Optimality conditions for optimal impulsive control problems with multipoint state constraints

  • Olga N. SamsonyukEmail author


This paper addresses an optimal impulsive control problem whose trajectories are functions of bounded variation and impulsive controls are regular vector measures. This problem is characterized by two main features. First, the dynamical control system to be considered may not possess the so-called well-posedness property. Second, the constraints on the one-sided limits of states are presented. Such constraints are interpreted as multipoint state constraints. For this problem, we derive global optimality conditions based on using of compound Lyapunov type functions which possess strongly monotone properties with respect to the control system. As a motivating case, a model of advertising expenses optimization for mutually complementary products is considered. For this model, we propose four variants of resolving sets of Lyapunov type functions and explain the technique of applying the optimality conditions.


Measure-driven differential equations Impulsive control Trajectories of bounded variation Global optimality conditions 

Mathematics Subject Classification

34A37 34H05 49K21 



  1. 1.
    Arutyunov, A., Karamzin, D., Pereira, F.L.: Optimal Impulsive Control: The Extension Approach. Lecture Notes in Control and Information Sciences, vol. 477. Springer, Berlin (2019)CrossRefGoogle Scholar
  2. 2.
    Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control. Applied Mathematics, vol. 2. American Institute of Mathematical Sciences (AIMS), Springfield (2007)zbMATHGoogle Scholar
  3. 3.
    Bressan, A., Rampazzo, F.: On differential systems with vector-valued impulsive controls. Boll. Un. Mat. Ital. Ser. B 3, 641–656 (1988)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol. 264. Springer, London (2013)CrossRefGoogle Scholar
  5. 5.
    Clarke, F., Ledyaev, Yu., Stern, R., Wolenski, P.: Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics, vol. 178. Springer, New York (1998)zbMATHGoogle Scholar
  6. 6.
    Daryin, A.N., Kurzhanski, A.B.: Dynamic programming for impulse control. Ann. Rev. Control. 32, 213–227 (2008)CrossRefGoogle Scholar
  7. 7.
    Dorroh, J.R., Ferreyra, G.: A multistate, multicontrol problem with unbounded controls. SIAM J. Control Optim. 32, 1322–1331 (1994)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dykhta, V.A.: Lyapunov–Krotov inequality and sufficient conditions in optimal control. J. Math. Sci. 121, 2156–2177 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dykhta, V.A., Samsonyuk, O.N.: Optimal Impulsive Control with Applications, 2nd edn. Fizmatlit, Moscow (2003). (in Russian)zbMATHGoogle Scholar
  10. 10.
    Dykhta, V.A., Samsonyuk, O.N.: A maximum principle for smooth optimal impulsive control problems with multipoint state constraints. Comp. Math. Math. Phys. 49, 942–957 (2009)CrossRefGoogle Scholar
  11. 11.
    Dykhta, V., Samsonyuk, O.: Some applications of Hamilton-Jacobi inequalities for classical and impulsive optimal control problems. Eur. J. Control. 17, 55–69 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dykhta, V.A., Samsonyuk, O.N.: Hamilton–Jacobi Inequalities and Variational Optimality Conditions. ISU, Irkutsk (2015). (in Russian)zbMATHGoogle Scholar
  13. 13.
    Fraga, S.L., Pereira, F.L.: On the feedback control of impulsive dynamic systems. In: 47th IEEE Conference on Decision and Control. pp. 2135–2140 (2008)Google Scholar
  14. 14.
    Gurman, V.: The Extension Principle in Optimal Control Problems, 2nd edn. Fizmatlit, Moscow (1997). (in Russian)zbMATHGoogle Scholar
  15. 15.
    Karamzin, DYu., Oliveira, V.A., Pereira, F.L., Silva, G.N.: On some extension of optimal control theory. Eur. J. Control. 20(6), 284–291 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Krotov, V.F.: Global Methods in Optimal Control Theory. Marcel Dekker, New York (1996)zbMATHGoogle Scholar
