Controllability of dynamical systems with Play-type hysteresis via approximation by delayed relays

Fabio Bagagiolo; Vincenzo Recupero; Marta Zoppello; Fabio Bagagiolo; Vincenzo Recupero; Marta Zoppello

doi:10.3934/mine.2025018

Mathematics in Engineering

2025, Volume 7, Issue 4: 439-463. doi: 10.3934/mine.2025018

Previous Article Next Article

Research article Special Issues

Controllability of dynamical systems with Play-type hysteresis via approximation by delayed relays

1.
Department of Mathematics, University of Trento, Italy
2.
Department of Mathematical Sciences, Politecnico di Torino, Italy; vincenzo.recupero@polito.it, marta.zoppello@polito.it

Received: 06 February 2025 Revised: 09 June 2025 Accepted: 29 June 2025 Published: 08 July 2025

In this paper we study the controllability problem for systems exhibiting hysteresis represented by play-type operators. To this end we first formalize and study in a functional setting the approximation of the Play operators by a finite weighted sum of delayed relay. Then we prove the controllability for the case with the Play operator, by the controllability result for the case with the weighted sum of delayed relay. Finally, we discuss potential applications of our approach to the sweeping process.
- controllability of differential systems,
- hysteresis
Citation: Fabio Bagagiolo, Vincenzo Recupero, Marta Zoppello. Controllability of dynamical systems with Play-type hysteresis via approximation by delayed relays[J]. Mathematics in Engineering, 2025, 7(4): 439-463. doi: 10.3934/mine.2025018

Related Papers:

Abstract

In this paper we study the controllability problem for systems exhibiting hysteresis represented by play-type operators. To this end we first formalize and study in a functional setting the approximation of the Play operators by a finite weighted sum of delayed relay. Then we prove the controllability for the case with the Play operator, by the controllability result for the case with the weighted sum of delayed relay. Finally, we discuss potential applications of our approach to the sweeping process.

