On the Existence of a Lipschitz Feedback Control in a Control Problem with State Constraints



We consider a nonlinear control system with state constraints given as a solution set for a finite system of nonlinear inequalities. The problem of constructing a feedback control that ensures the viability of trajectories of the closed system in a small neighborhood of the boundary of the state constraints is studied. Under some assumptions, the existence of a feedback control in the form of a Lipschitz function of the state of the system is proved.


state constraints feedback control viability problem invariance. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.-P. Aubin, Viability Theory (Birkhäuser, Boston, 1991).MATHGoogle Scholar
  2. 2.
    A. B. Kurzhanski and T. F. Filippova, “On the theory of trajectory tubes—A mathematical formalism for uncertain dynamics, viability and control,” in Advances in Nonlinear Dynamics and Control: A Report from Russia, Ed. by A. B. Kurzhanski (Birkhäuser, Boston, 1993), Ser. Progress in Systems and Control Theory 17, pp. 122–188.CrossRefGoogle Scholar
  3. 3.
    N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].MATHGoogle Scholar
  4. 4.
    F. L. Chernousko, State Estimation for Dynamic Systems (Nauka, Moscow, 1988; CRC, Boca Raton, 1994).Google Scholar
  5. 5.
    A. B. Kurzhanski and I. Valyi, Ellipsoidal Calculus for Estimation and Control (Birkhäuser, Boston, 1997).CrossRefMATHGoogle Scholar
  6. 6.
    Yu. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions (Gordon and Breach, London, 1995).MATHGoogle Scholar
  7. 7.
    M. Gusev, “On reachability analysis for nonlinear control systems with state constraints,” in Proceedings of the 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Madrid, Spain, 2014 (AIMS, Springfield, MO, 2015), pp. 579–587.Google Scholar
  8. 8.
    M. I. Gusev, “Application of penalty function method to computation of reachable sets for control systems with state constraints, AIP Conf. Proc. 1773, paper 050003 (2016).Google Scholar
  9. 9.
    F. Forcellini and F. Rampazzo, “On nonconvex differential inclusions whose state is constrained in the closure of an open set,” Differential Integral Equations 12 (4), 471–497 (1999).MathSciNetMATHGoogle Scholar
  10. 10.
    H. Frankowska and R. B. Vinter, “Existence of neighboring feasible trajectories: Applications to dynamic programming for state-constrained optimal control problems,” J. Optim. Theory Appl. 104 (1), 21–40 (2000).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A. Bressan and G. Facchi, “Trajectories of differential inclusions with state constraints,” J. Differential Equations 250 (2), 2267–2281 (2011).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    A. Ornelas, “Parametrization of Caratheodory multifunctions,” Rend. Sem. Mat. Univ. Padova 83, 33–44 (1990).MathSciNetMATHGoogle Scholar
  13. 13.
    S. M. Robinson, “Stability theory for systems of inequalities. I. Linear systems,” SIAM J. Numer. Anal. 12 (5), 754–769 (1975).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    O. L. Mangasarian and T. H. Shiau, “Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems,” SIAM J. Control Optim. 25 (3), 583–595 (1987).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Ural Federal UniversityYekaterinburgRussia

Personalised recommendations