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

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Abstract

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.

Supported by the Russian Foundation for Basic Research, Project no. 18-01-00026.

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References

  1. Benabdellah, H.: Existence of solutions to the nonconvex sweeping process. J. Differential Equ. 164, 286–295 (2000). https://doi.org/10.1006/jdeq.1999.3756

    Article  MathSciNet  MATH  Google Scholar 

  2. Bensoussan, A., Turi, J.: Optimal control of variational inequalities. Commun. Inf. Syst. 10(4), 203–220 (2010). https://doi.org/10.4310/CIS.2010.v10.n4.a3

    Article  MathSciNet  MATH  Google Scholar 

  3. Brokate, M.: Optimal Streuerungen von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ.Peter D. Lang Verlag, Frankfurt am Main (1987)

    Google Scholar 

  4. Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Series of Applied Mathematical Sciences, vol. 121. Springer, New York (1996). https://doi.org/10.1007/978-1-4612-4048-8

    Book  MATH  Google Scholar 

  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). https://doi.org/10.3934/dcdsb.2013.18.331

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.3934/dcdsb.2017014

    Article  MATH  Google Scholar 

  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). ftp://ftp4.de.freesbie.org/pub/EMIS/journals/PM/53f1/pm53f106.pdf

    MathSciNet  MATH  Google Scholar 

  8. Colombo, G., Goncharov, V.V.: The sweeping processes without convexity. Set-Valued Anal. 7, 357–374 (1999). https://doi.org/10.1023/A:100877452

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.jde.2015.10.039

    Article  MathSciNet  MATH  Google Scholar 

  10. Dorroh, J.R., Ferreyra, G.: A multistate, multicontrol problem with unbounded controls. SIAM J. Control Optim. 32, 1322–1331 (1994). https://doi.org/10.1137/S0363012992229823

    Article  MathSciNet  MATH  Google Scholar 

  11. Dykhta, V.A., Samsonyuk, O.N.: Optimal Impulsive Control with Applications, 2nd edn. Fizmatlit, Moscow (2003)

    MATH  Google Scholar 

  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). https://doi.org/10.1134/S0965542509060050

    Article  MathSciNet  MATH  Google Scholar 

  13. Dykhta, V.A., Samsonyuk, O.N.: Hamilton-Jacobi Inequalities and Variational Optimality Conditions. ISU, Irkutsk (2015)

    MATH  Google Scholar 

  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). https://doi.org/10.3934/dcdss.2018061

    Article  MathSciNet  Google Scholar 

  15. Gudovich, A., Quincampoix, M.: Optimal control with hysteresis nonlinearity and multidimensional play operator. SIAM J. Control. Optim. 49(2), 788–807 (2011). https://doi.org/10.1137/090770011

    Article  MathSciNet  MATH  Google Scholar 

  16. Gurman, V.I.: The Extension Principle in Optimal Control Problems, 2nd edn. Fizmatlit, Moscow (1997)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1016/j.ejcon.2014.09.003

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1051/cocv/2014053

    MathSciNet  MATH  Google Scholar 

  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. 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). https://doi.org/10.1016/j.jde.2016.08.025

    Article  MathSciNet  MATH  Google Scholar 

  22. Krasnoselskii, M.A., Pokrovskii, A.V.: Systems with Hysteresis. Springer, Heidelberg (1989). https://doi.org/10.1007/978-3-642-61302-9

    Book  Google Scholar 

  23. Krejčí, P.: Vector hysteresis models. Eur. J. Appl. Math. 2, 281–292 (1991). https://doi.org/10.1017/S0956792500000541

    Article  MathSciNet  MATH  Google Scholar 

  24. Krejčí, P., Liero, M.: Rate independent Kurzweil process. Appl. Math. 54, 117–145 (2009). https://doi.org/10.1007/s10492-009-0009-5

    Article  MathSciNet  MATH  Google Scholar 

  25. Krejčí, P., Recupero, V.: Comparing BV solutions of rate independent processes. J. Convex. Anal. 21, 121–146 (2014)

    MathSciNet  MATH  Google Scholar 

  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). http://ncmm.karlin.mff.cuni.cz/preprints/1042115110pr12.pdf

