Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition

Computational Optimization and Applications

Volumn 39, Issue 3, 2008, Pages 319-345

  Hinze, M ^a   Volkwein, S ^b  

^a UNIVERSITY OF HAMBURG   (Germany)

^b UNIVERSITY OF GRAZ   (Austria)

Author keywords

Error estimates; Evolution problems; Optimal control; Proper orthogonal decomposition

Indexed keywords

ABSTRACTING; ESTIMATION; INTERSECTIONS; OPTIMAL CONTROL SYSTEMS; QUADRATIC PROGRAMMING;

ADJOINT SYSTEMS; CONTROL CONSTRAINTS; DISCRETIZATIONS; ERROR ESTIMATES; EVOLUTION PROBLEMS; LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS; NUMERICAL EXAMPLES; OPTIMAL CONTROL; PROPER ORTHOGONAL DECOMPOSITION;

ERROR ANALYSIS;

EID: 49149100096     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-007-9058-4     Document Type: Conference Paper

References (31)

1. Afanasiev, K., Hinze, M.: Adaptive control of a wake flow using proper orthogonal decomposition. In: Lecture Notes in Pure and Applied Mathematics, vol. 216. pp. 317-332. Decker, New York (2001)
2. Arian, E., Fahl, M., Sachs, E.W.: Trust-region proper orthogonal decomposition for flow control. Technical report 2000-25, ICASE (2000)
3. J.A. Atwell J.T. Borggaard B.B. King 2001 Reduced order controllers for Burgers' equation with a nonlinear observer Int. J. Appl. Math. Comput. Sci. 11 1311 1330
4. N. Aubry P. Holmes J.L. Lumley E. Stone 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer J. Fluid Mech. 192 115 173
5. H.T. Banks M.L. Joyner B. Winchesky W.P. Winfree 2000 Nondestructive evaluation using a reduced-order computational methodology Inverse Problems 16 1 17
6. Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Evolution Problems I. Springer, Berlin (1992)
7. Deckelnick, K., Hinze, M.: Error estimates in space and time for tracking-type control of the instationary Stokes system. In: International Series of Numerical Mathematics, vol. 143, pp. 87-103. Birkhäuser, Basel (2002)
8. K. Deckelnick M. Hinze 2004 Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations Numer. Math. 97 297 320
9. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)
10. Fukunaga, K.: Introduction to Statistical Pattern Recognition. Academic, New York (1990)
11. R. Griesse S. Volkwein 2005 A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system SIAM J. Control Optim. 44 467 494
12. Henri, T., Yvon, M.: Convergence estimates of POD Galerkin methods for parabolic problems. Technical report No. 02-48, Institute of Mathematical Research of Rennes (2002)
13. M. Hinze 2005 A variational discretization concept in control constrained optimization: the linear-quadratic case Comput. Optim. Appl. 30 45 61
14. M. Hinze K. Kunisch 2000 Three control methods for time-dependent fluid flow Flow Turbul. Combust. 65 273 298
15. Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Benner, P., Mehrmann, V., Sorensen, D.C. (eds.) Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45, pp. 261-306. Springer, Berlin (2005)
16. Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1996)
17. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)
18. K. Kunisch S. Volkwein 2001 Galerkin proper orthogonal decomposition methods for parabolic problems Numer. Math. 90 117 148
19. K. Kunisch S. Volkwein 2002 Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics SIAM J. Numer. Anal. 40 492 515
20. S. Lall J.E. Marsden S. Glavaski 2002 A subspace approach to balanced truncation for model reduction of nonlinear control systems Int. J. Robust Nonlinear Control 12 519 535
21. Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
22. H.V. Ly H.T. Tran 2001 Modeling and control of physical processes using proper orthogonal decomposition Math. Comput. Model. 33 223 236
23. B. Noack K. Afanasiev M. Morzynski G. Tadmor F. Thiele 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake J. Fluid. Mech. 497 335 363
24. M. Rathinam L. Petzold 2002 Dynamic iteration using reduced order models: a method for simulation of large scale modular systems SIAM J. Numer. Anal. 40 1446 1474
25. S.S. Ravindran 2002 Adaptive reduced order controllers for a thermal flow system using proper orthogonal decomposition SIAM J. Sci. Comput. 28 1924 1942
26. D. Rempfer H.F. Fasel 1994 Dynamics of three-dimensional coherent structures in a flat-plate boundary layer J. Fluid Mech. 275 257 283
27. C.W. Rowley 2005 Model reduction for fluids using balanced proper orthogonal decomposition Int. J. Bifurc. Chaos 15 997 1013
28. S.Y. Shvartsman Y. Kevrikidis 1998 Nonlinear model reduction for control of distributed parameter systems: a computer-assisted study AIChE J. 44 1579 1595
29. L. Sirovich 1987 Turbulence and the dynamics of coherent structures parts I-III Q. Appl. Math. XLV 561 590
30. K.Y. Tang W.R. Graham J. Peraire 1999 Optimal control of vortex shedding using low-order models I: open loop model development. II. Model based control Int. J. Numer. Methods Eng. 44 945 990
31. K. Willcox J. Peraire 2002 Balanced model reduction via the proper orthogonal decomposition AIAA J. 40 2323 2330