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
Aliprantis, C .D., Border, K .C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (2006)
Borggaard, J., Burns, J.: A PDE sensitivity equation method for optimal aerodynamic design. J. Comput. Phys. 136(2), 366–384 (1997)
Chowdhury, S., Ramaswamy, M.: Optimal control of linearized compressible Navier–Stokes equations. ESAIM Control Optim. Calc. Var. 19(2), 587–615 (2013)
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)
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)
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)
Ekeland, I., Turnbull, T.: Infinite-Dimensional Optimization and Convexity. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1983)
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)
Fattorini, H., Sritharan, S.S.: Existence of optimal controls for viscous flow problems. Proc. R. Soc. Lond. Ser. A 439, 81–102 (1992)
Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics. OUP, Oxford (2004)
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)
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)
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)
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)
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)
Fursikov, A.V.: Optimal Control of Distributed Systems. Theory and Applications. American Mathemtical Society, Providence (2000)
Gunzburger, M.D.: Perspectives in Flow Control and Optimization. SIAM’s Advances in Design and Control series, Philadelphia (2003)
Jameson, A., Pierce, N.A., Martinelli, L.: Optimum aerodynamic design using the Navier–Stokes equations. Theor. Comput. Fluid Dyn. 10(1), 213–237 (1998)
Lions, J .L.: Optimal control of systems governed by partial differential equations. Springer, Berlin (1971)
Lions, J.L.: Control of Distributed Singular Systems. Bordas, Paris (1985)
Lions, P.L.: Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models. Mathematical Topics in Fluid Mechanics. Clarendon Press, Oxford (1998)
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)
Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and Its Applications. OUP, Oxford (2004)
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)
Sritharan, S.S.: An optimal control problem in exterior hydrodynamics. Proc. R. Soc. Edinb. 121A, 5–32 (1992)
Sritharan, S.S. (ed.): Optimal Control of Viscous Flow. SIAM Frontiers in Applied Mathematics, Philadelphia (1998)
Sritharan, S.S.: Deterministic and stochastic control of Navier–Stokes equations with linear, monotone and hyper viscosities. Appl. Math. Optim. 41, 255–308 (2000)
Vishik, M.J., Fursikov, A.V.: Mathematical Problems of Statistical Hydromechanics. Kluwer Academic Publishers, Boston (1988)
Wang, G.: Optimal controls of 3-dimensional Navier-Stokes equations with state constraints. SIAM J. Control Optim. 41(2), 583–606 (2002)
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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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DOI: https://doi.org/10.1007/s00021-017-0318-5