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1007/3-540-45501-9_1

    Chapter  Google Scholar 

  28. Miller, B.M.: The generalized solutions of nonlinear optimization problems with impulse control. SIAM J. Control Optim. 34, 1420–1440 (1996). https://doi.org/10.1137/S0363012994263214

    Article  MathSciNet  MATH  Google Scholar 

  29. Miller, B.M., Rubinovich, E.Ya.: Impulsive Controls in Continuous and Discrete-Continuous Systems. Kluwer Academic Publishers, New York (2003). https://doi.org/10.1007/978-1-4615-0095-7

    Book  Google Scholar 

  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). https://doi.org/10.1134/S0005117913120047

    Article  MathSciNet  Google Scholar 

  31. Moreau, J.-J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differential Equ. 26, 347–374 (1977). https://doi.org/10.1016/0022-0396(77)90085-7

    Article  MathSciNet  MATH  Google Scholar 

  32. Moreau, J.-J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3–4), 329–349 (1999). https://doi.org/10.1016/S0045-7825(98)00387-9

    Article  MathSciNet  MATH  Google Scholar 

  33. Motta, M., Rampazzo, F.: Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls. Differential Integral Equ. 8, 269–288 (1995)

    MathSciNet  MATH  Google Scholar 

  34. Recupero, V., Santambrogio, F.: Sweeping processes with prescribed behavior on jumps. ArXiv:1707.09765 (2017)

  35. Recupero, V.: BV continuous sweeping processes. J. Differential Equ. 259, 4253–4272 (2015). https://doi.org/10.1016/j.jde.2015.05.019

    Article  MathSciNet  MATH  Google Scholar 

  36. Samsonyuk, O.N.: Invariant sets for nonlinear impulsive control systems. Autom. Remote Control 76(3), 405–418 (2015). https://doi.org/10.1134/S0005117915030054

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1109/CNSA.2017.7974010

  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). https://doi.org/10.1109/STAB.2018.8408396

  39. Sesekin, A.N., Zavalishchin, S.T.: Dynamic Impulse Systems: Theory and Applications. Kluwer Academic Publishers, Dordrecht (1997). https://doi.org/10.1007/978-94-015-8893-5

    Book  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-1-4613-0263-6_15

    Chapter  Google Scholar 

  41. Staritsyn, M.: On “discontinuous” continuity equation and impulsive ensemble control. Syst. Control Lett. 118, 77–83 (2018). https://doi.org/10.1016/j.sysconle.2018.06.001

    Article  MathSciNet  MATH  Google Scholar 

  42. Thibault, L.: Regularization of nonconvex sweeping process in Hilbert space. Set-Valued Anal. 16, 319–333 (2008). https://doi.org/10.1007/s11228-008-0083-y

    Article  MathSciNet  MATH  Google Scholar 

  43. Thibault, L.: Moreau sweeping process with bounded truncated retraction. J. Convex Anal. 23(4), 1051–1098 (2016). http://www.heldermann.de/JCA/JCA23/JCA234/jca23039.htm

    MathSciNet  MATH  Google Scholar 

  44. Valadier, M.: Lipschitz approximation of the sweeping (or Moreau) process. J. Differential Equ. 88, 248–264 (1990). https://doi.org/10.1016/0022-0396(90)90098-A

    Article  MathSciNet  MATH  Google Scholar 

  45. Visintin, A.: Differential Models of Hysteresis. Series in Applied Mathematical Sciences, vol. 111. Springer, Berlin (1994). https://doi.org/10.1007/978-3-662-11557-2

    Book  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russia

    Olga N. Samsonyuk

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  1. Olga N. Samsonyuk
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Correspondence to Olga N. Samsonyuk .

Editor information

Editors and Affiliations

  1. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Yury Evtushenko

  2. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  3. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  4. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    Yury Kochetov

  5. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Vlasta Malkova

  6. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Mikhail Posypkin

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Samsonyuk, O.N. (2019). The Space-Time Representation for Optimal Impulsive Control Problems with Hysteresis. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_25

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  • DOI: https://doi.org/10.1007/978-3-030-10934-9_25

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  • Online ISBN: 978-3-030-10934-9

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