A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system

SIAM Journal on Control and Optimization

Volumn 44, Issue 2, 2006, Pages 467-494

  Griesse, R ^a,b   Volkwein, S ^a,b  

^a UNIVERSITY OF GRAZ   (Austria)

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

Author keywords

Bilateral and integral control constraints; Optimal boundary control; Primal dual active set strategy; Reaction diffusion equations; Semismooth Newton methods

Indexed keywords

CONSTRAINT THEORY; DIFFUSION; OPTIMIZATION; PROBLEM SOLVING;

BILATERAL AND INTEGRAL CONTROL CONSTRAINTS; OPTIMAL BOUNDARY CONTROL; REACTION-DIFFUSION EQUATIONS;

NONLINEAR CONTROL SYSTEMS;

EID: 33644943921     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012903438696     Document Type: Article

References (37)

1. M. BERGOUNIOUX, M. HADDOU, M. HINTERMÜLLER, AND K. KUNISCH, A comparison of interior point methods and a Moreau-Yosida based active set strategy for constrained optimal control problems, SIAM J. Optim., 11 (2000), pp. 495-521.
2. M. BERGOUNIOUX, K. ITO, AND K. KUNISCH, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (1999), pp. 1176-1194.
3. D. P. BERTSEKAS, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.
4. J. F. BONNANS AND H. ZIDANI, Optimal control problems with partially polyhedric constraints, SIAM J. Control Optim., 37 (1999), pp. 1726-1741.
5. E. CASAS, J.-P. RAYMOND, AND H. ZIDANI, Pontryagin's principle for local solutions of controlproblems with mixed control-state constraints, SIAM J. Control Optim., 39 (2000), pp. 1182-1203.
6. L. C. EVANS, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, RI, 1998.
7. C. GEIGER AND C. KANZOW, Theorie und Numerik restringierter Optimierungsaufgaben, Springer-Verlag, Berlin, 2002.
8. R. GRIESSE, Parametric sensitivity analysis in optimal control of a reaction diffusion system-Part II: Practical methods and examples, Optim. Methods Softw., 19 (2004), pp. 217-242.
9. R. GRIESSE AND S. VOLKWEIN, Analysis for Optimal Boundary Control for a Three-Dimensional Reaction-Diffusion System, Technical report 286, Special Research Center F 003 Optimization and Control, Project area Continuous Optimization and Control, University of Graz & Technical University of Graz, 2003 (see http://www.unigraz.at/imawww/volkwein/publist.html).
10. R. GRIESSE AND S. VOLKWEIN, A semi-smooth Newton method for optimal boundary control of a nonlinear reaction-diffusion system, in Proceedings of the Sixteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS), Leuven, Belgium, 2004.
11. W. W. HAGER AND G. IANCULESCU, Dual approximations in optimal control, SIAM J. Control Optim., 22 (1984), pp. 423-465.
12. M. HINTERMÜLLER, On a globalized augmented Lagrangian-SQP algorithm for nonlinear optimal control problems with box constraints, in Fast Solution of Discretized Optimization Problems, K.-H. Hoffmann, R. H. W. Hoppe, and V. Schulz, eds., Internat. Ser. Numer. Math. 138, Birkhäuser, Basel, 2001, pp. 139-153.
13. M. HINTERMÜLLER, A primal-dual active set algorithm for bilaterally control constrained optimal control problems, Quart. Appl. Math., 61 (2003), pp. 131-160.
14. M. HINTERMÜLLER AND M HINZE, Globalization of SQP methods in control of the instationary Navier-Stokes equations, Math. Model. Numer. Anal., 36 (2002), pp. 725-746.
15. M. HINTERMÜLLER, K. ITO, AND K. KUNISCH, The primal-dual active set strategy as a semi-smooth Newton method, SIAM J. Optim., 13 (2003), pp. 865-888.
16. M. HINTERMÜLLER AND G. STADLER, A semi-smooth Newton method for constrained linear-quadratic control problems, ZAMM Z. Angew. Math. Mech., 83 (2003), pp. 219-237.
17. D. HÖMBERG AND J. SOKOLOWSKI, Optimal control of laser hardening, Adv. Math. Sci. Appl., 8 (1998), pp. 911-928.
18. K. ITO AND K. KUNISCH, Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces, Nonlinear Anal., 41 (2000), pp. 573-589.
19. K. ITO AND K. KUNISCH, The primal-dual active set method for nonlinear problems with bilateral constraints, SIAM J. Control Optim., 43 (2004), pp. 357-376.
20. K. KUNISCH AND J. C. DE LOS REYES, A semismooth Newton method for control-constrained boundary optimal control of the Navier-Stokes equations, Nonlinear Anal., to appear.
21. K. KUNISCH AND A. RÖSCH, Primal-dual strategy for constrained parabolic optimal control problems, SIAM J. Optim., 13 (2002), pp. 321-334.
22. F. LEIBFRITZ AND E. W. SACHS, Inexact SQP interior point methods and large scale optimal control problems, SIAM J. Control Optim., 38 (1999), pp. 272-299.
23. J. L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, 1972.
24. Z. Q. LUO, J. S. PANG, AND D. RALPH, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996.
25. 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.
26. J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, Springer Ser. Oper. Res., Springer-Verlag, New York, 1999.
27. J. P. RAYMOND AND H. ZIDANI, Hamiltonian Pontyagin's principles for control problems governed by semilinear equations, Appl. Math. Optim., 39 (1999), pp. 143-177.
28. S. M. ROBINSON, Stability theorems for systems of inequalities, Part II: Differentiable nonlinear systems, SIAM J. Numer. Anal., 13 (1976), pp. 497-513.
29. R. T. ROCKAFELLAR, Conjugate Duality and Optimization, Regional Conf. Ser. in Appl. Math., 1974.
30. A. RÖSCH AND F. TRÖLTZSCH, Sufficient second-order optimality conditions for a parabolic optimal control problem with pointwise control-state constraints, SIAM J. Control Optim., 42 (2003), pp. 138-154.
31. G. STADLER, Semi-smooth Newton methods and augmented Lagrangian methods for a simplified friction problem, SIAM J. Optim., 15 (2004), pp. 39-62.
32. H. TANABE, Functional Analytic Methods for Partial Differential Equations, Dekker, New York, 1996.
33. R. TEMAM, Navier-Stokes Equations, North-Holland, Amsterdam, 1979.
34. F. TRÖLTZSCH AND S. VOLKWEIN, The SQP method for bilaterally control constrained optimal control of the Burgers equation, ESAIM Control Optim. Calc. Var., 6 (2001), pp. 649-674.
35. M. ULBRICH, Semismooth Newton methods for operator equations in function spaces, SIAM J. Optim., 13 (2003), pp. 805-842.
36. S. VOLKWEIN, Second-order conditions for boundary control problems of the Burgers equation, Control Cybernet., 30 (2001), pp. 249-278.
37. S. VOLKWEIN, Optimal Control of Laser Surface Hardening by Utilizing a Nonlinear Primal-Dual Active Set Strategy, Technical report 277, Special Research Center F 003 Optimization and Control, Project area Continuous Optimization and Control, University of Graz & Technical University of Graz, 2003, submitted.