References

[1]	L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford university press, 2000.
[2]	A. A. Agrachev, Y. L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, Vol. 87, Springer-Verlag, 2004. https://doi.org/10.1007/978-3-662-06404-7
[3]	F. Bagagiolo, Dynamic programming for some optimal control problems with hysteresis, Nonlinear Differ. Equ. Appl., 9 (2002), 149–174. https://doi.org/10.1007/s00030-002-8122-0 doi: 10.1007/s00030-002-8122-0
[4]	F. Bagagiolo, On the controllability of the semilinear heat equation with hysteresis, Phys. B, 407 (2012), 1401–1403. https://doi.org/10.1016/j.physb.2011.10.014 doi: 10.1016/j.physb.2011.10.014
[5]	F. Bagagiolo, K. Danieli, Infinite horizon optimal control problems with multiple thermostatic hybrid dynamics, Nonlinear Anal., 6 (2012), 824–838. https://doi.org/10.1016/j.nahs.2011.09.001 doi: 10.1016/j.nahs.2011.09.001
[6]	F. Bagagiolo, R. Maggistro, M. Zoppello, Swimming by switching, Meccanica, 52 (2017), 3499–3511. https://doi.org/10.1007/s11012-017-0620-6
[7]	F. Bagagiolo, R. Maggistro, M. Zoppello, A hybrid differential game with switching thermostatic-type dynamics and costs, Minimax Theory Appl., 5 (2020), 151–180.
[8]	F. Bagagiolo, M. Zoppello, Hysteresis and controllability of affine driftless systems: some case studies, Math. Model. Nat. Phenom., 15 (2020), 55. https://doi.org/10.1051/mmnp/2020023 doi: 10.1051/mmnp/2020023
[9]	M. Brokate, J. Sprekels, Hysteresis and phase transitions, Applied Mathematical Sciences, Springer-Verlag, 1996. https://doi.org/10.1007/978-1-4612-4048-8
[10]	J. M. Coron, Control and nonlinearity: mathematical surveys and monographs, American Mathematical Society, 2007.
[11]	E. Feleqi, F. Rampazzo, Iterated Lie brackets for nonsmooth vector fields, Nonlinear Differ. Equ. Appl., 24 (2017), 61. https://doi.org/10.1007/s00030-017-0484-4 doi: 10.1007/s00030-017-0484-4
[12]	C. Gavioli, P. Krejčí, Control and controllability of PDEs with hysteresis, Appl. Math. Optim., 84 (2021), 829–847. https://doi.org/10.1007/s00245-020-09663-6 doi: 10.1007/s00245-020-09663-6
[13]	I. Gudoshnikov, O. Makarenkov, D. Rachinskii, Finite-time stability of polyhedral sweeping processes with application to elastoplastic systems, SIAM J. Control Optim., 60 (2022), 1320–1346. https://doi.org/10.1137/20M1388796 doi: 10.1137/20M1388796
[14]	M. A. Krasnosel'skiǐ, A. V. Pokrovskiǐ, Systems with hysteresis, Springer-Verlag, 1989.
[15]	M. I. Krastanov, M. Quincampoix, On the small-time controllability of discontinuous piece-wise linear systems, Syst. Control Lett., 62 (2013), 218–223. https://doi.org/10.1016/j.sysconle.2012.11.006 doi: 10.1016/j.sysconle.2012.11.006
[16]	P. Krejčí, Hysteresis, convexity and dissipation in hyperbolic equations, Gakuto International Series Mathematical Sciences and Applications, Vol. 8, Tokyo: Gakkotosho, 1997.
[17]	P. Krejčí, The Kurzweil integral and hysteresis, J. Phys.: Conf. Ser., 55 (2006), 144–154. https://doi.org/10.1088/1742-6596/55/1/014 doi: 10.1088/1742-6596/55/1/014
[18]	P. Krejčí, G. A. Monteiro, V. Recupero, Non-convex sweeping processes in the space of regulated functions, Commun. Pure Appl. Anal., 21 (2022), 2999–3029. https://doi.org/10.3934/cpaa.2022087 doi: 10.3934/cpaa.2022087
[19]	J. Liesen, V. Mehrmann, Linear algebra, Springer, 2015. https://doi.org/10.1007/978-3-319-24346-7
[20]	I. D. Mayergoyz, Mathematical models of hysteresis, Springer-Verlag, 1991. https://doi.org/10.1007/978-1-4612-3028-1
[21]	A. Mielke, T. Roubíček, Rate-independent systems: theory and application, Springer, 2015. https://doi.org/10.1007/978-1-4939-2706-7
[22]	J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations, 26 (1977), 347–374. https://doi.org/10.1016/0022-0396(77)90085-7 doi: 10.1016/0022-0396(77)90085-7
[23]	T. Murphey, J. Burdick, A local controllability test for nonlinear multiple model systems, Proceedings of the 2002 American Control Conference, 6 (2002), 4657–4661. https://doi.org/10.1109/ACC.2002.1025390 doi: 10.1109/ACC.2002.1025390
[24]	F. Rampazzo, H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields, J. Differ. Equations, 232 (2007), 134–175. https://doi.org/10.1016/j.jde.2006.04.016 doi: 10.1016/j.jde.2006.04.016
[25]	V. Recupero, $BV$-extension of rate independent operators, Math. Nachr., 282 (2009), 86–98. https://doi.org/10.1002/mana.200610723 doi: 10.1002/mana.200610723
[26]	V. Recupero, Sweeping processes and rate independence, J. Convex Anal., 23 (2016), 921–946.
[27]	S. Tarbouriech, I. Queinnec, C. Prieur, Stability analysis and stabilization of systems with input backlash, IEEE Trans. Autom. Control, 59 (2014), 488–494. https://doi.org/10.1109/TAC.2013.2273279 doi: 10.1109/TAC.2013.2273279
[28]	A. Visintin, Differential models of hsyteresis, Applied Mathematical Sciences, Vol. 111, Springer-Verlag, 1994. https://doi.org/10.1007/978-3-662-11557-2

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)