Skip to main content
SpringerLink
Log in

A Maximum Principle for the Controlled Sweeping Process

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

We consider the free endpoint Mayer problem for a controlled Moreau process, the control acting as a perturbation of the dynamics driven by the normal cone, and derive necessary optimality conditions of Pontryagin’s Maximum Principle type. The results are also discussed through an example. We combine techniques from Sene and Thibault (Journal of Nonlinear and Convex Analysis 15, 647–663, 2014) and from Brokate and Krejčí (Discrete and Continuous Dynamical Systems Series B 18, 331–348, 2013), which in particular deals with a different but related control problem. Our assumptions include the smoothness of the boundary of the moving set C(t), but, differently from Brokate and Krejčí, do not require strict convexity and time independence of C(t). Rather, a kind of inward/outward pointing condition is assumed on the reference optimal trajectory at the times where the boundary of C(t) is touched. The state space is finite dimensional.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adly, S., Nacry, F., Thibault, L.: Preservation of prox-regularity of sets with application to constrained optimization. SIAM J. Optim. 26, 448–473 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Soner, H.M.: Level set approach to mean curvature flow in arbitrary codimension. J. Differential Geom. 43, 693–737 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aubin, J.-P., Cellina, A.: Differential Inclusions. Set-valued Maps and Viability Theory. Springer (1984)

  4. Barles, G., Briani, A, Trélat, E.: Value function and optimal trajectories for regional control problems via dynamic programming and pontryagin maximum principles. arXiv:1605.04079

  5. Bressan, A., Piccoli, B.: Introduction to the mathematical theory of control. AIMS (2007)

  6. Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Discrete and Continuous Dynamical Systems Series B 18, 331–348 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control. Birkhäuser, Boston (2004)

  8. Cao, T.H., Mordukhovich, B.Sh.: Optimal control of a perturbed sweeping process via discrete approximations, to appear in disc. Cont. Dyn. Syst., Ser B. arXiv:1511.08922

  9. Cao, T.H., Mordukhovich, B.Sh: OptiMality conditions for a controlled sweeping process with applications to the crowd motion model, to appear in disc. Cont. Dyn. Syst., Ser B. arXiv:1511.08923

  10. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory Graduate Texts in Mathematics, vol. 178. Springer, New York (1998)

    Google Scholar 

  11. Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.Sh.: Optimal control of the sweeping process. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19(1-2), 117–159 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.Sh.: Discrete approximations of a controlled sweeping process. Set-Valued and Variational Analysis 23, 69–86 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.Sh.: Optimal control of the sweeping process over polyhedral controlled sets. J. Differential Equations 260, 3397–3447 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Colombo, G., Palladino, M.: The minimum time function for the controlled Moreau’s sweeping process. SIAM J. Control 54, 2036–2062 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Colombo, G., Thibault, L.: Prox-regular sets and applications. In: Handbook of Nonconvex Analysis and Applications, pp 99–182. International Press (2010)

  16. Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction. Basel, Birkhäuser (1993)

    Book  MATH  Google Scholar 

  17. Rindler, F.: Optimal control for nonconvex rate-independent evolution processes. SIAM J. Control Optim. 47, 2773–2794 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rindler, F.: Approximation of rate-independent optimal control problems. SIAM J. Numer. Anal. 47, 3884–3909 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sene, M., Thibault, L.: Regularization of dynamical systems associated with prox-regular moving sets. Journal of Nonlinear and Convex Analysis 15, 647–663 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Serea, O.S.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42, 559–575 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Serea, O.S.: Optimality conditions for reflecting boundary control problems. Nonlinear Differ. Equ. Appl. 20, 1225–1242 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sussmann, H.J.: A pontryagin maximum principle for systems of flows. In: Blondel, V.D., Boyd, S.P., Kimura, H. (eds.) Recent Advances in Learning and Control. Lecture Notes in Control and Information Sciences, vol. 371. Springer, London (2008)

    Google Scholar 

  23. Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193, 1–26 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tolstonogov, A.A.: Control sweeping processes. J. Convex Anal. 23(4) (2016)

  25. Vinter, R.B.: Optimal Control. Birkhäuser, Boston (2000)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jijel University, Jijel, Algeria

    Chems Eddine Arroud

  2. Mila University Center, Mila, Algeria

    Chems Eddine Arroud

  3. Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Padova, Italy

    Giovanni Colombo

  4. I.N.d.A.M Research Unit, via Trieste 63, 35121, Padova, Italy

    Giovanni Colombo

Authors
  1. Chems Eddine Arroud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giovanni Colombo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Colombo.

Additional information

This work was done while the first author was visiting the Department of Mathematics of Padova University, funded by Programme Boursier “PNE” du Ministère de l’Einsegnement Supérieur et de la Recherche Scientifique, République Algérienne. The second author is partially supported by Padova University Research Project PRAT 2015 “Control of dynamics with active constraints”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arroud, C.E., Colombo, G. A Maximum Principle for the Controlled Sweeping Process. Set-Valued Var. Anal 26, 607–629 (2018). https://doi.org/10.1007/s11228-017-0400-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-017-0400-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation