Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier-Stokes Flow

  • Matthias Heinkenschloss
Part of the Applied Optimization book series (APOP, volume 15)


The optimal boundary control of Navier-Stokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constraints during the iterations, care must be taken to avoid a possible incompatibility of Dirichlet boundary conditions and incompressibility constraint. In this paper, compatibility is enforced by choosing appropriate function spaces. The resulting optimization problem is analyzed. Differentiability of the constraints and surjectivity of linearized constraints are verified and adjoints are computed. An SQP method is applied to the optimization problem and compared with other approaches.


Optimal flow control Navier-Stokes equations sequential quadratic programming. 


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  1. [1]
    Abergel, F. and Casas, E. (1993), “Some control problems of multistage equations appearing in fluid mechanics,” Mathematical Modeling and Numerical Analysis, Vol. 27, 223–247.MathSciNetMATHGoogle Scholar
  2. [2]
    Abergel, F. and Temam, R. (1990), “On some control problems in fluid mechanics,” Theoretical and Computational Fluid Dynamics, Vol. 1, 303–325.MATHCrossRefGoogle Scholar
  3. [3]
    Adams, R.A. (1975), Sobolev Spaces, Academic Press, New York.MATHGoogle Scholar
  4. [4]
    Alt, W. and Malanowski, K. (1993), “The Lagrange-Newton method for nonlinear optimal control problems,” Computational Optimization and Applications, Vol. 2, 77–100.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Berggren, M. (1995), Optimal Control of Time Evolution Systems: Controllability Investigations and Numerical Algorithms, Ph.D. dissertation, Department of Computational and Applied Mathematics, Rice University, Houston, Texas.Google Scholar
  6. [6]
    Biegler, L.T., Nocedal, J. and Schmid, C. (1995), “A reduced Hessian method for large-scale constrained optimization,” SIAM J. on Optimization, Vol. 5, 314–347.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Boggs, P.T. (1995), “Sequential quadratic programming,” in Acta Numerica 1995, A. Iserles, ed., Cambridge University Press, Cambridge, 1–51.Google Scholar
  8. [8]
    Byrd, R.H. and Nocedal, J. (1991), “An analysis of reduced Hessian methods for constrained optimization,” Math. Programming, Vol. 49, 285–323.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Dennis, J.E., Heinkenschloss, M. and Vicente, L.N. (1994), Trust-Region Interior-Point Algorithms for a Class of Nonlinear Programming Problems, Technical Report TR94–45, Department of Computational and Applied Mathematics, Rice University, Houston, Texas.Google Scholar
  10. [10]
    Desai, M. and Ito, K. (1994), “Optimal control of Navier-Stokes equations,” SIAM J. Control and Optimization, Vol. 32, 1332–1363.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Ghattas, O. and Bark, J.H. (1997), “Optimal control of two-and three-dimensional Navier-Stokes flow,” Journal of Computational Physics, Vol. 136, 231–244.MATHCrossRefGoogle Scholar
  12. [12]
    Girault, V. and Raviart, P.A. (1986), Finite Element Methods for the Navier-Stokes Equations, Springer-Verlag, New York.MATHCrossRefGoogle Scholar
  13. [13]
    Gunzburger, M.D. and Hou, L.S. (1992), “Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses,” SIAM J. Numer. Anal., Vol. 29, 390–424.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Gunzburger, M.D., Hou, L.S. and Svobotny, T.P. (1991), “Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls,” Mathematical Modelling and Numerical Analysis, Vol. 25, 711–748.MATHGoogle Scholar
  15. [15]
    Gunzburger, M.D., Hou, L.S. and Svobotny, T.P. (1992), “Boundary velocity control of incompressible flow with an application to drag reduction,” SIAM J. Numer. Anal., Vol. 30, 167–181.MATHGoogle Scholar
  16. [16]
    Gunzburger, M.D., Hou, L.S. and Svobotny, T.P. (1993), “Optimal control and optimization of viscous, incompressible flows,” in Incompressible Computational Fluid Dynamics, M. D. Gunzburger and R. A. Nicolaides, eds., Cambridge University Press, Cambridge, 109–150.CrossRefGoogle Scholar
  17. [17]
    Heinkenschloss, M. “Numerical solution of optimal control problems governed by the Navier-Stokes equations using sequential quadratic programming,” in preparation.Google Scholar
  18. [18]
    Heinkenschloss, M. (1996), “Projected sequential quadratic programming methods,” SIAM J. Optimization, Vol. 6, 373–417.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Ito, K. (1994), “Boundary temperature control for thermally coupled Navier-Stokes equations,” in Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena, W. Desch, F. Kappel and K. Kunisch, eds., Int. Series of Numer. Math., Vol. 118, BirkhäuserVerlag, Basel, 211–230.CrossRefGoogle Scholar
  20. [20]
    Ito, K. and Kunisch, K. (1996), “Augmented Lagrangian-SQP methods for nonlinear optimal control problems of tracking type,” SIAM J. Control and Optimization, Vol. 34, 874–891.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Ito, K. and Ravindran, S.S. (1996), A Reduced Order Method for Simulation and Control of Fluid Flow, Technical Report CSRC-TR96–27, Center for Research in Scientific Computation, North Carolina State University, Raleigh.Google Scholar
  22. [22]
    Kupfer, F.S. (1996), “An infinite dimensional convergence theory for reduced SQP methods in Hilbert space,” SIAM J. Optimization, Vol. 6, 126–163.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Kupfer, F.S. and Sachs, E.W. (1991), “A prospective look at SQP methods for semilinear parabolic control problems,” in Optimal Control of Partial Differential Equations, Irsee 1990, K. H. Hoffmann and W. Krabs, eds., Lect. Notes in Control and Information Sciences, Vol. 149, Springer-Verlag, Berlin, 143–157.CrossRefGoogle Scholar
  24. [24]
    Maurer, H. and Zowe, J. (1979), “First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems,” Math. Programming, Vol. 16, 98–110.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Nocedal, J. (1980), “Updating quasi-Newton matrices with limited storage,” Math. Comp., Vol. 35, 773–782.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Temam, R. (1979), Navier Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam.MATHGoogle Scholar
  27. [27]
    Tröltzsch, F. (1994), “An SQP method for optimal control of a nonlinear heat equation,” Control and Cybernetics, Vol. 23, 268–288.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Matthias Heinkenschloss
    • 1
  1. 1.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

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