General Nonlinear Impulsive Control Problems

  • Aram Arutyunov
  • Dmitry KaramzinEmail author
  • Fernando Lobo Pereira
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 477)


In this concluding chapter, an extension of the classical control problem is given in the most general nonlinear case. The essential matter is that now the control variable is not split into conventional and impulsive types, while the dependence on this unified control variable is not necessarily affine. By combining the two approaches, the one based on the Lebesgue discontinuous time variable change, and the other based on the convexification of the problem by virtue of the generalized controls proposed by Gamkrelidze, a fairly general extension of the optimal control problem is constructed founded on the concept of generalized impulsive control. A generalized Filippov-like existence theorem for a solution is proved. The Pontryagin maximum principle for the generalized impulsive control problem with state constraints is presented. Within the framework of the proposed approach, a number of classic examples of essentially nonlinear problems of calculus of variations which allow for discontinuous optimal arcs are also examined. The chapter ends with seven exercises.


  1. 1.
    Arutyunov, A., Karamzin, D., Pereira, F.: The maximum principle for optimal control problems with state constraints by R.V. Gamkrelidze: revisited. J. Optim. Theory Appl. 149(3), 474–493 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arutyunov, A.V.: Optimality conditions. Abnormal and degenerate problems. In: Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  3. 3.
    Gamkrelidze, R.: Optimal control processes for bounded phase coordinates. Izv. Akad. Nauk SSSR. Ser. Mat. 24, 315–356 (1960)MathSciNetGoogle Scholar
  4. 4.
    Gamkrelidze, R.: On sliding optimal states. Soviet Math. Dokl. 3, 390–395 (1962)Google Scholar
  5. 5.
    Gamkrelidze, R.: On some extremal problems in the theory of differential equations with applications to the theory of optimal control. J Soc. Ind. Appl. Math. Ser. A Control 3(1), 106–128 (1965)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gamkrelidze, R.: Principles of Optimal Control Theory. Plenum Press, New York (1978)CrossRefGoogle Scholar
  7. 7.
    Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: Mathematical theory of optimal processes. Translated from the Russian ed. by L.W. Neustadt. Interscience Publishers, Wiley, 1st edn (1962)Google Scholar
  8. 8.
    Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, New York, London (1972)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Aram Arutyunov
    • 1
    • 2
    • 3
  • Dmitry Karamzin
    • 4
    Email author
  • Fernando Lobo Pereira
    • 5
  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Institute of Control Sciences of the Russian Academy of SciencesMoscowRussia
  3. 3.RUDN UniversityMoscowRussia
  4. 4.Federal Research Center “Computer Science and Control” of the Russian Academy of SciencesMoscowRussia
  5. 5.FEUP/DEECPorto UniversityPortoPortugal

Personalised recommendations