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Stochastic Linear Quadratic Optimal Control Problems in Infinite Horizon

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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.

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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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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Jingrui Sun

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Jiongmin Yong

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  1. Jingrui Sun
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  2. Jiongmin Yong
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Correspondence to Jingrui Sun.

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Sun, J., Yong, J. Stochastic Linear Quadratic Optimal Control Problems in Infinite Horizon. Appl Math Optim 78, 145–183 (2018). https://doi.org/10.1007/s00245-017-9402-8

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