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Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach
On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator
Department of Mathematical Sciences, The University of Texas at Dallas, USA |
The sweeping process was proposed by J. J. Moreau as a general mathematical formalism for quasistatic processes in elastoplastic bodies. This formalism deals with connected Prandtl's elastic-ideal plastic springs, which can form a system with an arbitrarily complex topology. The model describes the complex relationship between stresses and elongations of the springs as a multi-dimensional differential inclusion (variational inequality). On the other hand, the Prandtl-Ishlinskii model assumes a very simple connection of springs. This model results in an input-output operator, which has many good mathematical properties and admits an explicit solution for an arbitrary input. It turns out that the sweeping processes can be reducible to the Prandtl-Ishlinskii operator even if the topology of the system of springs is complex. In this work, we analyze the conditions for such reducibility.
References:
[1] |
S. Adly, T. Haddad and L. Thibault,
Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program Ser. B, 148 (2014), 5-47.
doi: 10.1007/s10107-014-0754-4. |
[2] |
F. Al-Bender, W. Symens, J. Swevers and H. Van Brussel,
Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems, Int. J. Non-Linear Mechanics, 39 (2004), 1721-1735.
doi: 10.1016/j.ijnonlinmec.2004.04.005. |
[3] |
M. Brokate and J. Sprekels,
Hysteresis and Phase Transitions, Applied Mathematical Sciences, 121. Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[4] |
M. Brokate, P. Krejci and D. Rachinskii,
Some analytical properties of the multidimensional continuous Mroz model of plasticity, Control Cybernetics, 27 (1998), 199-215.
|
[5] |
G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich,
Optimal control of the sweeping process: The polyhedral case, J. Differ. Equ., 260 (2016), 3397-3447.
doi: 10.1016/j.jde.2015.10.039. |
[6] |
G. Colombo and M. D. P. Monteiro Marques,
Sweeping by a continuous prox-regular set, J. Diff. Equ., 187 (2003), 46-62.
doi: 10.1016/S0022-0396(02)00021-9. |
[7] |
G. Colombo and M. Palladino,
The minimum time function for the controlled Moreau's sweeping process, SIAM J. Control Optimization, 54 (2016), 2036-2062.
doi: 10.1137/15M1043364. |
[8] |
D. Davino, P. Krejci and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility,
Smart Mater. Struct. , 22 (2013), 095009. |
[9] |
D. Davino, P. Krejci, A. Pimenov, D. Rachinskii and C. Visone,
Analysis of an operator-differential model for magnetostrictive energy harvesting, Communications Nonlinear Science and Numerical Simulation, 39 (2016), 504-519.
doi: 10.1016/j.cnsns.2016.04.004. |
[10] |
W. Desch and J. Turi,
The Stop operator related to a convex polyhedron, J. Diff. Equ., 157 (1999), 329-347.
doi: 10.1006/jdeq.1998.3601. |
[11] |
S. Di Marino, B. Maury and F. Santambrogio,
Measure sweeping processes, J. Convex Anal., 23 (2016), 567-601.
|
[12] |
M. Eleuteri, J. Kopfova and P. Krejčı,
A new phase field model for material fatigue in an oscillating elastoplastic beam, Discrete Continuous Dynam. Systems - A, 35 (2015), 2465-2495.
doi: 10.3934/dcds.2015.35.2465. |
[13] |
G.-Y. Gu, L.-M. Zhu and C.-Y. Su,
Modeling and compensation of asymmetric hysteresis nonlinearity for piezoceramic actuators with a modified Prandtl-Ishlinskii model, IEEE Trans Industrial Electronics, 61 (2014), 1583-1595.
doi: 10.1109/TIE.2013.2257153. |
[14] |
A. Yu. Ishlinskii,
Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR Techn. Ser., 9 (1944), 583-590.
|
[15] |
M. A. Janaideh and P. Krejci,
Inverse rate-dependent Prandtl-Ishlinskii model for feedforward compensation of hysteresis in a piezomicropositioning actuator, IEEE/ASME Trans. Mechatronics, 18 (2013), 1498-1507.
doi: 10.1109/TMECH.2012.2205265. |
[16] |
B. Kaltenbacher and P. Krejci,
A thermodynamically consistent phenomenological model for ferroelectric and ferroelastic hysteresis, ZAMM - Z. Angew. Math. Mech., 96 (2016), 874-891.
doi: 10.1002/zamm.201400292. |
[17] |
M. Kamenskii and O. Makarenkov,
On the response of autonomous sweeping processes to periodic perturbations, Set-Valued and Variational Analysis, 24 (2016), 551-563.