  17. 17.
    Kurzhanski, A.B., Daryin, A.N.: Dynamic Programming for Impulse Feedback and Fast Controls: The Linear Systems Case. Lecture Notes in Control and Information Sciences, vol. 468. Springer, London (2020)CrossRefGoogle Scholar
  18. 18.
    Lakshmikantham, V., Matrosov, V., Sivasundaram, S.: Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems. Mathematics and Its Applications, vol. 63. Kluwer Academic Publishers, Dordrecht (1991)CrossRefGoogle Scholar
  19. 19.
    Miller, B.M.: The generalized solutions of nonlinear optimization problems with impulse control. SIAM J. Control Optim. 34, 1420–1440 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Miller, B.M., Rubinovich, EYa.: Impulsive Control in Continuous and Discrete-Continuous Systems. Kluwer Academic Publishers, New York (2003)CrossRefGoogle Scholar
  21. 21.
    Miller, B.M., Rubinovich, E.Ya.: Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations. Autom. Remote Control 74, 1969–2006 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Milyutin, A.A., Osmolovskii, N.P.: Calculus of Variations and Optimal Control. Am. Math. Soc, Providence (1998)CrossRefGoogle Scholar
  23. 23.
    Motta, M., Rampazzo, F.: Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls. Differ. Integral Equ. 8, 269–288 (1995)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Motta, M., Rampazzo, F.: Dynamic programming for nonlinear systems driven by ordinary and impulsive control. SIAM J. Control Optim. 34, 199–225 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Motta, M., Rampazzo, F.: Nonlinear systems with unbounded controls and state constraints: a problem of proper extension. NoDEA 3, 191–216 (1996)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Motta, M., Rampazzo, F., Vinter, R.: Normality and gap phenomena in optimal unbounded control. ESAIM Control Optim. Calc. Var. 24(4), 1645–1673 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Motta, M., Sartori, C.: Exit time problems for nonlinear unbounded control systems. Discrete Contin. Dyn. Syst. Ser. A 5(1), 137–156 (1999)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pereira, F.L., Matos, A.C., Silva, G.N.: Hamilton–Jacobi conditions for an impulsive control problem. IFAC Nonlinear Control Syst. 34(6), 1297–1302 (2001)Google Scholar
  29. 29.
    Pereira, F.L., Silva, G.N.: Necessary conditions of optimality for vector-valued impulsive control problems. Syst. Control Lett. 40, 205–215 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Samsonyuk, O.N.: Compound Lyapunov type functions in control problems of impulsive dynamical systems. Trudy Inst. Mat. Mekh. UrO RAN. 16(5), 170–178 (2010)Google Scholar
  31. 31.
    Samsonyuk, O.N.: Invariant sets for nonlinear impulsive control systems. Autom. Remote Control 76(3), 405–418 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Samsonyuk, O., Sorokin, S., Staritsyn, M.: Feedback optimality conditions with weakly invariant functions for nonlinear problems of impulsive control. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) Mathematical Optimization Theory and Operations Research. Lecture Notes in Computer Science, vol. 11548, pp. 513–526. Springer, Switzerland (2019)CrossRefGoogle Scholar
  33. 33.
    Sesekin, A.N., Zavalishchin, S.T.: Dynamic Impulse Systems: Theory and Applications. Kluwer Academic Publishers, Dordrecht (1997)zbMATHGoogle Scholar
  34. 34.
    Silva, G.N., Vinter, R.B.: Necessary conditions for optimal impulsive control problems. SIAM J. Control Optim. 35, 1829–1846 (1997)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sorokin, S., Staritsyn, M.: Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numer. Algebra Control Optim. 7(2), 201–210 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Vinter, R.B.: Optimal Control. Birkhäuser, Berlin (2000)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control Theory of SB RASIrkutskRussia

Personalised recommendations