Advertisement

Set-Valued and Variational Analysis

, Volume 25, Issue 1, pp 177–210 | Cite as

Second-Order Optimality Conditions for a Semilinear Elliptic Optimal Control Problem with Mixed Pointwise Constraints

  • B. T. KienEmail author
  • V. H. Nhu
  • N. H. Son
Article

Abstract

This paper studies second-order optimality conditions for a semilinear elliptic optimal control problem with mixed pointwise constraints. We show that in some cases, there is a common critical cone under which the second-order necessary and sufficient optimality conditions for the problem are valid. Our results approach to a theory of no-gap second-order conditions. In order to obtain such results, we reduce the problem to a special mathematical programming problem with polyhedricity constraint set. We then use some tools of variational analysis and techniques of semilinear elliptic equations to analyze second-order conditions.

Keywords

Second-order necessary optimality condition Second-order sufficient optimality condition Optimal control Semilinear elliptic equation Mixed pointwise constraint Strongly extended polyhedricity condition 

Mathematics Subject Classification (2010)

49K20 35J25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Aubin, J.-P., Frankowska, H.: Set-valued Analysis. Birkhäuser (1990)Google Scholar
  3. 3.
    Bayen, T., Bonnans, J.F., Silva, F.J.: Characterization of locall quadractic growth for strong minima in the optimal control of semilinear elliptic equations. Trans. Amer. Math. Soc. 366, 2063–2087 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bonnans, J.F.: Second-order analysis for control constrained optimal control problems of semilinear elliptic systems. Appl. Math. Optim. 38, 303–325 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonnans, J.F., Zidani, H.: Optimal control problem with partially polyhedric constraints. SIAM J. Control Optim. 37, 1726–1741 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problems. Springer (2000)Google Scholar
  7. 7.
    Bonnans, J.F., Hermant, A.: No-gap second-order optimality conditions for optimal control problems with a single state constraint and control. Math. Program. Ser. B 117, 21–50 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bonnans, J.F., Hermant, A.: Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Ann. I. H. Poincar - AN 26, 561–598 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brezis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Casas, E.: Second order analysis for bang-bang control problems of PDEs. SIAM J. Control Optim. 50, 2355–2372 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Casas, E., Reyes, J. C.D.L., Tröltzsch, F.: Sufficient second-order optiMality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19, 616–643 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Casas, E., Tröltzsch, F.: First- and second-order optiMality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM. J. Control Optim. 48, 688–718 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Casas, E., Mateos, M.: Second order optiMality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40, 1431–1454 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21, 265-287 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Clarke, F.H.: Optimization and nonsmooth analysis. SIAM, Philadelphia (1990)CrossRefzbMATHGoogle Scholar
  16. 16.
    Dacorogna, B.: Direct methods in calculus of variations. Springer Science+Business Media LLC (2008)Google Scholar
  17. 17.
    Giner, E.: Etudes des Fonctionnelles Integrables. Thesis, Université de Pau, France (1985)Google Scholar
  18. 18.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)zbMATHGoogle Scholar
  19. 19.
    Kien, B.T., Nhu, V.H.: Second-order necessary optimality conditions for a class of semilinear elliptic optimal control problems with mixed pointwise constraints. SIAM J. Control Optim. 52, 1166–1202 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Meyer, C., Tröltzsch, F.: On an elliptic optimal control problem with pointwise mixed control-state constraints. In: Seeger, A. (ed.) Recent Advances in Optimization, volume 563 of Lecture Notes in Economics and Mathematical Systems, p 187204. Springer Berlin Heidelberg (2006)Google Scholar
  21. 21.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation I, Basis theory. Springer (2006)Google Scholar
  22. 22.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation II, Applications. Springer (2006)Google Scholar
  23. 23.
    Penot, J.-P.: Calculus without derivatives. Springer (2013)Google Scholar
  24. 24.
    Rösch, A., Tröltzsch, F.: On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Optim. 46, 1098–1115 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer (1997)Google Scholar
  26. 26.
    Rockafellar, R.T.: Conjugate duality and optimization, regional conference series in applied mathematics. SIAM, Philadelphia, PA (1974)CrossRefGoogle Scholar
  27. 27.
    Robinson, S.M.: Stability theory for systems of inequalities, part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 12, 497–513 (1976)CrossRefzbMATHGoogle Scholar
  28. 28.
    Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5, 49–62 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Optimization and Control TheoryInstitute of Mathematics, Vietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Department of Scientific FundamentalsPosts and Telecommunications Institute of TechnologyHanoiVietnam
  3. 3.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHanoiVietnam

Personalised recommendations