Advertisement

Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 256–289 | Cite as

Extended Euler–Lagrange and Hamiltonian Conditions in Optimal Control of Sweeping Processes with Controlled Moving Sets

  • Nguyen D. Hoang
  • Boris S. MordukhovichEmail author
Article
  • 88 Downloads

Abstract

This paper concerns optimal control problems for a class of sweeping processes governed by discontinuous unbounded differential inclusions that are described via normal cone mappings to controlled moving sets. Largely motivated by applications to hysteresis, we consider a general setting where moving sets are given as inverse images of closed subsets of finite-dimensional spaces under nonlinear differentiable mappings dependent on both state and control variables. Developing the method of discrete approximations and employing generalized differential tools of first-order and second-order variational analysis allow us to derive nondegenerate necessary optimality conditions for such problems in extended Euler–Lagrange and Hamiltonian forms involving the Hamiltonian maximization. The latter conditions of the Pontryagin Maximum Principle type are the first in the literature for optimal control of sweeping processes with control-dependent moving sets.

Keywords

Optimal control Sweeping process Variational analysis Discrete approximations Generalized differentiation Euler–Lagrange and Hamiltonian formalisms Maximum principle Rate-independent operators 

Mathematics Subject Classification

49J52 49J53 49K24 49M25 90C30 

Notes

Acknowledgements

Research of the second author was partly supported by the USA National Science Foundation under grants DMS-1512846 and DMS-1808978, and by the USA Air Force Office of Scientific Research under Grant #15RT0462.

References

  1. 1.
    Moreau, J.J.: On unilateral constraints, friction and plasticity. In: Capriz, G., Stampacchia, G. (eds.) Proceedings from CIME New Variational Techniques in Mathematical Physics, pp. 173–322. Cremonese, Rome (1974)Google Scholar
  2. 2.
    Colombo, G., Thibault, L.: Prox-regular sets and applications. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis, pp. 99–182. International Press, Boston (2010)Google Scholar
  3. 3.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process. Dyn. Contin. Discrete Impuls. Syst. Ser. B 19, 117–159 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Clarke, F.H.: Necessary conditions in dynamics optimization. Mem. Am. Math. Soc. 344(816), 299 (2005)MathSciNetGoogle Scholar
  5. 5.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)CrossRefGoogle Scholar
  6. 6.
    Vinter, R.B.: Optimal Control. Birkhaüser, Boston (2000)zbMATHGoogle Scholar
  7. 7.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. J. Differ. Equ. 260, 3397–3447 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mordukhovich, B.S.: Discrete approximations and refined Euler–Lagrange conditions for differential inclusions. SIAM J. Control Optim. 33, 882–915 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cao, T.H., Mordukhovich, B.S.: Optimal control of a perturbed sweeping process via discrete approximations. Disc. Contin. Dyn. Syst. Ser. B 21, 3331–3358 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cao, T.H., Mordukhovich, B.S.: Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Disc. Contin. Dyn. Syst. Ser. B 22, 267–306 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cao, T.H., Mordukhovich, B.S.: Optimal control of a nonconvex perturbed sweeping process. J. Differ. Equ. (2018).  https://doi.org/10.1016/j.jde.2018.07.066 zbMATHGoogle Scholar
  12. 12.
    Donchev, T., Farkhi, E., Mordukhovich, B.S.: Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces. J. Differ. Equ. 243, 301–328 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Briceño-Arias, L.M., Hoang, N.D., Peypouquet, J.: Existence, stability and optimality for optimal control problems governed by maximal monotone operators. J. Differ. Equ. 260, 733–757 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)Google Scholar
  15. 15.
    Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Disc. Contin. Dyn. Syst. Ser. B 18, 331–348 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Arroud, C.E., Colombo, G.: A maximum principle of the controlled sweeping process. Set Valued Var. Anal. 26, 607–629 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    de Pinho, M.D.R., Ferreira, M.M.A., Smirnov, G.V.: Optimal control involving sweeping processes. Set-Valued Var. Anal. (to appear)Google Scholar
  18. 18.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mordukhovich, B.S.: Variational Analysis and Applications. Springer, New York (2018)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mordukhovich, B.S., Rockafellar, R.T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22, 953–986 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Henrion, R., Outrata, J.V., Surowiec, T.: On the coderivative of normal cone mappings to inequality systems. Nonlinear Anal. 71, 1213–1226 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mordukhovich, B.S., Outrata, J.V.: Coderivative analysis of quasi-variational inequalities with mapplications to stability and optimization. SIAM J. Optim. 18, 389–412 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Edmond, J.F., Thibault, L.: Relaxation of an optimal control problem involving a perturbed sweeping process. Math. Program. 104, 347–373 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tolstonogov, A.A.: Control sweeping process. J. Convex Anal. 23, 1099–1123 (2016)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Dubovitskii, A.Y., Milyutin, A.A.: Extremum problems in the presence of restrictions. USSR Comput. Math. Math. Phys. 5, 1–80 (1965)CrossRefzbMATHGoogle Scholar
  26. 26.
    Arutyunov, A.V., Aseev, S.M.: Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints. SIAM J. Control Optim. 35, 930–952 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Arutyunov, A.V., Karamzin, DYu.: Nondegenerate necessary optimality conditions for the optimal control problems with equality-type state constraints. J. Global Optim. 64, 623–647 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Brogliato, B.: Nonsmooth Mechanics: Models, Dynamics and Control, 3rd edn. Springer, Cham (2016)CrossRefzbMATHGoogle Scholar
  29. 29.
    Razavy, M.: Classical and Quantum Dissipative Systems. World Scientific, Singapore (2005)zbMATHGoogle Scholar
  30. 30.
    Adly, S., Haddad, T., Thibault, L.: Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. Math. Program. 148, 5–47 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Herzog, R., Meyer, C., Wachsmuth, G.: B-and strong stationarity for optimal control of static plasticity with hardening. SIAM J. Optim. 23, 321–352 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA

Personalised recommendations