Constrained Dirichlet boundary control in L2 for a class of evolution equations

SIAM Journal on Control and Optimization

Volumn 46, Issue 5, 2007, Pages 1726-1753

  Kunisch, K ^a   Vexler, B ^b  

^a UNIVERSITY OF GRAZ   (Austria)

^b JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

Author keywords

Dirichlet boundary control; Inequality constraints; Parabolic equations; Very weak solution

Indexed keywords

BOUNDARY CONTROLS; DIRICHLET BOUNDARY CONTROL; DIRICHLET BOUNDARY CONTROLS; INEQUALITY CONSTRAINTS; PARABOLIC EQUATIONS; STATE EQUATIONS; VERY WEAK SOLUTION;

EQUATIONS OF STATE;

EID: 52649085253     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/060670110     Document Type: Article

References (43)

1. R. A. ADAMS, Sobolev Spaces, Academic Press, Amsterdam, 2005.
2. N, ARADA AND J.-P. RAYMOND, Dirichlet boundary control of semilinear parabolic equations, Part 1: Problems with no state constraints, Appl. Math. Optim., 45 (2002), pp. 125-143.
3. R. BECKER, D. MEIDNER, AND B. VEXLER, Efficient numerical solution of parabolic optimization problems by finite element methods, Optim. Methods Softw., to appear.
4. R. BECKER AND B. VEXLER, Optimal control of the convection-diffusion equation using stabilized finite element methods, Numer. Math., 106 (2007), pp. 349-367.
5. R. BECKER, Mesh adaptation for Dirichlet flow control via Nitsche's method, Comm. Numer. Methods Engrg., 18 (2002), pp. 669-680.
6. F. B. BELGACEM, H. E. FEKIH, AND H. METOUI, Singular perturbation for the Dirichlet boundary control of elliptic problems, M2AN Math. Model. Numer. Anal., 37 (2003), pp. 883-850.
7. M. BERGGREN, Approximations of very weak solutions to boundary-value problems, SIAM J. Numer. Anal., 42 (2004), pp. 860-877.
8. M. BERGOUNIOUX, K. ITO, AND K. KUNISCH, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (1999), pp. 1176-1194.
9. M. BERGOUNIOUX, M. HADDOU, M. HINTERMULLER, AND K. KUNISCH, A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems, SIAM J. Optim., 11 (2000), pp. 495-521.
10. E. BURMAN AND P. HANSBO, Edge stabilization for Galerkin approximations of convectiondiffusion-reaction problems, Comput. Methods Appl. Mech. Engrg., 193 (2004), pp. 1437-1453.
11. E. CASAS AND J.-P. RAYMOND, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), pp. 1586-1611.
12. P. G. CIARLET, The Finite Element Method for Elliptic Problems, Classics Appl. Math. 40, SIAM, Philadelphia, 2002.
13. S. S. COLLIS AND M. HEINKENSCHLOSS, Analysis of the Streamline Upwind/Petrov Galerkin Method Applied to the Solution of Optimal Control Problems, CAAM TR02-01, Rice University, Houston, TX, 2002.
14. R. DAUTRAY AND J.-L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology: Evolution Problems I, Vol. 5, Springer-Verlag, Berlin, 1992.
15. J. DE LOS REYES AND K. KUNISCH, A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations, Nonlinear Anal., 62 (2005), pp. 1289-1316.
16. K. ERIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Computational Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
17. K. ERIKSSON, C. JOHNSON, AND V. THOMÉE, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér., 19 (1985), pp. 611-643.
18. K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal., 28 (1991), pp. 43-77.
19. D. FRENCH AND J. KING, Analysis of a robust finite element approximation for a parabolic equation with rough boundary data, Math. Comp., 60 (1993), pp. 79-104.
20. A. FURSIKOV, Optimal Control of Distributed Systems: Theory and Applications, Transl. Math. Monogr. 187, AMS, Providence, RI, 1999.
21. P. GRISVARD, Commutativié de deux foncteurs d'interpolation et applications, J. Math. Pures Appl., 45 (1966), pp. 143-206.
22. J.-L. GUERMOND, Stabilization of Galerkin approximations of transport equations by subgrid modeling, Modél. Math. Anal. Numér., 36 (1999), pp. 1293-1316.
23. M. GUNZBURGER, L. HOU, AND T. SVOBODNY, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls, RAIRO Modél. Math. Anal. Numér., 25 (1991), pp. 711-748.
24. M. D. GUNZBURGER AND S. MANSERVISI, The velocity tracking problem for Navier-Stokes flows with boundary control, SIAM J. Control Optim., 39 (2000), pp. 594-634.
25. M. HINTERMÜLLER, K. ITO, AND K. KUNISCH, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13 (2003), pp. 865-888.
26. M. HINTERMÜLLER AND K. KUNISCH, Feasible and noninterior path-following in constrained minimization with low multiplier regularity, SIAM J. Control Optim., 45 (2006), pp. 1198-1221.
27. K. ITO AND K. KUNISCH, The primal-dual active set method for nonlinear optimal control problems with bilateral constraints, SIAM J. Control Optim., 43 (2004), pp. 357-376.
28. C. JOHNSON, Numerical Solution of Partial Differential Equations by Finite Element Method, Cambridge University Press, Cambridge, UK, 1987.
29. F. KAPPEL AND K. KUNISCH, Invariance results for delay and Volterra equations in fractional order Sobolev spaces, Trans. Amer. Math. Soc, 304 (1987), pp. 1-51.
30. A. KUNOTH, Adaptive wavelet schemes for an elliptic control problem with Dirichlet boundary control, Numer. Algorithms, 39 (2005), pp. 199-200.
31. O. A. LADYẐENSKAJA, V. A. SOLONNIKOV, AND N. N. URALĆEVA, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968.
32. I. LASIECKA, Galerkin approximation of abstract parabolic boundary value problems with rough boundary data-ιP theory, Math. Comp., 175 (1986), pp. 55-75.
33. H.-C. LEE, Analysis and computational methods of Dirichlet boundary optimal control problems for 2d Boussinesq equations, Adv. Comput. Math., 19 (2003), pp. 255-275.
34. J. L. LIONS AND E. MAGENES, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin, 1972.
35. J. L. LIONS AND E. MAGENES, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, Berlin, 1972.
36. J.-L. LIONS, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren Math. Wiss. 170, Springer-Verlag, Berlin, 1971.
37. H. MAURER AND J. ZOWE, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Program., 16 (1979), pp. 98-110.
38. D. MEIDNER AND B. VEXLER, Adaptive space-time finite element methods for parabolic optimization problems, SIAM J. Control Optim., 46 (2007), pp. 116-142.
39. A. QUARTERONI AND A. VALLI, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer-Verlag, Berlin, 1994.
40. H.-G. ROOS, M. STYNES, AND L. TOBISKA, Numerical Methods for Singularly Perturbed Differential Equations, Springer Ser. Comput. Math. 24, Springer-Verlag, Berlin, 1996.
41. M. SCHMICH AND B. VEXLER, Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations, SIAM J. Sci. Comput., to appear.
42. H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, J. A. Barth Verlag, Heidelberg, 1995.
43. B. VEXLER, Finite element approximation of elliptic Dirichlet optimal control problems, Numer. Funct. Anal. Optim., to appear.