Differential Equations

, Volume 55, Issue 4, pp 532–540 | Cite as

Hamiltonian Formalism in Team Control Problems

  • A. B. KurzhanskiiEmail author
Control Theory


We consider the target control synthesis problem for a team of single-type plants performing a common motion to a target set under the condition that the team members do not collide with each other. In the process of motion, the team members must stay inside a virtual container forming a standard motion (tube) and avoiding the obstacles known in advance by reconfiguration. The general scheme for solving this problem, which reduces the original problem to a series of subproblems, is given, and Hamiltonian formalism is used for a detailed consideration of the subproblems of terminal control of ellipsoidal tubes, the container motions between moving external obstacles, and the evolution of the team inside the virtual container.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kurzhanski, A.B., On a team control problem under obstacles, Proc. Steklov Inst. Math., 2015, vol. 291, suppl. no. 1, pp. S128–S142.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kurzhanski, A.B., Problem of collision avoidance for a team motion with obstacles, Proc. Steklov Inst. Math., 2016, vol. 293, suppl. no. 1, pp. S120–S136.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kurzhanski, A.B. and Mesyatz, A.I., Optimal control of ellipsoidal motions, Differ. Equations, 2012, vol. 48, no. 11, pp. 1502–1509.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kurzhanski, A.B. and Mesyatz, A.I., Control of ellipsoidal trajectories: theory and numerical results, Comput. Math. Math. Phys., 2014, vol. 54, no. 3, pp. 418–428.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kurzhanski, A.B. and Varaiya, P., Dynamics and Control of Trajectory Tubes. Theory and Computation, in Systems & Control: Foundations & Applications, Boston: Birkhäuser/Springer, 2014.zbMATHGoogle Scholar
  6. 6.
    Krasovskii, N.N. and Subbotin, A.I., Game-Theoretical Control Problems, New York; Berlin; Heidelberg: Springer, 1988.CrossRefGoogle Scholar
  7. 7.
    Subbotin, A.I., Generalized Solutions of First-Order PDE’s. The Dynamic Optimization Perspective, Systems & Control: Foundations & Applications, Boston: Birkhäuser/Springer, 1995.CrossRefGoogle Scholar
  8. 8.
    Subbotina, N.N., Kolpakova, E.A., Tokmantsev, T.B., and Shagalova, L.G., Metod kharakteristik dlya uravneniya Gamil’tona-Yakobi-Bellmana (Method of Characteristics for the Hamiton-Jacobi-Bellman Equation), Yekaterinburg: RIO URO RAN, 2013.Google Scholar
  9. 9.
    Kurzhanski, A.B. and Dar’in, A.N., Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty, Comput. Math. Math. Phys., 2013, vol. 53, no. 1, pp. 34–43.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Crandall, M.G. and Lions, P-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 1983, vol. 277, pp. 1–41.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Filippov, A.F., Differentsial’nye uravneniya s razryvnoi pravoi chast’yu (Differential Equations with Discontinuous Right-Hand Side), Moscow: Nauka, 1985.Google Scholar
  12. 12.
    Rockafellar, R.T. and Wets, R.J-B., Variational Analysis, New York: Springer, 2004.zbMATHGoogle Scholar
  13. 13.
    Dem’yanov, V.F., Minimaks: differentsiruemost’ po napravleniyam (Minimax: Differentiability in Directions), Leningrad: Leningrad Gos. Univ., 1974.Google Scholar
  14. 14.
    Chang, D.E., Shadden, S., Marsden, J.E., and Olfati-Saber, R., Collision avoidance for multiple agent systems, Proc. of 42nd IEEE Conf. on Decision and Control, Maui, 2003, vol. 1, pp. 539–543.CrossRefGoogle Scholar
  15. 15.
    Olfati-Saber, R., Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Trans. Autom. Control, 2006, vol. 51, no. 3, pp. 401–420.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kurzhanski, A.B., Mitchell, I.M., and Varaiya, P., Optimization techniques for state-constrained control and obstacle problems, Proc. of the 6th IFAC Symposium NOLCOS-2004, Stuttgart, 2004, pp. 813–818.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations