The Space-Time Representation for Optimal Impulsive Control Problems with Hysteresis

  • Olga N. SamsonyukEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)


An optimal control problem for a sweeping process driven by impulsive controls is considered. The control system we study is described by both a measure-driven differential equation and a differential inclusion. This system is the impulsive-trajectory relaxation of an ordinary control system with nonlinearity of hysteresis type, in which the hysteresis is modeled by the play operator and considered as a particular case of a nonconvex sweeping process. The concept of a sweeping process for the so-called graph completions of functions of bounded variation, defining the corresponding moving set, is developed. The space-time representation based on the singular space-time transformation and a method to obtain optimality conditions for impulsive processes are proposed. By way of motivation, an example from mathematical economics is considered.


Measure-driven differential equations Sweeping process Rate independent hysteresis Impulsive control Space-time representation Optimal control 



The work is partially supported by the Russian Foundation for Basic Research, Project no. 18-01-00026.


  1. 1.
    Benabdellah, H.: Existence of solutions to the nonconvex sweeping process. J. Differential Equ. 164, 286–295 (2000). Scholar
  2. 2.
    Bensoussan, A., Turi, J.: Optimal control of variational inequalities. Commun. Inf. Syst. 10(4), 203–220 (2010). Scholar
  3. 3.
    Brokate, M.: Optimal Streuerungen von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ.Peter D. Lang Verlag, Frankfurt am Main (1987)Google Scholar
  4. 4.
    Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Series of Applied Mathematical Sciences, vol. 121. Springer, New York (1996). Scholar
  5. 5.
    Brokate, M., Krejc̆í, P.: Optimal control of ODE systems involving a rate independent variational inequality. Disc. Cont. Dyn. Syst. Ser. B. 18(2), 331–348 (2013). Scholar
  6. 6.
    Cao, T.H., Mordukhovich, B.S.: Optimality conditions for a controlled sweeping processwith applications to the crowd motion model. Discrete Contin. Dyn. Syst. Ser. B 21, 267–306 (2017). Scholar
  7. 7.
    Castaing, C., Monteiro Marques, M.D.P.: Evolution problems associated with nonconvex closed moving sets with bounded variation. Port. Math. 53, 73–87 (1996). Scholar
  8. 8.
    Colombo, G., Goncharov, V.V.: The sweeping processes without convexity. Set-Valued Anal. 7, 357–374 (1999). Scholar
  9. 9.
    Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. J. Differential Equ. 260, 3397–3447 (2016). Scholar
  10. 10.
    Dorroh, J.R., Ferreyra, G.: A multistate, multicontrol problem with unbounded controls. SIAM J. Control Optim. 32, 1322–1331 (1994). Scholar
  11. 11.
    Dykhta, V.A., Samsonyuk, O.N.: Optimal Impulsive Control with Applications, 2nd edn. Fizmatlit, Moscow (2003)zbMATHGoogle Scholar
  12. 12.
    Dykhta, V.A., Samsonyuk, O.N.: A maximum principle for smooth optimal impulsive control problems with multipoint state constraints. Comput. Math. Math. Phys. 49, 942–957 (2009). Scholar
  13. 13.
    Dykhta, V.A., Samsonyuk, O.N.: Hamilton-Jacobi Inequalities and Variational Optimality Conditions. ISU, Irkutsk (2015)zbMATHGoogle Scholar
  14. 14.
    Goncharova, E., Staritsyn, M.: On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations. Disc. Cont. Dyn. Syst. Ser. 11(6), 1061–1070 (2018). Scholar
  15. 15.
    Gudovich, A., Quincampoix, M.: Optimal control with hysteresis nonlinearity and multidimensional play operator. SIAM J. Control. Optim. 49(2), 788–807 (2011). Scholar
  16. 16.
    Gurman, V.I.: The Extension Principle in Optimal Control Problems, 2nd edn. Fizmatlit, Moscow (1997)zbMATHGoogle Scholar
  17. 17.
    Henry, C.: An existence theorem for a class of differential equations with multivalued right-hand side. J. Math. Anal. Appl. 41, 179–186 (1973)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Karamzin, D.Yu., Oliveira, V.A., Pereira, F.L., Silva, G.N.: On some extension of optimal control theory. Eur. J. Control 20(6), 284–291 (2014). Scholar
  19. 19.
    Karamzin, D.Yu., Oliveira, V.A., Pereira, F.L., Silva, G.N.: On the properness of the extension of dynamic optimization problems to allow impulsive controls. ESAIM Control Optim. Calculus Var. 21(3), 857–875 (2015). Scholar
