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Existence of Optimal Controls for Compressible Viscous Flow

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Abstract

We formulate a control problem for a distributed parameter system where the state is governed by the compressible Navier–Stokes equations. Introducing a suitable cost functional, the existence of an optimal control is established within the framework of strong solutions in three dimensions.

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References

  1. Aliprantis, C .D., Border, K .C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Borggaard, J., Burns, J.: A PDE sensitivity equation method for optimal aerodynamic design. J. Comput. Phys. 136(2), 366–384 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Chowdhury, S., Ramaswamy, M.: Optimal control of linearized compressible Navier–Stokes equations. ESAIM Control Optim. Calc. Var. 19(2), 587–615 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chowdhury, S., Ramaswamy, M., Raymond, J.-P.: Controllability and stabilizability of the linearized compressible Navier–Stokes system in one-dimension. SIAM J. Control Optim. 50(5), 2959–2987 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Collis, S., Ghayour, K., Heinkenschloss, M., Ulbrich, M., Ulbrich, S.: Numerical solution of optimal control problems governed by the compressible Navier-Stokes equations. In: Optimal Control of Complex Structures: International Conference in Oberwolfach. June 4–10, 2000, pp. 43–55. Birkhäuser Basel, Basel (2002)

  6. Collis, S., Ghayour, K., Heinkenschloss, M., Ulbrich, M., Ulbrich, S.: Optimal control of unsteady compressible viscous flows. Int. J Numer. Methods Fluids 40(11), 1401–1429 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ekeland, I., Turnbull, T.: Infinite-Dimensional Optimization and Convexity. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1983)

    MATH  Google Scholar 

  8. Ervedoza, S., Glass, O., Guerrero, S., Puel, J.-P.: Local exact controllability for the one-dimensional compressible Navier–Stokes equation. Arch. Ration. Mech. Anal. 206, 189–238 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fattorini, H., Sritharan, S.S.: Existence of optimal controls for viscous flow problems. Proc. R. Soc. Lond. Ser. A 439, 81–102 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics. OUP, Oxford (2004)

    Google Scholar 

  11. Feireisl, E., Jin, B.J., Novotný, A.: Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech. 14(4), 717–730 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Feireisl, E., Novotný, A., Petzeltová, H.: On the domain dependence of solutions to the compressible Navier–Stokes equations of a barotropic fluid. Math. Methods Appl. Sci. 25(12), 1045–1073 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Feireisl, E., Novotný, A., Sun, Y.: Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60(2), 611–632 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fursikov, A.V.: Control problems and theorems concerning the unique solvability of a mixed boundary value problem for the three-dimensional Navier–Stokes and Euler equations. Math. USSR Sb. 43(2), 251 (1982)

    Article  MATH  Google Scholar 

  16. Fursikov, A.V.: Optimal Control of Distributed Systems. Theory and Applications. American Mathemtical Society, Providence (2000)

    MATH  Google Scholar 

  17. Gunzburger, M.D.: Perspectives in Flow Control and Optimization. SIAM’s Advances in Design and Control series, Philadelphia (2003)

    MATH  Google Scholar 

  18. Jameson, A., Pierce, N.A., Martinelli, L.: Optimum aerodynamic design using the Navier–Stokes equations. Theor. Comput. Fluid Dyn. 10(1), 213–237 (1998)

    Article  MATH  Google Scholar 

  19. Lions, J .L.: Optimal control of systems governed by partial differential equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  20. Lions, J.L.: Control of Distributed Singular Systems. Bordas, Paris (1985)

    MATH  Google Scholar 

  21. Lions, P.L.: Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models. Mathematical Topics in Fluid Mechanics. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  22. Matsumura, A., Nishida, T.: Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and Its Applications. OUP, Oxford (2004)

    MATH  Google Scholar 

  24. Solonnikov, V.A.: Solvability of the initial-boundary-value problem for the equations of motion of a viscous compressible fluid. Zap. Nauchn. Semin. LOMI 56, 128–142 (1976)

    MathSciNet  MATH  Google Scholar 

  25. Sritharan, S.S.: An optimal control problem in exterior hydrodynamics. Proc. R. Soc. Edinb. 121A, 5–32 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sritharan, S.S. (ed.): Optimal Control of Viscous Flow. SIAM Frontiers in Applied Mathematics, Philadelphia (1998)

    MATH  Google Scholar 

  27. Sritharan, S.S.: Deterministic and stochastic control of Navier–Stokes equations with linear, monotone and hyper viscosities. Appl. Math. Optim. 41, 255–308 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vishik, M.J., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer Academic Publishers, Boston (1988)

    Book  MATH  Google Scholar 

  29. Wang, G.: Optimal controls of 3-dimensional Navier-Stokes equations with state constraints. SIAM J. Control Optim. 41(2), 583–606 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Sivaguru S. Sritharan.

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Communicated by A.V. Fursikov.

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Doboszczak, S., Mohan, M.T. & Sritharan, S.S. Existence of Optimal Controls for Compressible Viscous Flow. J. Math. Fluid Mech. 20, 199–211 (2018). https://doi.org/10.1007/s00021-017-0318-5

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