FINITE ELEMENT ANALYSIS OF OPTIMAL CONTROL PROBLEM GOVERNED BY STOKES EQUATIONS WITH L 2 -NORM STATE-CONSTRAINTS *

An optimal control problem governed by the Stokes equations with L 2 -norm state constraints is studied. Finite element approximation is constructed. The optimality conditions of both the exact and discretized problems are discussed, and the a priori error estimates of the optimal order accuracy in L 2 -norm and H 1 -norm are given. Some numerical experiments are presented to verify the theoretical results. Mathematics subject classification: 49J20, 65N30

[1]  R. Temam.  Navier-Stokes Equations , 1977 .

[2]  H. K. Moffatt.  Magnetic Field Generation in Electrically Conducting Fluids , 1978 .

[3]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[4]  L. Hou,et al.  FINITE-DIMENSIONAL APPROXIMATION OFA CLASS OFCONSTRAINED NONLINEAR OPTIMAL CONTROL PROBLEMS , 1996 .

[5]  Karl Kunisch,et al.  Primal-Dual Strategy for State-Constrained Optimal Control Problems , 2002, Comput. Optim. Appl..

[6]  M. Hinze.  Optimal and instantaneous control of the instationary Navier-Stokes equations , 2002 .

[7]  L. Hou,et al.  Boundary velocity control of incompressible flow with an application to viscous drag reduction , 1992 .

[8]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[9]  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 .

[10]  R. Temam,et al.  On some control problems in fluid mechanics , 1990 .

[11]  Karl Kunisch,et al.  Augemented Lagrangian Techniques for Elliptic State Constrained Optimal Control Problems , 1997 .

[12]  Danping Yang,et al.  A PRIORI ERROR ESTIMATE AND SUPERCONVERGENCE ANALYSIS FOR AN OPTIMAL CONTROL PROBLEM OF BILINEAR TYPE , 2008 .

[13]  S. Shvartsman,et al.  Nonlinear model reduction for control of distributed systems: A computer-assisted study , 1998 .

[14]  Lei,et al.  A POSTERIORI ERROR ESTIMATE OF OPTIMAL CONTROL PROBLEM OF PDE WITH INTEGRAL CONSTRAINT FOR STATE , 2009 .

[15]  Wenbin Liu,et al.  A Posteriori Error Estimates for Control Problems Governed by Stokes Equations , 2002, SIAM J. Numer. Anal..

[16]  Wenbin Liu.  Adaptive Multi-Meshes in Finite Element Approximation of Optimal Control , 2005 .

[17]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[18]  Wenbin Liu,et al.  Adaptive Finite Element Methods for Optimal Control Governed by PDEs: C Series in Information and Computational Science 41 , 2008 .

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

[20]  Wei,et al.  A NEW FINITE ELEMENT APPROXIMATION OF A STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM , 2009 .

[21]  Eduardo Casas.  Error Estimates for the Numerical Approximation of Semilinear Elliptic Control Problems with Finitely Many State Constraints , 2002 .

[22]  P. Hood,et al.  A numerical solution of the Navier-Stokes equations using the finite element technique , 1973 .

[23]  Michael Hinze,et al.  Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem , 2007, SIAM J. Numer. Anal..

[24]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[25]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[26]  L. Steven Hou,et al.  Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls , 1995 .

[27]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.