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Maximum Principle for Some Optimal Control Problems Governed by 2D Nonlocal Cahn–Hillard–Navier–Stokes Equations

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Abstract

This work concerns some optimal control problems associated with the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The Cahn–Hilliard–Navier–Stokes model consists of a Navier–Stokes equation governing the fluid velocity field coupled with a convective Cahn–Hilliard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study an another optimal control problem of finding the unknown optimal initial data. Optimal data initialization problem is also known as the data assimilation problems in meteorology, where determining the correct initial condition is very crucial for the future predictions.

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Acknowledgements

Tania Biswas would like to thank the Indian Institute of Science Education and Research, Thiruvananthapuram, for providing financial support and stimulating environment for the research. The authors sincerely would like to thank the reviewer for his/her valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics, Indian Institute of Science Education and Research, Trivandrum (IISER-TVM), Maruthamala PO, Vithura, Thiruvananthapuram, Kerala,  695 551, India

    Tania Biswas & Sheetal Dharmatti

  2. Department of Mathematics, Indian Institute of Technology Roorkee - IIT Roorkee, Haridwar Highway, Uttarakhand, Roorkee, 247 667, India

    Manil T. Mohan

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  1. Tania Biswas
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  2. Sheetal Dharmatti
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  3. Manil T. Mohan
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Correspondence to Sheetal Dharmatti.

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Biswas, T., Dharmatti, S. & Mohan, M.T. Maximum Principle for Some Optimal Control Problems Governed by 2D Nonlocal Cahn–Hillard–Navier–Stokes Equations. J. Math. Fluid Mech. 22, 34 (2020). https://doi.org/10.1007/s00021-020-00493-8

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  • DOI: https://doi.org/10.1007/s00021-020-00493-8

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