Mathematics of Control, Signals, and Systems

, Volume 31, Issue 4, pp 503–544 | Cite as

Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times

  • Loïc Bourdin
  • Gaurav DharEmail author
Original Article


In a recent paper by Bourdin and Trélat, a version of the Pontryagin maximum principle (in short, PMP) has been stated for general nonlinear finite-dimensional optimal sampled-data control problems. Unfortunately, their result is only concerned with fixed sampling times, and thus, it does not take into account the possibility of free sampling times. The present paper aims to fill this gap in the literature. Precisely, we establish a new version of the PMP that can handle free sampling times. As in the aforementioned work by Bourdin and Trélat, we obtain a first-order necessary optimality condition written as a nonpositive averaged Hamiltonian gradient condition. Furthermore, from the freedom of choosing sampling times, we get a new and additional necessary optimality condition which happens to coincide with the continuity of the Hamiltonian function. In an autonomous context, even the constancy of the Hamiltonian function can be derived. Our proof is based on the Ekeland variational principle. Finally, a linear–quadratic example is numerically solved using shooting methods, illustrating the possible discontinuity of the Hamiltonian function in the case of fixed sampling times and highlighting its continuity in the instance of optimal sampling times.


Sampled-data control Optimal control Optimal sampling times Pontryagin maximum principle Hamiltonian continuity Hamiltonian constancy Ekeland variational principle 

