Taiwanese Journal of Mathematics

HÖLDER CONTINUITY OF THE SOLUTION MAP TO AN ELLIPTIC OPTIMAL CONTROL PROBLEM WITH MIXED CONSTRAINTS

V. Nhu, N. Auh, and B. T. Kien

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Abstract

The goal of the paper is to investigate the Hölder continuity of the solution map to a parametric optimal control problem which is governed by elliptic equations with mixed control-state constraints and convex cost functions. By reducing the problem to a programming problem and parametric variational inequality, we get sufficient conditions under which the solution map is Hölder continuous in parameters.

Article information

Source
Taiwanese J. Math., Volume 17, Number 4 (2013), 1245-1266.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706115

Digital Object Identifier
doi:10.11650/tjm.17.2013.2617

Mathematical Reviews number (MathSciNet)
MR3085509

Zentralblatt MATH identifier
1275.49067

Keywords
parametric optimal control Hölder continuity variational inequality elliptic equation

Citation

Nhu, V.; Auh, N.; Kien, B. T. HÖLDER CONTINUITY OF THE SOLUTION MAP TO AN ELLIPTIC OPTIMAL CONTROL PROBLEM WITH MIXED CONSTRAINTS. Taiwanese J. Math. 17 (2013), no. 4, 1245--1266. doi:10.11650/tjm.17.2013.2617. https://projecteuclid.org/euclid.twjm/1499706115


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References

  • R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  • W. Alt, R. Griesse, N. Metla and A. Rösch, Lipschitz stability for elliptic optimal control problems with mixed control-state constraints, Optimization, 59(6) 2010, 833-849.
  • J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis, Springer, Berlin, Heidelberg and New York, 2005.
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010.
  • E. Casas, J. C. D. L. Reyes and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643.
  • E. Casas and F. Tröltzsch, Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems, ESAIM: Control, Optim. Caculus of Variations, 16 (2010), 581-600.
  • M. Chipot, Elliptic Equations: An Introduction Course, Birkhäuser Verlag AG, Basel-Boston-Berlin, 2009.
  • A. Domokos, Solution Sensitivity of Variational Inequalities, J. Math. Anal. Appl., 230 (1999), 382-389.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer-Verlag Berlin Heidelberg, 2001.
  • R. Griesse, Lipschitz stability of solutions to some state-constrained elliptic optimal control problems, J. Anal. Appl., 25 (2006), 435-455.
  • P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
  • G. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge, At The University Press, 1934.
  • A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holand Publishing Company, 1979.
  • B. T. Kien, Lower semicontinuity of the solution set to a parametric generalized variational inequality in reflexive Banach spaces, Set-Valued Analysis, 16 (2008), 1089-1105.
  • K. Malanowski and F. Tröltzsch, Lipschitz stability of solutions to parametric optimal control for elliptic equations, Control Cybern., 29 (2000), 237-256.
  • C. Meyer, U. Prüfert and F. Tröltzsch, On two numerical method for state-constrained elliptic control problems, Opt. Meth. Software, 22 (2007), 871-889.
  • T. R. Rockafellar and R. J.-B. West, Variational Analysis, Springer-Verlag, 1997.
  • N. D. Yen, Hölder continuity of solutions to a parametric variational inequality, Appl, Math. Optim., 31 (1995), 245-255.
  • E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators, Springer-Verlag, Berlin, 1990.
  • E. Zeilder, Nonlinear Functional Analysis and Its Applications III: Variational Methods and Optimization, Springer-Verlag, New York 1985.