# American Institute of Mathematical Sciences

September  2013, 2(3): 495-516. doi: 10.3934/eect.2013.2.495

## Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations

 1 Lavryentyev Institute of Hydrodynamics, Siberian Division of Russian Academy of Sciences, Lavryentyev pr. 15, Novosibirsk 630090, Russian Federation 2 Institut Élie Cartan Nancy, UMR7502 Université Lorraine, CNRS, INRIA, Laboratoire de Mathématiques, 54506 Vandoeuvre-lès-Nancy Cedex, France

Received  March 2013 Revised  May 2013 Published  July 2013

The flow around a rigid obstacle is governed by the compressible Navier-Stokes equations. The nonhomogeneous Dirichlet problem is considered in a bounded domain in two spatial dimensions with a compact obstacle in its interior. The flight of the airflow is characterized by the work shape functional, to be minimized over a family of admissible obstacles. The lift of the airfoil is a given function of temporal variable and should be maintain closed to the flight scenario. The continuity of the work functional with respect to the shape of obstacle in two spatial dimensions is shown for a wide class of admissible obstacles compact with respect to the Kuratowski-Mosco convergence.
The dependence of small perturbations of approximate solutions to the governing equations with respect to the boundary variations of obstacles is analyzed for the nonstationary state equation.
Citation: Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations and Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495
##### References:
 [1] E. Feireisl and E. Friedmann, Continuity of drag and domain stability in the low Mach number limits, J. Math. Fluid Mech., 14 (2012), 731-750. doi: 10.1007/s00021-012-0106-1. [2] G. Frémiot, W. Horn, A. Laurain, M. Rao and J. Sokołowski, On the analysis of boundary value problems in nonsmooth domains, Dissertationes Math., 462 (2009), 1-149. doi: 10.4064/dm462-0-1. [3] A. Kaźmierczak, P. I. Plotnikov, J. Sokołowski and A. .Zochowski, Numerical method for drag minimization in compressible flows, in "15th International Conference on Methods and Models in Automation and Robotics," MMAR'10. 97-101. Avalaible from: http://ieeexplore.ieee.org/stamp/stamp.jsp\$\protect\unhbox\voidb@x\hbox{?}\$tp=\$\$arnumber=5587258 [4] P.-L. Lions, "Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models," Clarendon Press, Oxford, 1998. [5] M. Moubachir and J.-P. Zolésio, "Moving Shape Analysis and Control: Applications to Fluid Structure Interactions," Chapman & Hall/CRC, Boca Raton, FL, 2006. doi: 10.1201/9781420003246. [6] P. I. Plotnikov and J. Sokolowski, "Compressible Navier-Stokes Equations. Theory and Shape Optimization," Birkhäuser, Basel, 2012. [7] P. I. Plotnikov, E. V. Ruban and J. Sokołowski, Inhomogeneous boundary value problems for compressible Navier-Stokes equations, well-posedness and sensitivity analysis, SIAM J. Math. Anal., 40 (2008), 1152-1200. doi: 10.1137/070694272. [8] P. I. Plotnikov, E. V. Ruban and J. Sokołowski, Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations, J. Math. Pures Appl., 92 (2009), 113-162. doi: 10.1016/j.matpur.2009.02.001. [9] P. I. Plotnikov and J. Sokołowski, On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations, J. Math. Fluid Mech., 7 (2005), 529-573. doi: 10.1007/s00021-004-0134-6. [10] P. I. Plotnikov and J. Sokołowski, Concentrations of solutions to time-discretized compressible Navier -Stokes equations, Comm. Math. Phys., 258 (2005), 567-608. doi: 10.1007/s00220-005-1358-x. [11] P. I. Plotnikov and J. Sokołowski, Domain dependence of solutions to compressible Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 1165-1197. doi: 10.1137/050635304. [12] P. I. Plotnikov and J. Sokołowski, Stationary boundary value problems for compressible Navier-Stokes equations, in "Handbook of Differential Equations: Stationary Partial Differential Equations," VI, Elsevier/North-Holland, Amsterdam, (2008), 313-410. doi: 10.1016/S1874-5733(08)80022-8. [13] P. I. Plotnikov and J. Sokołowski, Shape derivative of drag functional, SIAM J. Control Optim., 48 (2010), 4680-4706. doi: 10.1137/090758179. [14] P. Plotnikov, J. Sokołowski and A. .Zochowski, Numerical experiments in drag minimization for compressible Navier-Stokes flows in bounded domains, in "Proc. 14th International IEEE/IFAC Conference on Methods and Models in Automation and Robotics MMAR'09" (2009), 37-40. Avalaible from: http://www.ifac-papersonline.net/Detailed/41051.html. [15] J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer Ser. Comput. Math., 16, Springer, Berlin, 1992. [16] V. Šverák, On optimal shape design, J. Math. Pures Appl., 72 (1993), 537-551.

