An adaptive numerical method for semi-infinite elliptic control problems based on error estimates

References

• N. Arada, E. Casas, and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl. 23 (2002), pp. 201–229. doi: 10.1023/A:1020576801966
   Web of Science ®Google Scholar
• R. Becker and R. Rannacher, An optimal control approach to a-posteriori error estimation in finite element methods, Acta Numer. 10 (2001), pp. 1–102.
   Google Scholar
• O. Benedix and B. Vexler, A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints, Comput. Optim. Appl. 44(1) (2009), pp. 3–25. doi: 10.1007/s10589-008-9200-y
   Web of Science ®Google Scholar
• J.C. De los Reyes, P. Merino, J. Rehberg, and F. Tröltzsch, Optimality conditions for state-constrained PDE control problems with time-dependent controls, Control Cybernet. 37 (2008), pp. 7–38.
   Google Scholar
• A.L. Dontchev, W.W. Hager, A.B. Poore, and B. Yang, Optimality, stability, and convergence in nonlinear control, Appl. Math. Optim. 31 (1995), pp. 297–326. doi: 10.1007/BF01215994
   Web of Science ®Google Scholar
• K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numer. 4 (1995), pp. 105–158. doi: 10.1017/S0962492900002531
   Google Scholar
• D. Gilbarg and N.S. Trudinger. Elliptic Partial Differential Equations of Second Order, 3rd ed., Springer, Berlin, Heidelberg, 2001.
   Google Scholar
• A. Günther, M. Hinze, and M.H. Tber, A posteriori error representation for elliptic optimal control problems with control and state constraints, in Constrained Optimization and Optimal Control for Partial Differential Equations, G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, and S. Ulbrich, eds., volume 160 of ISNM, Birkhäuser Verlag, Basel, 2012, pp. 303–318.
   Google Scholar
• R. Hettich and K.O. Kortanek, Semi-infinite programming: Theory, methods, and applications, SIAM Rev. 35(3) (1993), pp. 380–429. doi: 10.1137/1035089
   Web of Science ®Google Scholar
• M. Hintermüller, R.H.W. Hoppe, Y. Iliash, and M. Kieweg, An a-posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints, ESAIM. Control Optim. Calc. Var. 14(3) (2008), pp. 540–560. doi: 10.1051/cocv:2007057
   Web of Science ®Google Scholar
• M. Hinze and A. Rösch, Discretization of optimal control problems, in Constrained Optimization and Optimal Control for Partial Differential Equations, G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, and S. Ulbrich, eds., volume 160 of ISNM, Birkhäuser Verlag, Basel, 2012, pp. 391–430.
   Google Scholar
• K. Kohls, A. Rösch, and K.G. Siebert, A posteriori error estimators for control constrained optimal control problems, in Constrained Optimization and Optimal Control for Partial Differential Equations, G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, and S. Ulbrich, eds., volume 160 of ISNM, Birkhäuser Verlag, Basel, 2012, pp. 431–444.
   Google Scholar
• D.G. Luenberger, Optimization by Vector Space Methods, Wiley, New York, 1969.
   Google Scholar
• K. Malanowski, Ch. Büskens, and H. Maurer, Convergence of approximations to nonlinear optimal control problems, in Mathematical Programming with Data Perturbations, volume 195 of Lecture Notes in Pure and Applied Mathematics, Marcel-Dekker, Inc., New York, 1997, pp. 253–284.
   Google Scholar
• P. Merino, I. Neitzel, and F. Tröltzsch, Error estimates for the finite-element discretization of semi-infinite elliptic control problems, Discuss. Math. Differ. Incl. Control Optim. 30 (2010), pp. 221–236. doi: 10.7151/dmdico.1121
   Google Scholar
• P. Merino, I. Neitzel, and F. Tröltzsch, On linear-quadratic elliptic control problems of semi-infinite type, Appl. Anal. 90 (2011), pp. 1047–1074. doi: 10.1080/00036811.2010.489187
   Web of Science ®Google Scholar
• P. Merino, F. Tröltzsch, and B. Vexler, Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space, ESAIM Math. Model. Numer. Anal. 44 (2010), pp. 167–188. doi: 10.1051/m2an/2009045
   Web of Science ®Google Scholar
• C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital. (2) 3 (1940), pp. 5–7.
   Google Scholar
• R.H. Nochetto, A. Schmidt, K.G. Siebert, and A. Veeser, Pointwise a-posteriori error estimates for monotone semi-linear equations, Numer. Math. 104(4) (2006), pp. 515–538. doi: 10.1007/s00211-006-0027-0
   Web of Science ®Google Scholar
• R. Rannacher and B. Vexler, A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements, SIAM J. Control Optim. 44 (2005), pp. 1844–1863. doi: 10.1137/040611100
   Web of Science ®Google Scholar
• R. Reemtsen and S. Görner, Numerical methods for semi-infinite programming: A survey, in Semi-infinite Programming, R. Reemtsen and J-J. Rückmann, eds., volume 25 of Nonconvex Optimization and Applications, Kluwer Acad. Publ., Boston, MA, 1998, pp. 195–275.
   Google Scholar
• A. Rösch and D. Wachsmuth, Numerical verification of optimality conditions, SIAM J. Control Optim. 47(5) (2008), pp. 2557–2581. doi: 10.1137/060663714
   Web of Science ®Google Scholar
• U. Schäfer, A fixed point theorem based on Miranda, Fixed Point Theory Appl. 5 (2007), Art. ID 78706.
   Google Scholar
• T. Tonn, K. Urban, and S. Volkwein, Optimal control of parameter-dependent convection-diffusion problems around rigid bodies, SIAM J. Sci. Comput. 32 (2010), pp. 1237–1260. doi: 10.1137/08074194X
   Web of Science ®Google Scholar
• F. Tröltzsch and S. Volkwein, The SQP method for control constrained optimal control of the Burgers equation, ESAIM Control Optim. Calc. Var. 6 (2001), pp. 649–674. doi: 10.1051/cocv:2001127
   Web of Science ®Google Scholar