Journal of Mathematical Sciences

, Volume 199, Issue 6, pp 646–653 | Cite as

The Canonical Theory of the Impulse Process Optimality



The paper is devoted to the development of the canonical theory of the Hamilton–Jacobi optimality for nonlinear dynamical systems with controls of the vector measure type and with trajectories of bounded variation. Infinitesimal conditions of the strong and weak monotonicity of continuous Lyapunov-type functions with respect to the impulsive dynamical system are formulated. Necessary and sufficient conditions of the global optimality for the problem of the optimal impulsive control with general end restrictions are represented. The conditions include the sets of weak and strong monotone Lyapunov-type functions and are based on the reduction of the original problem of the optimal impulsive control a finite-dimensional optimization problem on an estimated set of connectable points.


Canonical Theory Impulsive Control Impulsive System Weak Monotonicity Connectable Point 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. V. Arguchintsev, V. A. Dykhta, and V. A. Srochko, “Optimal control: nonlocal conditions, computational methods, and the variational principle of maximum,” Russian Math., 53, No. 1, 1–35 (2009).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    A. V. Arutyunov, D. Yu. Karamzin, and F. L. Pereira, “A nondegenerate maximum principle for the impulse control problem with state constraints,” SIAM J. Control Optim., 43, 1812–1843 (2005).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    F. Clarke, Yu. Ledyaev, R. Stern, and P. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New-York (1998).MATHGoogle Scholar
  4. 4.
    V. A. Dykhta, “Lyapunov–Krotov inequality and sufficient conditions in optimal control,” J. Math. Sci. (N.Y.), 121, No. 2, 2156–2177 (2004).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    V. A. Dykhta, “Hamilton–Jacobi inequalities in optimal control: smooth duality and improvement,” Vest. Tambov Univ., Ser. Estestv. Tekhn. Nauki, 15, No. 1, 405–426 (2010).MathSciNetGoogle Scholar
  6. 6.
    V. A. Dykhta and O. N. Samsonyuk, “Some applications of Hamilton–Jacobi inequalities for classical and impulsive optimal control problems,” Eur. J. Control., To appear.Google Scholar
  7. 7.
    B. M. Miller and E. Ya. Rubinovich, Optimization of Dynamic Systems with Pulse Controls, Nauka, Moscow (2005).Google Scholar
  8. 8.
    M. Motta and F. Rampazzo, “Space–time trajectories of nonlinear systems driven by ordinary and impulsive controls,” Differential and Integral Equations, 8, 269–288 (1995).MATHMathSciNetGoogle Scholar
  9. 9.
    F. L. Pereira and G. N. Silva, “Necessary conditions of optimality for vector-valued impulsive control problems,” Systems Control Lett., 40, 205–215 (2000).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    A. N. Sesekin and S. T. Zavalishin, Dynamic Impulse Systems: Theory and Applications, Kluwer Acad. Publ., Dordrecht (1997).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of System Dynamics and Control Theory, SB RAS, Russia; Institute of MathematicsEconomics and Informatics of Irkutsk State UniversityIrkutskRussia

Personalised recommendations