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

Set-Valued and Variational Analysis volume 25, pages 177–210 (2017)Cite this 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.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Aubin, J.-P., Frankowska, H.: Set-valued Analysis. Birkhäuser (1990)

  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)

    Article  MATH  Google Scholar 

  4. Bonnans, J.F.: Second-order analysis for control constrained optimal control problems of semilinear elliptic systems. Appl. Math. Optim. 38, 303–325 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  5. Bonnans, J.F., Zidani, H.: Optimal control problem with partially polyhedric constraints. SIAM J. Control Optim. 37, 1726–1741 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  6. Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problems. Springer (2000)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  9. Brezis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)

    MathSciNet  MATH  Google Scholar 

  10. Casas, E.: Second order analysis for bang-bang control problems of PDEs. SIAM J. Control Optim. 50, 2355–2372 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  14. Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21, 265-287 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  15. Clarke, F.H.: Optimization and nonsmooth analysis. SIAM, Philadelphia (1990)

    Book  MATH  Google Scholar 

  16. Dacorogna, B.: Direct methods in calculus of variations. Springer Science+Business Media LLC (2008)

  17. Giner, E.: Etudes des Fonctionnelles Integrables. Thesis, Université de Pau, France (1985)

    Google Scholar 

  18. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

  21. Mordukhovich, B.S.: Variational analysis and generalized differentiation I, Basis theory. Springer (2006)

  22. Mordukhovich, B.S.: Variational analysis and generalized differentiation II, Applications. Springer (2006)

  23. Penot, J.-P.: Calculus without derivatives. Springer (2013)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  25. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer (1997)

  26. Rockafellar, R.T.: Conjugate duality and optimization, regional conference series in applied mathematics. SIAM, Philadelphia, PA (1974)

    Book  Google Scholar 

  27. Robinson, S.M.: Stability theory for systems of inequalities, part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 12, 497–513 (1976)

    Article  MATH  Google Scholar 

  28. Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5, 49–62 (1979)

    MathSciNet  Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Optimization and Control Theory, Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet road, Hanoi, Vietnam

    B. T. Kien

  2. Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam

    V. H. Nhu

  3. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam

    N. H. Son

Authors
  1. B. T. Kien
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  2. V. H. Nhu
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  3. N. H. Son
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Correspondence to B. T. Kien.

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Kien, B.T., Nhu, V.H. & Son, N.H. Second-Order Optimality Conditions for a Semilinear Elliptic Optimal Control Problem with Mixed Pointwise Constraints. Set-Valued Var. Anal 25, 177–210 (2017). https://doi.org/10.1007/s11228-016-0373-8

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  • DOI: https://doi.org/10.1007/s11228-016-0373-8

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