Applied Mathematics & Optimization

, Volume 68, Issue 1, pp 1–19 | Cite as

State-Constrained Optimal Control Problems of Impulsive Differential Equations



The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption.


Optimal control problems Impulsive differential equations Time-dependent state constraints Time measurable Hamilton-Jacobi equations 


  1. 1.
    Altarovici, A., Bokanowski, O., Zidani, H.: A general Hamilton-Jacobi framework for nonlinear state-constrained control problems. ESAIM Control Optim. Calc. Var. (2012). doi: 10.1051/cocv/2012011 Google Scholar
  2. 2.
    Arutyunov, A., Dykhta, V., Lobo Pereira, L.: Necessary conditions for impulsive nonlinear optimal control problems without a priori normality assumptions. J. Optim. Theory Appl. 124(1), 55–77 (2005) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bechhofer, J., Johnson, B.: A simple model for Faraday waves. Am. J. Phys. 64, 1482–1487 (1996) CrossRefGoogle Scholar
  4. 4.
    Bokanowski, O., Forcadel, N., Zidani, H.: Deterministic state constrained optimal control problems without controllability assumptions. ESAIM Control Optim. Calc. Var. 17(04), 975–994 (2011) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brach, R.M.: Mechanical Impact Dynamics. Wiley, New York (1991) Google Scholar
  6. 6.
    Bressan, A.: Impulsive control systems. In: Mordukhovich, B., Sussmann, H. (eds.) Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, pp. 1–22. Springer, New York (1996) CrossRefGoogle Scholar
  7. 7.
    Bressan, A., Rampazzo, F.: On differential systems with vector-valued impulsive controls. Boll. Unione Mat. Ital. 7(2-B), 641–656 (1988) MathSciNetGoogle Scholar
  8. 8.
    Bressan, A., Rampazzo, F.: Impulsive control-systems with commutativity assumptions. J. Optim. Theory Appl. 71(1), 67–83 (1991) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bressan, A., Rampazzo, F.: Impulsive control-systems without commutativity assumptions. J. Optim. Theory Appl. 81(3), 435–457 (1994) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Briani, A.: A Hamilton-Jacobi equation with measures arising in Γ-convergence of optimal control problems. Differ. Integral Equ. 12(6), 849–886 (1999) MathSciNetMATHGoogle Scholar
  11. 11.
    Briani, A., Zidani, H.: Characterisation of the value function of final state constrained control problems with BV trajectories. Commun. Pure Appl. Anal. 10(6), 1567–1587 (2011) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Brogliato, B.: Nonsmooth Impact Mechanics: Models, Dynamics and Control. Lecture Notes in Control and Information Sciences, vol. 220. Springer, New York (1996) MATHGoogle Scholar
  13. 13.
    Capuzzo-Dolcetta, I., Lions, P.-L.: Hamilton-Jacobi equations with state constraints. Trans. Am. Math. Soc. 318(2), 643–683 (1990) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Catllá, A., Porter, J., Silber, M.: Weakly nonlinear analysis of impulsively-forced Faraday waves. Phys. Rev. E 72(3) (2005) Google Scholar
  15. 15.
    Catllá, A., Schaeffer, D., Witelski, T., Monson, E., Lin, A.: On spiking models for synaptic activity and impulsive differential equations. SIAM Rev. 50(3), 553–569 (2005) CrossRefGoogle Scholar
  16. 16.
    Dal Maso, G., Rampazzo, F.: On systems of ordinary differential equations with measures as controls. Differ. Integral Equ. 4(4), 738–765 (1991) MathSciNetGoogle Scholar
  17. 17.
    Forcadel, N., Rao, Z., Zidani, H.: Optimal control problems of BV trajectories with pointwise state constraints. In: Proceedings of the 18th IFAC World Congress, Milan, vol. 18 (2011) Google Scholar
  18. 18.
    Frankowska, H., Plaskacz, S.: Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints. J. Math. Anal. Appl. 251(2), 818–838 (2000) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Frankowska, H., Plaskacz, S., Rzeuchowski, T.: Measurable viability theorems and Hamilton-Jacobi-Bellman equation. J. Differ. Equ. 116, 265–305 (1995) MATHCrossRefGoogle Scholar
  20. 20.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer, New York (1990) Google Scholar
  21. 21.
    Hsu, C., Cheng, W.: Applications of the theory of impulsive parametric excitation and new treatments of general parametric excitation problems. Trans. ASME J. Appl. Mech. 40, 551–558 (1973) CrossRefGoogle Scholar
  22. 22.
    Huepe, C., Ding, Y., Umbanhowar, P., Silber, M.: Forcing function control of Faraday wave instabilities in viscous shallow fluids. Phys. Rev. E 73 (2006) Google Scholar
  23. 23.
    Ishii, H.: Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets. Bull. Fac. Sci. Eng. 28, 33–77 (1985) Google Scholar
  24. 24.
    Motta, M., Rampazzo, F.: Dynamic programming for nonlinear systems driven by ordinary and impulsive controls. SIAM J. Control Optim. 44(1), 199–225 (1996) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Raymond, J.-P.: Optimal control problems in spaces of functions of bounded variation. Differ. Integral Equ. 10, 105–136 (1997) MathSciNetMATHGoogle Scholar
  26. 26.
    Silva, G.N., Vinter, R.B.: Measure-driven differential inclusions. J. Math. Anal. Appl. 202, 746–767 (1996) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Silva, G.N., Vinter, R.B.: Necessary conditions for optimal impulsive control systems. SIAM J. Control Optim. 35, 1829–1846 (1998) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Soner, H.M.: Optimal control with state-space constraint. I. SIAM J. Control Optim. 24(3), 552–561 (1986) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Soner, H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24(6), 1110–1122 (1986) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Wolenski, P.R., Žabić, S.: A differential solution concept for impulsive systems. Differ. Equ. Dyn. Syst. 2, 199–210 (2006) Google Scholar
  31. 31.
    Wolenski, P.R., Žabić, S.: A sampling method and approximations results for impulsive systems. SIAM J. Control Optim. 46, 983–998 (2007) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.CeremadeUniversité Paris-DauphineParis Cedex 16France
  2. 2.Equipe COMMANDSENSTA ParisTech & INRIA-SaclayParis CedexFrance

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