Mathematics Subject Classification

34K35 34H05 49J15 49K15 93C15 93C57 93C62 93C83 



  1. 1.
    Ackermann JE (1985) Sampled-data control systems: analysis and synthesis, robust system design. Springer, Berlin CrossRefGoogle Scholar
  2. 2.
    Aström KJ (1963) On the choice of sampling rates in optimal linear systems. Engineering Studies, IBM ResearchGoogle Scholar
  3. 3.
    Aström KJ, Wittenmark B (1997) Computer-controlled systems. Prentice Hall, Upper Saddle RiverGoogle Scholar
  4. 4.
    Bakir T, Bonnard B, Bourdin L, Rouot J (2019) Pontryagin-type conditions for optimal muscular force response to functional electrical stimulations. Accepted for publication in J Optim Theory ApplGoogle Scholar
  5. 5.
    Bergounioux M, Bourdin L (2019) Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints. Accepted for publication in ESAIM Control Optim Calc VarGoogle Scholar
  6. 6.
    Bini E, Buttazzo GM (2014) The optimal sampling pattern for linear control systems. IEEE Trans Automat Control 59(1):78–90MathSciNetCrossRefGoogle Scholar
  7. 7.
    Biryukov RS (2016) Generalized \(H_\infty \)-optimal control of linear continuous-discrete plant. Avtomat I Telemekh 77(3):33–51MathSciNetGoogle Scholar
  8. 8.
    Boltyanski VG, Poznyak AS (2012) The robust maximum principle. Systems & control: foundations & applications. Birkhäuser/Springer, New York. Theory and applicationsGoogle Scholar
  9. 9.
    Boltyanskii VG (1978) Optimal control of discrete systems. Wiley, New York-TorontoGoogle Scholar
  10. 10.
    Bourdin L (2016) Note on Pontryagin maximum principle with running state constraints and smooth dynamics—proof based on the Ekeland variational principle. Research notes, hal-01302222Google Scholar
  11. 11.
    Bourdin L, Trélat E (2013) Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales. SIAM J Control Optim 51(5):3781–3813MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bourdin L, Trélat E (2015) Pontryagin maximum principle for optimal sampled-data control problems. In: Proceedings of the IFAC workshop CAOGoogle Scholar
  13. 13.
    Bourdin L, Trélat E (2016) Optimal sampled-data control, and generalizations on time scales. Math Control Relat Fields 6(1):53–94MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bourdin L, Trélat E (2017) Linear-quadratic optimal sampled-data control problems: convergence result and Riccati theory. Automatica 79:273–281MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chen T, Francis B (1996) Optimal sampled-data control systems. Springer, London Ltd, LondonzbMATHGoogle Scholar
  16. 16.
    Dmitruk AV, Kaganovich AM (2011) Maximum principle for optimal control problems with intermediate constraints. Comput Math Model 22(2):180–215. Translation of Nelineĭnaya Din. Upr. No. 6(2008):101–136MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ekeland I (1974) On the variational principle. J Math Anal Appl 47:324–353MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fadali MS, Visioli A (2013) Digital control engineering: analysis and design. Elsevier, AmsterdamGoogle Scholar
  19. 19.
    Fattorini HO (1999) Infinite-dimensional optimization and control theory, vol 62. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  20. 20.
    Grasse KA, Sussmann HJ (1990) Global controllability by nice controls. In: Nonlinear controllability and optimal control, vol 133 of Monogr. Textbooks Pure Appl Math pp 33–79. Dekker, New YorkGoogle Scholar
  21. 21.
    Grüne L, Pannek J (2017) Nonlinear model predictive control. Communications and Control Engineering Series. Springer, Cham. Theory and algorithms, Second edition [of MR3155076]Google Scholar
  22. 22.
    Haberkorn T, Trélat E (2011) Convergence results for smooth regularizations of hybrid nonlinear optimal control problems. SIAM J Control Optim 49(4):1498–1522MathSciNetCrossRefGoogle Scholar
  23. 23.
    Halkin H (1966) A maximum principle of the Pontryagin type for systems described by nonlinear difference equations. SIAM J Control 4(1):90–111MathSciNetCrossRefGoogle Scholar
  24. 24.
    Holtzman JM, Halkin H (1966) Discretional convexity and the maximum principle for discrete systems. SIAM J Control 4:263–275MathSciNetCrossRefGoogle Scholar
  25. 25.
    Landau ID, Zito G (2006) Digital control systems: design, identification and implementation. Springer, LondonGoogle Scholar
  26. 26.
    Levis AH, Schlueter RA (1971) On the behaviour of optimal linear sampled-data regulators. Int J Control 13(2):343–361CrossRefGoogle Scholar
  27. 27.
    Margaliot M (2006) Stability analysis of switched systems using variational principles: an introduction. Autom J IFAC 42(12):2059–2077MathSciNetCrossRefGoogle Scholar
  28. 28.
    Melzer SM, Kuo BC (1971) Sampling period sensitivity of the optimal sampled data linear regulator. Autom J IFAC 7:367–370MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, New YorkzbMATHGoogle Scholar
  30. 30.
    Santina MS, Stubberud AR (2005) Basics of sampling and quantization. In: Handbook of networked and embedded control systems, control eng. Birkhauser, Boston, pp 45–69CrossRefGoogle Scholar
  31. 31.
    Schlueter RA (1973) The optimal linear regulator with constrained sampling times. IEEE Trans Autom Control AC–18(5):515–518MathSciNetCrossRefGoogle Scholar
  32. 32.
    Schlueter RA, Levis AH (1973) The optimal linear regulator with state-dependent sampling. IEEE Trans Autom Control AC–18(5):512–515MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sethi SP, Thompson GL (2000) Optimal control theory. Kluwer Academic Publishers, Boston, MA, second edition. Applications to management science and economicsGoogle Scholar
  34. 34.
    Sussmann HJ (1999) A maximum principle for hybrid optimal control problems. In: Proceedings of the 38th IEEE conference on decision and control (Cat. No. 99CH36304), vol 1. IEEE, pp 425–430Google Scholar
  35. 35.
    Vinter R (2010) Optimal control. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA. Paperback reprint of the 2000 editionGoogle Scholar
  36. 36.
    Volz RA, Kazda LF (1966) Design of a digital controller for a tracking telescope. IEEE Trans Autom Control AC–12(4):359–367CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de recherche XLIM, UMR CNRS 7252Université de LimogesLimogesFrance

Personalised recommendations