From Pure State and Input Constraints to Mixed Constraints in Nonlinear Systems

Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 473)


We survey the results on the problem of pure/mixed state and input constrained control, with multidimensional constraints, for finite dimensional nonlinear differential systems with focus on the so-called admissible set and its boundary. The admissible set is the set of initial conditions for which there exist a control and an integral curve satisfying the constraints for all time. Its boundary is made of two disjoint parts: the subset of the state constraint boundary on which there are trajectories pointing towards the interior of the admissible set or tangentially to it; and a barrier, namely a semipermeable surface which is constructed via a generalized minimum-like principle with nonsmooth terminal conditions. Comparisons between pure state constraints and mixed ones are presented on a series of simple academic examples.


Input and state constraints Mixed constraints Nonlinear systems Barriers Admissible sets 


  1. 1.
    Aubin, J.P.: Viability Theory. Systems and Control Foundations, Birkhäuser (1991)MATHGoogle Scholar
  2. 2.
    Chutinan, A.C., Krogh, B.H.: Computational techniques for hybrid system verification. IEEE Trans. Autom. Control 64–75 (2003)Google Scholar
  3. 3.
    Clarke, F.H., de Pinho, M.: Optimal control problems with mixed constraints. SIAM J. Control Optim. 48, 4500–4524 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Clarke, F.H., Ledyaev, Yu.S., Stern, R.J.,Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)Google Scholar
  5. 5.
    Danskin, J.: The Theory of Max-Min. Springer (1967)Google Scholar
  6. 6.
    De Dona, J.A., Lévine, J.: On barriers in state and input constrained nonlinear systems. SIAM J. Control Optim. 51(4), 3208–3234 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Esterhuizen, W., Lévine, J.: Barriers in nonlinear control systems with mixed constraints (2015). arXiv:1508.01708 [math.OC]
  8. 8.
    Hartl, R.F., Sethi, S.P., Vickson, R.J.: A survey of the maximal principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. Wiley (1966)Google Scholar
  10. 10.
    Isaacs, R.: Differential Games. Wiley (1965)Google Scholar
  11. 11.
    Kaynama, S., Maidens, J., Oishi, M., Mitchell, I., Dumont, G.: Computing the viability kernel using maximal reachable sets. In: Proceedings of the 15th ACM HSCC’12, New York, NY, USA, pp. 55–64. ACM (2012)Google Scholar
  12. 12.
    Lhommeau, M., Jaulin, L., Hardouin, L.: Capture basin approximation using interval analysis. Int. J. Adapt. Control Signal Process. 25(3), 264–272 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lygeros, J., Tomlin, C., Sastry, S.: Controllers for reachability specifications for hybrid systems. Automatica 35(3), 349–370 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50(7), 947–957 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: The Mathematical Theory of Optimal Processes. Wiley (1965)Google Scholar
  16. 16.
    Prajna, S.: Barrier certificates for nonlinear model validation. Automatica 42(1), 117–126 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Tee, K.P., Ge, S.S., Tay, E.H.: Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4), 918–927 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Tomlin, C.J., Lygeros, J., Sastry, S.S.: A game theoretic approach to controller design for hybrid systems. Proc. IEEE 88(7), 949–970 (2000)Google Scholar
  19. 19.
    Tomlin, C.J., Mitchell, I., Bayen, A.M., Oishi, M.: Computational techniques for the verification of hybrid systems. Proc. IEEE 91(7), 986–1001 (2003)CrossRefGoogle Scholar
  20. 20.
    van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Lecture Notes in Control and Information Sciences, vol. 251. Springer (2000)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CAS, Mathématiques et Systèmes, MINES-ParisTech, PSL Research UniversityFontainebleauFrance

Personalised recommendations