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Applied Mathematics & Optimization

, Volume 78, Issue 1, pp 145–183 | Cite as

Stochastic Linear Quadratic Optimal Control Problems in Infinite Horizon

  • Jingrui Sun
  • Jiongmin Yong
Article

Abstract

This paper is concerned with stochastic linear quadratic (LQ, for short) optimal control problems in an infinite horizon with constant coefficients. It is proved that the non-emptiness of the admissible control set for all initial state is equivaleznt to the \(L^{2}\)-stabilizability of the control system, which in turn is equivalent to the existence of a positive solution to an algebraic Riccati equation (ARE, for short). Different from the finite horizon case, it is shown that both the open-loop and closed-loop solvabilities of the LQ problem are equivalent to the existence of a static stabilizing solution to the associated generalized ARE. Moreover, any open-loop optimal control admits a closed-loop representation. Finally, the one-dimensional case is worked out completely to illustrate the developed theory.

Keywords

Stochastic linear quadratic optimal control Stabilizability Open-loop solvability Closed-loop solvability Algebraic Riccati equation Static stabilizing solution Closed-loop representation 

AMS Subject Classifications

49N10 49N35 93D15 93E20 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their suggestive comments, which lead to an improvement of the paper. Jiongmin Yong was partially supported by NSF Grant DMS-1406776.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Department of MathematicsUniversity of Central FloridaOrlandoUSA

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