetric Riccati equations; finally, the necessary and sufficient solvability condition for the time-inconsistent LQ control problem is presented explicitly. The key techniques adopted are the maximum principle and the solution to the FBSDE developed in this paper.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61120106011, 61573221, 61633014). Qingyuan QI was supported by the Program for Outstanding Ph.D. Candidate of Shandong University.

Author information

Correspondence to Huanshui Zhang.

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Cite this article

Qi, Q., Zhang, H. Time-inconsistent stochastic linear quadratic control for discrete-time systems. Sci. China Inf. Sci. 60, 120204 (2017). https://doi.org/10.1007/s11432-017-9167-3

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Keywords

  • time-inconsistency
  • open-loop equilibrium control
  • maximum principle
  • FBSDE
  • nonsymmetric Riccati equations