Optimality conditions for reflecting boundary control problems



We consider a control problem with reflecting boundary and obtain necessary optimality conditions in the form of the maximum Pontryagin principle. To derive these results we transform the constrained problem in an unconstrained one or we use penalization techniques of Morreau-Yosida type to approach the original problem by a sequence of optimal control problems with Lipschitz dynamics. Then nonsmooth analysis theory is used to study the convergence of the penalization in order to obtain optimality conditions.

Mathematics Subject Classification

34K35 49J24 49L20 49L25 93B18 93C15 


Boundary reflection Pontryagin’s principle Necessary conditions Dynamic programming Synthesis for optimal solutions 


  1. 1.
    Aubin J.P., Cellina A.: Differential Inclusions, Set-valued Maps and Viability Theory. Springer, Berlin (1984)MATHCrossRefGoogle Scholar
  2. 2.
    Aubin J.P., Frankowska H.: Set-valued Analysis. Birkhäuser, Boston (1990)MATHGoogle Scholar
  3. 3.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. In: Systems and Control: Foundations and Applications. Birkhäuser, BostonGoogle Scholar
  4. 4.
    Barles G.: Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. J. Differ. Equ. 154(1), 191–224 (1999)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Barron E.N., Jensen R.: Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15(12), 1713–1742 (1990)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Benoist J.: On ergodic problem for Hamilton-Jacobi-Isaacs equations. Comptes rendus de l’Académie des sciences. Série 1 Mathématique 315(8), 941–944 (1992)MathSciNetMATHGoogle Scholar
  7. 7.
    Bergounioux M., Zidani H.: Pontryagin maximum principle for optimal control of variational inequalities.. SIAM J. Control Optim. 37, 1273–1290 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bonnans F., Tiba D.: Control problems with mixed constraints and application to an optimal investment problem. Math. Rep. (Rom. Acad Sci.) 11(4), 293–306 (2009)MathSciNetGoogle Scholar
  9. 9.
    Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problems. In: Springer Series in Operations Research. Springer, New York (2000)Google Scholar
  10. 10.
    Brogliato B., Ten Dam A.A., Paoli L., Génot F., Abadie M.: Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. ASME Appl. Mech. Rev. 55(2), 107–150 (2002)CrossRefGoogle Scholar
  11. 11.
    Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton-Jacobi equations, and optimal control. In: Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser, Boston (2004)Google Scholar
  12. 12.
    Clarke F.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)MATHGoogle Scholar
  13. 13.
    Cornet B.: Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96, 130–147 (1983)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Crandall M.G., Ishii H., Lions P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. New Ser. 27(1), 1–67 (1992)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    de Pinho M.R., Rosenblueth J.F.: Necessary conditions for constrained problems under mangasarian-fromowitz conditions. SIAM J. Control Optim. 47(1), 535–552 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Dmitruk, A.V.: Maximum principle for the general optimal control problem with phase and regular mixed constraints. Comput. Math. Model. 4(4), 364–377. Software and models of systems analysis. Optimal control of dynamical systems (1993)Google Scholar
  17. 17.
    Frankowska H.: A viability approach to the Skorohod problem. Stochastics 14, 227–244 (1985)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Frankowska H.: The maximum principle for an optimal solution to a differential inclusion with end points constraints. SIAM J. Control Optim. 25, 145–157 (1987)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Frankowska H., Cernea A.: A connection between the maximum principle and dynamic programming for constrained control problems. SIAM J. Control Optim. 44(2), 673–703 (2005)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Hiriart-Urruty J.B., Strodiot J.J., Nguyen V.H.: Generalized hessian matrix and second-order optimality conditions for problems with c 1,1 data. Appl. Math. Optim. 11, 43–56 (1984)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Lions P.L.: Generalized Solutions of Hamilton-Jacobi Equations. Pitman Advanced Publishing Program, Boston (1982)MATHGoogle Scholar
  22. 22.
    Lions P.L.: Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J. 52, 793–820 (1985)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Lions P.L., Sznitman A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511–537 (1984)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Milyutin, A.A., Osmolovskii, N.P.: Calculus of variations and optimal control. Translations of Mathematical Monographs. American Mathematical Society, vol 180. Providence (1998). Translated from the Russian manuscript by Dimitrii ChibisovGoogle Scholar
  25. 25.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation. I: Basic theory. II: Applications. Springer, Berlin (2005)Google Scholar
  26. 26.
    Moreau J.J.: Liaisons unilatérales sans frottement et chocs inélastiques. C. R. Acad. Sci. Paris, Sr. II 296, 1473–1476 (1938)MathSciNetGoogle Scholar
  27. 27.
    Poliquin R.A., Rockafellar R.T.: Prox-regular functions in variational analysis. Trans. Am. Math. Soc. 348(5), 1805–1838 (1996)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Poliquin R.A., Rockafellar R.T., Thibault L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352(11), 5231–5249 (2000)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Polovinkin E.S., Smirnov G.V.: An approach to differentiation of many-valued mapping and necessary optimality conditions for optimization of solutions of differential inclusions. Differ. Equ. 22, 660–668 (1986)MATHGoogle Scholar
  30. 30.
    Rockafellar R.T., Wets R.J.-B.: Variational Analysis. Springer, Berlin (1998)MATHCrossRefGoogle Scholar
  31. 31.
    Serea O.S.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42(2), 559–575 (2003)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Thibault L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193(1), 1–26 (2003)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Vinter, R.: Optimal control. In: Modern Birkhäuser Classics. Birkhäuser, Boston (2010)Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Laboratoire de Mathematiques et PhysiqueUniversity of PerpignanPerpignanFrance

Personalised recommendations