Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 207–234 | Cite as

Strong Local Optimality for Generalized L1 Optimal Control Problems

  • Francesca C. Chittaro
  • Laura PoggioliniEmail author


In this paper, we analyze control-affine optimal control problems with a cost functional involving the absolute value of the control. The Pontryagin extremals associated with such systems are given by (possible) concatenations of bang arcs with singular arcs and with zero control arcs, that is, arcs where the control is identically zero. Here, we consider Pontryagin extremals given by a bang-zero control-bang concatenation. We establish sufficient optimality conditions for such extremals, in terms of some regularity conditions and of the coerciveness of a suitable finite-dimensional second variation.


Optimal control Sufficient conditions Hamiltonian methods Sparse control 

Mathematics Subject Classification

49K15 49K30 



This work was supported by the projects “Dinamiche non autonome, sistemi Hamil- toniani e teoria del controllo” (GNAMPA 2015), “Dinamica topologica, sistemi Hamiltoniani e teoria del controllo” (GNAMPA 2016), “Sistemi Dinamici, Teoria del Controllo e Applicazioni” (GNAMPA 2017) by Istituto Nazionale di Alta Matematica “Francesco Severi” and by UTLN–Appel à projet “Chercheurs invités”, Université de Toulon, years 2015, 2016 and 2017.


  1. 1.
    Berret, B., Darlot, C., Jean, F., Pozzo, T., Papaxanthis, C., Gauthier, J.P.: The inactivation principle: mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements. PLoS Comput. Biol. 4(10), e1000194 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, Z., Caillau, J.B., Chitour, Y.: L\(^1\) minimization for mechanical systems. SIAM J. Control Optim. 54, 1245–1265 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Vossen, G., Maurer, H.: On L\(^1\)-minimization in optimal control and applications to robotics. Optim. Control Appl. Methods 27(6), 301–321 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boizot, N., Oukacha, O.: Consumption minimisation for an academic vehicle (2016). hal-01384651Google Scholar
  5. 5.
    Craig, A.J., Flügge-Lotz, I.: Investigation of optimal control with a minimum-fuel consumption criterion for a fourth-order plant with two control inputs; synthesis of an efficient suboptimal control. J. Basic Eng. 87, 39–58 (1965)CrossRefGoogle Scholar
  6. 6.
    Ross, I.M.: Space trajectory optimization and L\(^1\)-optimal control problems. In: Gurfil, P. (ed.) Modern Astrodynamics. Elsevier Astrodynamics Series, vol. 1, pp. 155–VIII. Butterworth-Heinemann, Oxford (2006)Google Scholar
  7. 7.
    Clason, C., Kunisch, K.: A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM COCV 17, 243–266 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kalise, D., Kunisch, K., Rao, Z.: Infinite horizon sparse optimal control. J. Optim. Theory Appl. 172(2), 481–517 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Stadler, G.: Elliptic optimal control problems with L\(^1\)-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44, 159–181 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Chittaro, F.C., Poggiolini, L.: Optimality conditions for extremals containing bang and inactivated arcs. In: 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pp. 1975–1980 (2017).
  12. 12.
    Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Stefani, G., Zezza, P.: Variational Methods in Imaging and Geometric Control, chap. A Hamiltonian Approach to Sufficiency in Optimal Control with Minimal Regularity Conditions: Part I. De Gruyter, Berlin (2016)Google Scholar
  14. 14.
    Agrachev, A., Stefani, G., Zezza, P.: Strong optimality for a bang–bang trajectory. SIAM J. Control Optim. 41(4), 991–1014 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Poggiolini, L., Stefani, G.: State-local optimality of a bang-bang trajectory: a Hamiltonian approach. Syst. Control Lett. 53(2), 269–279 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Agrachev, A., Stefani, G., Zezza, P.: An invariant second variation in optimal control. Int. J. Control 71(5), 689–715 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Poggiolini, L.: On local state optimality of bang-bang extremals in a free horizon Bolza problem. Rendiconti del seminario matematico italiano 64, 1–23 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Clarke, F.H.: On the inverse function theorem. Pac. J. Math. 64(1), 97–102 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Aix Marseille Univ, Université de ToulonCNRS, LISMarseilleFrance
  2. 2.Dipartimento di Matematica e Informatica “Ulisse Dini”Università degli Studi di FirenzeFlorenceItaly

Personalised recommendations