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Optimal Control with Partially Observed Regime Switching: Discounted and Average Payoffs

Author

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  • Beatris Adriana Escobedo-Trujillo

    (Facultad de Ingeniería Campus Coatzacoalcos, Universidad Veracruzana, Coatzacoalcos 96535, Veracruz, Mexico)

  • Javier Garrido-Meléndez

    (Facultad de Ingeniería Campus Coatzacoalcos, Universidad Veracruzana, Coatzacoalcos 96535, Veracruz, Mexico)

  • Gerardo Alcalá

    (Centro de Investigación en Recursos Energéticos y Sustentables, Universidad Veracruzana, Coatzacoalcos 96535, Veracruz, Mexico)

  • J. D. Revuelta-Acosta

    (Facultad de Ingeniería Campus Coatzacoalcos, Universidad Veracruzana, Coatzacoalcos 96535, Veracruz, Mexico)

Abstract

We consider an optimal control problem with the discounted and average payoff. The reward rate (or cost rate) can be unbounded from above and below, and a Markovian switching stochastic differential equation gives the state variable dynamic. Markovian switching is represented by a hidden continuous-time Markov chain that can only be observed in Gaussian white noise. Our general aim is to give conditions for the existence of optimal Markov stationary controls. This fact generalizes the conditions that ensure the existence of optimal control policies for optimal control problems completely observed. We use standard dynamic programming techniques and the method of hidden Markov model filtering to achieve our goals. As applications of our results, we study the discounted linear quadratic regulator (LQR) problem, the ergodic LQR problem for the modeled quarter-car suspension, the average LQR problem for the modeled quarter-car suspension with damp, and an explicit application for an optimal pollution control.

Suggested Citation

  • Beatris Adriana Escobedo-Trujillo & Javier Garrido-Meléndez & Gerardo Alcalá & J. D. Revuelta-Acosta, 2022. "Optimal Control with Partially Observed Regime Switching: Discounted and Average Payoffs," Mathematics, MDPI, vol. 10(12), pages 1-28, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2073-:d:839482
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    References listed on IDEAS

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    1. Xianggang Lu & G. Yin & Xianping Guo, 2017. "Infinite Horizon Controlled Diffusions with Randomly Varying and State-Dependent Discount Cost Rates," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 535-553, February.
    2. Kawaguchi, Kazuhito & Morimoto, Hiroaki, 2007. "Long-run average welfare in a pollution accumulation model," Journal of Economic Dynamics and Control, Elsevier, vol. 31(2), pages 703-720, February.
    3. Borkar, V. S., 2003. "Dynamic programming for ergodic control with partial observations," Stochastic Processes and their Applications, Elsevier, vol. 103(2), pages 293-310, February.
    4. Kunita, Hiroshi, 1971. "Asymptotic behavior of the nonlinear filtering errors of Markov processes," Journal of Multivariate Analysis, Elsevier, vol. 1(4), pages 365-393, December.
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    Cited by:

    1. Francisco Germán Badía & María D. Berrade, 2023. "Special Issue “Probability Theory and Stochastic Modeling with Applications”," Mathematics, MDPI, vol. 11(14), pages 1-3, July.

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