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Pontryagin’s Maximum Principle for a State-Constrained System of Douglis-Nirenberg Type

Author

Listed:
  • Alexey S. Matveev

    (Saint Petersburg State University
    “Sirius University”)

  • Dmitrii V. Sugak

    (Saint Petersburg State University of Aerospace Instrumentation)

Abstract

This article is concerned with optimal control problems for plants described by systems of high order nonlinear PDE’s (whose linear approximation is elliptic in the sense of Douglis-Nirenberg), with a special attention being given to their particular case: the standard stationary system of non-linear Navier–Stokes equations. The objective is to establish an analog of the Pontryagin’s maximum principle. The major challenge stems from the presence of infinitely many point-wise constraints on the system’s state, which are imposed at any point from a given continuum set of independent variables. Necessary conditions for optimality in the form of an “abstract” maximum principle are first obtained for a general optimal control problem couched in the language of functional analysis. This result is targeted at a wide class of problems, with an idea to absorb, in its proof, a great deal of technical work needed for derivation of optimality conditions so that only an interpretation of the discussed result would be basically needed to handle a particular problem. The applicability of this approach is demonstrated via obtaining the afore-mentioned analog of the Pontryagin’s maximum principle for a state-constrained system of high-order elliptic equations and the Navier–Stokes equations.

Suggested Citation

  • Alexey S. Matveev & Dmitrii V. Sugak, 2024. "Pontryagin’s Maximum Principle for a State-Constrained System of Douglis-Nirenberg Type," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2370-2411, December.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:3:d:10.1007_s10957-024-02499-y
    DOI: 10.1007/s10957-024-02499-y
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    References listed on IDEAS

    as
    1. Harbir Antil & Deepanshu Verma & Mahamadi Warma, 2020. "Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the Dual of Fractional-Order Sobolev Spaces," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 1-23, July.
    2. Falcone, Maurizio & Kirsten, Gerhard & Saluzzi, Luca, 2023. "Approximation of optimal control problems for the Navier-Stokes equation via multilinear HJB-POD," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    3. B. D. Craven, 2009. "Pontryagin principle with a PDE: a unified approach," Springer Optimization and Its Applications, in: Charles Pearce & Emma Hunt (ed.), Optimization, edition 1, chapter 0, pages 135-141, Springer.
    4. Francisco Fuica & Felipe Lepe & Enrique Otárola & Daniel Quero, 2023. "An Optimal Control Problem for the Navier–Stokes Equations with Point Sources," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 590-616, February.
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