Optimal Control of the Stationary Navier--Stokes Equations with Mixed Control-State Constraints

In this paper we consider the distributed optimal control of the Navier-Stokes equations in the presence of pointwise mixed control-state constraints. After deriving a first order necessary condition, the regularity of the mixed constraint multiplier is investigated. Second order sufficient optimality conditions are studied as well. In the last part of the paper, a semismooth Newton method is applied for the numerical solution of the control problem. The convergence of the method is proved and numerical experiments are carried out.

[1]  Michael Ulbrich,et al.  Constrained optimal control of Navier-Stokes flow by semismooth Newton methods , 2003, Syst. Control. Lett..

[2]  H. Maurer.  First and second order sufficient optimality conditions in mathematical programming and optimal control , 1981 .

[3]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[4]  E. Casas.  Boundary control of semilinear elliptic equations with pointwise state constraints , 1993 .

[5]  Karl Kunisch,et al.  Second Order Methods for Optimal Control of Time-Dependent Fluid Flow , 2001, SIAM J. Control. Optim..

[6]  Fredi Tröltzsch,et al.  Second-order sufficient optimality conditions for the optimal control of navier-stokes equations , 2006 .

[7]  Gengsheng Wang,et al.  Optimal Controls of 3-Dimensional Navier--Stokes Equations with State Constraints , 2002, SIAM J. Control. Optim..

[8]  Kazufumi Ito,et al.  The primal-dual active set strategy as a semi-smooth Newton method for quadratic problems with affine constraints , 2002 .

[9]  M. Heinkenschloss.  Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier-Stokes Flow , 1998 .

[10]  E. Casas.  Control of an elliptic problem with pointwise state constraints , 1986 .

[11]  Sri Sritharan,et al.  Optimal Control Problems with State Constraints in Fluid Mechanics and Combustion , 1998 .

[12]  Michael Hintermüller,et al.  A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints , 2006, SIAM J. Optim..

[13]  Karl Kunisch,et al.  On the structure of Lagrange multipliers for state-constrained optimal control problems , 2003, Syst. Control. Lett..

[14]  Fredi Tröltzsch,et al.  Optimal Control of PDEs with Regularized Pointwise State Constraints , 2006, Comput. Optim. Appl..

[15]  L. Hou,et al.  Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls , 1991 .

[16]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[17]  Max D. Gunzburger,et al.  Analysis and Approximation of the Velocity Tracking Problem for Navier-Stokes Flows with Distributed Control , 2000, SIAM J. Numer. Anal..