IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i5p647-d1344064.html
   My bibliography  Save this article

Nonlinear Optimal Control for Stochastic Dynamical Systems

Author

Listed:
  • Manuel Lanchares

    (School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA)

  • Wassim M. Haddad

    (School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA)

Abstract

This paper presents a comprehensive framework addressing optimal nonlinear analysis and feedback control synthesis for nonlinear stochastic dynamical systems. The focus lies on establishing connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory within a unified perspective. We demonstrate that the closed-loop nonlinear system’s asymptotic stability in probability is ensured through a Lyapunov function, identified as the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation. This dual assurance guarantees both stochastic stability and optimality. Additionally, optimal feedback controllers for affine nonlinear systems are developed using an inverse optimality framework tailored to the stochastic stabilization problem. Furthermore, the paper derives stability margins for optimal and inverse optimal stochastic feedback regulators. Gain, sector, and disk margin guarantees are established for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton–Jacobi–Bellman controllers.

Suggested Citation

  • Manuel Lanchares & Wassim M. Haddad, 2024. "Nonlinear Optimal Control for Stochastic Dynamical Systems," Mathematics, MDPI, vol. 12(5), pages 1-30, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:647-:d:1344064
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/5/647/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/5/647/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Chang,Fwu-Ranq, 2009. "Stochastic Optimization in Continuous Time," Cambridge Books, Cambridge University Press, number 9780521541947, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Md. Azizul Baten & Anton Abdulbasah Kamil, 2013. "Optimal Consumption in a Stochastic Ramsey Model with Cobb-Douglas Production Function," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2013, pages 1-8, March.
    2. Cieślik, Andrzej & Goczek, Łukasz, 2018. "Control of corruption, international investment, and economic growth – Evidence from panel data," World Development, Elsevier, vol. 103(C), pages 323-335.
    3. W. A. Brock & A. Xepapadeas, 2015. "Modeling Coupled Climate, Ecosystems, and Economic Systems," Working Papers 2015.66, Fondazione Eni Enrico Mattei.
    4. A. Mantovi, 2016. "Stochastic and path dependence effects in the diffusion of ideas," Economics Department Working Papers 2016-EP02, Department of Economics, Parma University (Italy).
    5. Leong, Chee Kian & Huang, Weihong, 2010. "A stochastic differential game of capitalism," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 552-561, July.
    6. Posch, Olaf & Trimborn, Timo, 2013. "Numerical solution of dynamic equilibrium models under Poisson uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 37(12), pages 2602-2622.
    7. Kogan, Konstantin, 2012. "Ship-to-order supplies: Contract breachability and the impact of a manufacturer-owned direct channel," European Journal of Operational Research, Elsevier, vol. 218(1), pages 113-123.
    8. Fwu-Ranq Chang, 2008. "Property Insurance, Portfolio Selection and their Interdependence," CESifo Working Paper Series 2260, CESifo.
    9. Posch, Olaf, 2018. "Resurrecting the New-Keynesian Model: (Un)conventional Policy and the Taylor rule," VfS Annual Conference 2018 (Freiburg, Breisgau): Digital Economy 181616, Verein für Socialpolitik / German Economic Association.
    10. Leon-Santana, Miguel & Hernandez, Juan M., 2008. "Optimum management and environmental protection in the aquaculture industry," Ecological Economics, Elsevier, vol. 64(4), pages 849-857, February.
    11. Xiaoxue Du & Levan Elbakidze & Liang Lu & R. Garth Taylor, 2022. "Climate Smart Pest Management," Sustainability, MDPI, vol. 14(16), pages 1-19, August.
    12. Marín-Solano, Jesús & Navas, Jorge, 2010. "Consumption and portfolio rules for time-inconsistent investors," European Journal of Operational Research, Elsevier, vol. 201(3), pages 860-872, March.
    13. Müller-Plantenberg, Nikolas, 2012. "Boom-and-bust cycles marked by capital inflows, current account deterioration and a rise and fall of the real exchange rate," Working Papers in Economic Theory 2012/10, Universidad Autónoma de Madrid (Spain), Department of Economic Analysis (Economic Theory and Economic History).
    14. Lu, Liang & Elbakidze, Levan, 2012. "Application of Comparative Dynamics in Stochastic Invasive Species Management in Agricultural Production," 2012 Annual Meeting, August 12-14, 2012, Seattle, Washington 125226, Agricultural and Applied Economics Association.
    15. Tsuboi, Mizuki, 2020. "Growth, R&D, and uncertainty," Economic Modelling, Elsevier, vol. 87(C), pages 394-400.
    16. Razo-De-Anda, Jorge Omar & Cruz-Aké, Salvador & Venegas-Martínez, Francisco, 2022. "¿Can the stock market boost economic growth? evidence from the Mexican real estate investment trust (REIT)," Panorama Económico, Escuela Superior de Economía, Instituto Politécnico Nacional, vol. 17(36), pages 9-32, Primer se.
    17. Samuel Wills, 2015. "Seven Principles for Managing Resource Wealth," Economics Series Working Papers OxCarre Research Paper 15, University of Oxford, Department of Economics.
    18. Wen-Kai Wang & Christian-Oliver Ewald, 2010. "Dynamic voluntary provision of public goods with uncertainty: a stochastic differential game model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 33(2), pages 97-116, November.
    19. Tsuboi, Mizuki, 2019. "Resource scarcity, technological progress, and stochastic growth," Economic Modelling, Elsevier, vol. 81(C), pages 73-88.
    20. Tamai, Toshiki, 2018. "Dynamic provision of public goods under uncertainty," Economic Modelling, Elsevier, vol. 68(C), pages 409-415.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:647-:d:1344064. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.