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##### References:
 [1] E. Feireisl and E. Friedmann, Continuity of drag and domain stability in the low Mach number limits, J. Math. Fluid Mech., 14 (2012), 731-750. doi: 10.1007/s00021-012-0106-1. [2] G. Frémiot, W. Horn, A. Laurain, M. Rao and J. Sokołowski, On the analysis of boundary value problems in nonsmooth domains, Dissertationes Math., 462 (2009), 1-149. doi: 10.4064/dm462-0-1. [3] A. Kaźmierczak, P. I. Plotnikov, J. Sokołowski and A. .Zochowski, Numerical method for drag minimization in compressible flows, in "15th International Conference on Methods and Models in Automation and Robotics," MMAR'10. 97-101. Avalaible from: http://ieeexplore.ieee.org/stamp/stamp.jsp\$\protect\unhbox\voidb@x\hbox{?}\$tp=\$\$arnumber=5587258 [4] P.-L. Lions, "Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models," Clarendon Press, Oxford, 1998. [5] M. Moubachir and J.-P. Zolésio, "Moving Shape Analysis and Control: Applications to Fluid Structure Interactions," Chapman & Hall/CRC, Boca Raton, FL, 2006. doi: 10.1201/9781420003246. [6] P. I. Plotnikov and J. Sokolowski, "Compressible Navier-Stokes Equations. Theory and Shape Optimization," Birkhäuser, Basel, 2012. [7] P. I. Plotnikov, E. V. Ruban and J. Sokołowski, Inhomogeneous boundary value problems for compressible Navier-Stokes equations, well-posedness and sensitivity analysis, SIAM J. Math. Anal., 40 (2008), 1152-1200. doi: 10.1137/070694272. [8] P. I. Plotnikov, E. V. Ruban and J. Sokołowski, Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations, J. Math. Pures Appl., 92 (2009), 113-162. doi: 10.1016/j.matpur.2009.02.001. [9] P. I. Plotnikov and J. Sokołowski, On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations, J. Math. Fluid Mech., 7 (2005), 529-573. doi: 10.1007/s00021-004-0134-6. [10] P. I. Plotnikov and J. Sokołowski, Concentrations of solutions to time-discretized compressible Navier -Stokes equations, Comm. Math. Phys., 258 (2005), 567-608. doi: 10.1007/s00220-005-1358-x. [11] P. I. Plotnikov and J. Sokołowski, Domain dependence of solutions to compressible Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 1165-1197. doi: 10.1137/050635304. [12] P. I. Plotnikov and J. Sokołowski, Stationary boundary value problems for compressible Navier-Stokes equations, in "Handbook of Differential Equations: Stationary Partial Differential Equations," VI, Elsevier/North-Holland, Amsterdam, (2008), 313-410. doi: 10.1016/S1874-5733(08)80022-8. [13] P. I. Plotnikov and J. Sokołowski, Shape derivative of drag functional, SIAM J. Control Optim., 48 (2010), 4680-4706. doi: 10.1137/090758179. [14] P. Plotnikov, J. Sokołowski and A. .Zochowski, Numerical experiments in drag minimization for compressible Navier-Stokes flows in bounded domains, in "Proc. 14th International IEEE/IFAC Conference on Methods and Models in Automation and Robotics MMAR'09" (2009), 37-40. Avalaible from: http://www.ifac-papersonline.net/Detailed/41051.html. [15] J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis," Springer Ser. Comput. Math., 16, Springer, Berlin, 1992. [16] V. Šverák, On optimal shape design, J. Math. Pures Appl., 72 (1993), 537-551.
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