  20. 20.
    Kopfová, J.: BV-norm continuity of the play operator. In: 8th Workshop on Multi-Rate Processes and Hysteresis and the HSFS Workshop (Hysteresis and Slow-Fast Systems) (2016)Google Scholar
  21. 21.
    Kopfova, J., Recupero, V.: BV-norm continuity of sweeping processes driven by a set with constant shape. J. Differential Equ. 261(10), 5875–5899 (2016). Scholar
  22. 22.
    Krasnoselskii, M.A., Pokrovskii, A.V.: Systems with Hysteresis. Springer, Heidelberg (1989). Scholar
  23. 23.
    Krejčí, P.: Vector hysteresis models. Eur. J. Appl. Math. 2, 281–292 (1991). Scholar
  24. 24.
    Krejčí, P., Liero, M.: Rate independent Kurzweil process. Appl. Math. 54, 117–145 (2009). Scholar
  25. 25.
    Krejčí, P., Recupero, V.: Comparing BV solutions of rate independent processes. J. Convex. Anal. 21, 121–146 (2014)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Krejčí, P., Roche, T.: Lipschitz continuous data dependence of sweeping processes in BV spaces. Disc. Cont. Dyn. Syst. Ser. B. 15, 637–650 (2011). Scholar
  27. 27.
    Kunze, M., Marques, M.M.: An introduction to Moreau’s sweeping process. In: Brogliato, B. (ed.) Impacts in Mechanical Systems. Lecture Notes in Physics, vol. 551, pp. 1–60. Springer, Berlin (2000). Scholar
  28. 28.
    Miller, B.M.: The generalized solutions of nonlinear optimization problems with impulse control. SIAM J. Control Optim. 34, 1420–1440 (1996). Scholar
  29. 29.
    Miller, B.M., Rubinovich, E.Ya.: Impulsive Controls in Continuous and Discrete-Continuous Systems. Kluwer Academic Publishers, New York (2003). Scholar
  30. 30.
    Miller, B.M., Rubinovich, E.Ya.: Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations. Autom. Remote Control 74, 1969–2006 (2013). Scholar
  31. 31.
    Moreau, J.-J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differential Equ. 26, 347–374 (1977). Scholar
  32. 32.
    Moreau, J.-J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3–4), 329–349 (1999). Scholar
  33. 33.
    Motta, M., Rampazzo, F.: Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls. Differential Integral Equ. 8, 269–288 (1995)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Recupero, V., Santambrogio, F.: Sweeping processes with prescribed behavior on jumps. ArXiv:1707.09765 (2017)
  35. 35.
    Recupero, V.: BV continuous sweeping processes. J. Differential Equ. 259, 4253–4272 (2015). Scholar
  36. 36.
    Samsonyuk, O.N.: Invariant sets for nonlinear impulsive control systems. Autom. Remote Control 76(3), 405–418 (2015). Scholar
  37. 37.
    Samsonyuk, O.N., Timoshin, S.A.: Optimal impulsive control problems with hysteresis. In: Constructive Nonsmooth Analysis and Related Topics (dedicated to the Memory of V.F. Demyanov), CNSA–2017, pp. 276–280 (2017).
  38. 38.
    Samsonyuk, O.N., Tolkachev, D.E.: Approximation results for impulsive control systems with hysteresis. In: Tkhai, V.N. (ed.) 14th International Conference “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference) (STAB) (2018).
  39. 39.
    Sesekin, A.N., Zavalishchin, S.T.: Dynamic Impulse Systems: Theory and Applications. Kluwer Academic Publishers, Dordrecht (1997). Scholar
  40. 40.
    Siddiqi, A.H., Manchanda, P., Brokate, M.: On some recent developments concerning Moreau’s sweeping process. In: Siddiqi, A.H., Kocvara, M. (eds.) Trends in Industrial and Applied Mathematics. Applied Optimization, vol. 72, pp. 339–354. Springer, Boston (2002). Scholar
  41. 41.
    Staritsyn, M.: On “discontinuous” continuity equation and impulsive ensemble control. Syst. Control Lett. 118, 77–83 (2018). Scholar
  42. 42.
    Thibault, L.: Regularization of nonconvex sweeping process in Hilbert space. Set-Valued Anal. 16, 319–333 (2008). Scholar
  43. 43.
    Thibault, L.: Moreau sweeping process with bounded truncated retraction. J. Convex Anal. 23(4), 1051–1098 (2016). Scholar
  44. 44.
    Valadier, M.: Lipschitz approximation of the sweeping (or Moreau) process. J. Differential Equ. 88, 248–264 (1990). Scholar
  45. 45.
    Visintin, A.: Differential Models of Hysteresis. Series in Applied Mathematical Sciences, vol. 111. Springer, Berlin (1994). Scholar

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Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control Theory SB RASIrkutskRussia

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