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

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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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