Optimal Control Problems Governed by a Second Order Ordinary Differential Equation with m-Point Boundary Condition

  • Charles CastaingEmail author
  • Christiane Godet-Thobie
  • Le Xuan Truong
  • Bianca Satco
Part of the Advances in Mathematical Economics book series (MATHECON, volume 18)


Using a new Green type function we present a study of optimal control problem where the dynamic is governed by a second order ordinary differential equation (SODE) with m-point boundary condition.

Key words

Differential game Green function m-Point boundary Optimal control Pettis Strategy Sweeping process Viscosity 


  1. 1.
    Amrani, A., Castaing, C., Valadier, M.: Convergence in Pettis norm under extreme points condition. Vietnam J. Math. 26(4), 323–335 (1998)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Azzam, D.L., Castaing, C., Thibault, L.: Three point boundary value problems for second order differential inclusions in Banach spaces. Control Cybern. 31, 659–693 (2002). Well-posedness in optimization and related topics (Warsaw, 2001)Google Scholar
  3. 3.
    Azzam, D.L., Makhlouf, A., Thibault, L.: Existence and relaxation theorem for a second order differential inclusion. Numer. Funct. Anal. Optim. 31, 1103–1119 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997)CrossRefzbMATHGoogle Scholar
  5. 5.
    Castaing, C.: Topologie de la convergence uniforme sur les parties uniformément intégrables de L E 1 et théorème de compacité faible dans certains espaces du type Köthe-Orlicz. Sém. Anal. Convexe 10, 5.1–5.27 (1980)Google Scholar
  6. 6.
    Castaing, C.: Weak compactness and convergences in Bochner and Pettis integration. Vietnam J. Math. 24(3), 241–286 (1996)MathSciNetGoogle Scholar
  7. 7.
    Castaing, C., Marcellin, S.: Evolution inclusions with pln functions and application to viscosity and controls. J. Nonlinear Convex Anal. 8(2), 227–255 (2007)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Castaing, C., Raynaud de Fitte, P.: On the fiber product of Young measures with applications to a control problem with measures. Adv. Math. Econ. 6, 1–38 (2004)Google Scholar
  9. 9.
    Castaing, C., Truong, L.X.: Second order differential inclusions with m-points boundary condition. J. Nonlinear Convex Anal. 12(2), 199–224 (2011)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Castaing, C., Truong, L.X.: Some topological properties of solutions set in a second order inclusion with m-point boundary condition. Set Valued Var. Anal. 20, 249–277 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Castaing, C., Truong, L.X.: Bolza, relaxation and viscosity problems governed by a second order differential equation. J. Nonlinear Convex Anal. 14(2), 451–482 (2013)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. In: Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977)Google Scholar
  13. 13.
    Castaing, C., Jofre, A., Salvadori, A.: Control problems governed by functional evolution inclusions with Young measures. J. Nonlinear Convex Anal. 5(1), 131–152 (2004)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Castaing, C., Raynaud de Fitte, P., Valadier, M.: Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory. Kluwer Academic, Dordrecht (2004)Google Scholar
  15. 15.
    Castaing, C., Jofre, A., Syam, A.: Some limit results for integrands and Hamiltonians with application to viscosity. J. Nonlinear Convex Anal. 6(3), 465–485 (2005)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Castaing, C., Raynaud de Fitte, P., Salvadori, A.: Some variational convergence results with application to evolution inclusions. Adv. Math. Econ. 8, 33–73 (2006)Google Scholar
  17. 17.
    Castaing, C., Monteiro Marquès, M.D.P., Raynaud de Fitte, P.: On a optimal control problem governed by the sweeping process (2013). PreprintGoogle Scholar
  18. 18.
    Castaing, C., Monteiro Marquès, M.D.P., Raynaud de Fitte, P.: On a Skorohod problem (2013). PreprintGoogle Scholar
  19. 19.
    El Amri, K., Hess, C.: On the Pettis integral of closed valued multifunction. Set Valued Anal. 8, 329–360 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Elliot, R.J.: Viscosity Solutions and Optimal Control. Pitman, London (1977)Google Scholar
  21. 21.
    Elliot, R.J., Kalton, N.J.: Cauchy problems for certains Isaacs-Bellman equations and games of survival. Trans. Am. Soc. 198, 45–72 (1974)CrossRefGoogle Scholar
  22. 22.
    Evans, L.C., Souganides, P.E.: Differential games and representation formulas for solutions of Hamilton-Jacobi-Issacs equations. Indiana Univ. Math. J. 33, 773–797 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Godet-Thobie, C., Satco, B.: Decomposability and uniform integrability in Pettis integration. Questiones Math. 29, 39–58 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Grothendieck, A.: Espaces vectoriels topologiques. Publicação da Sociedade de Matemática de Sao Paulo (1964)Google Scholar
  25. 25.
    Henry, C.: An existence theorem for a class of differential equation with multivalued right-hand side. J. Math. Anal. Appl. 41, 179–186 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Moreau, J.J.: Evolution problem associated with a moving set in Hilbert space. J. Differ. Equ. 26, 347–374 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Satco, B.: Second order three boundary value problem in Banach spaces via Henstock and Henstock-Kurzweil-Pettis integral. J. Math. Appl. 332, 912–933 (2007)Google Scholar
  28. 28.
    Valadier, M.: Some bang-bang theorems. In: Multifunctions and Integrands, Stochastics Analysis, Approximations and Optimization Proceedings, Catania, 1983. Lecture Notes in Mathematics, vol. 1091, pp. 225–234Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Charles Castaing
    • 1
    Email author
  • Christiane Godet-Thobie
    • 2
  • Le Xuan Truong
    • 3
  • Bianca Satco
    • 4
  1. 1.Département de Mathématiques de Brest, Case 051Université Montpellier IIMontpellier cedexFrance
  2. 2.Laboratoire de Mathématiques de Brest, CNRS-UMR 6205Université de Bretagne OccidentaleBrest Cedex 3France
  3. 3.Department of Mathematics and StatisticsUniversity of Economics of HoChiMinh CityHoChiMinh CityVietnam
  4. 4.Stefan cel Mare University of SuceavaSuceavaRomania

Personalised recommendations