doi: 10.1007/s11228-015-0348-1. |
[18] |
J. Kopfova and V. Recupero,
BV-norm continuity of sweeping processes driven by a set with constant shape, J. Differ. Equ., 261 (2016), 5875-5899.
doi: 10.1016/j.jde.2016.08.025. |
[19] |
M. Krasnosel'skii and A. Pokrovskii,
Systems with Hysteresis, Springer, 1989.
doi: 10.1007/978-3-642-61302-9. |
[20] |
P. Krejci,
Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho Co. Ltd, Tokyo, 1996. |
[21] |
P. Krejci and J. Sprekels,
Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Math. Meth. Appl. Sci., 30 (2007), 2371-2393.
doi: 10.1002/mma.892. |
[22] |
P. Krejci, H. Lamba, S. Melnik and D. Rachinskii, Analytical solution for a class of network dynamics with mechanical and financial applications,
Phys. Rev. E, 90 (2014), 032822. |
[23] |
P. Krejci, H. Lamba, S. Melnik and D. Rachinskii,
Kurzweil integral representation of interacting Prandtl-Ishlinskii operators, Discrete Continuous Dynam. Systems - B, 20 (2015), 2949-2965.
doi: 10.3934/dcdsb.2015.20.2949. |
[24] |
P. Krejci, H. Lamba, G. A. Monteiro and D. Rachinskii,
The Kurzweil integral in financial market modeling, Mathematica Bohemica, 141 (2016), 261-286.
doi: 10.21136/MB.2016.18. |
[25] |
P. Krejci and V. Recupero,
Comparing BV solutions of rate independent processes, J. Convex Analysis, 21 (2014), 121-146.
|
[26] |
P. Krejci and T. Roche,
Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Continuous Dynam. Systems - B, 15 (2011), 637-650.
doi: 10.3934/dcdsb.2011.15.637. |
[27] |
P. Krejci and A. Vladimirov,
Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Analysis, 11 (2003), 91-110.
doi: 10.1023/A:1021980201718. |
[28] |
K. Kuhnen,
Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418.
doi: 10.3166/ejc.9.407-418. |
[29] |
M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in Impacts in Mechanical Systems (ed. B. Brogliato), Lecture Notes in Physics, 551, Springer, (2000), 1-60.
doi: 10.1007/3-540-45501-9_1. |
[30] |
B. Maury, A. Roudneff-Chupin and F. Santambrogio,
Congestion-driven dendritic growth, Discrete Continuous Dynam. Systems, 34 (2014), 1575-1604.
doi: 10.3934/dcds.2014.34.1575. |
[31] |
I. Mayergoyz,
Mathematical Models of Hysteresis and Their Applications, Elsevier, 2003. |
[32] |
A. Mielke,
Generalized Prandtl-Ishlinskii operators arising from homogenization and dimension reduction, Physica B, 407 (2012), 1330-1335.
doi: 10.1016/j.physb.2011.10.013. |
[33] |
A. Mielke and T. Roubiček,
Rate Independent Systems, Theory and Applications, Springer, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[34] |
J. J. Moreau, On unilateral constraints, friction and plasticity, in New Variational Techniques in Mathematical Physics (eds. G. Capriz and G. Stampacchia), Springer, (1974), 171-322. |
[35] |
J. J. Moreau,
Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equ., 26 (1977), 347-374.
doi: 10.1016/0022-0396(77)90085-7. |
[36] |
J. J. Moreau, Application of convex analysis to the treatment of elastoplastic systems, in Applications of Methods of Functional Analysis to Problems in Mechanics (eds. P. Germain and B. Nayroles), Springer, (1976), 56-89.
doi: 10.1007/BFb0088746. |
[37] |
L. Prandtl,
Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106.
|
[38] |
D. I. Rachinskii,
Equivalent combinations of stops, Automat. Remote Control, 59 (1998), 1370-1378.
|
[39] |
V. Recupero,
The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci., 31 (2008), 1283-1295.
doi: 10.1002/mma.968. |
[40] |
V. Recupero,
BV continuous sweeping processes, J. Differ. Equ., 259 (2015), 4253-4272.
doi: 10.1016/j.jde.2015.05.019. |
[41] |
D. D. Rizos and S. D. Fassois,
Friction identification based upon the LuGre and Maxwell slip models, IEEE Trans Control Systems Technology, 17 (2009), 153-160.
|
[42] |
M. Ruderman,
Presliding hysteresis damping of LuGre and Maxwell-slip friction models, Mechatronics, 30 (2015), 225-230.
doi: 10.1016/j.mechatronics.2015.07.007. |
[43] |
M. Ruderman, F. Hoffmann and T. Bertram,
Modeling and identification of elastic robot joints with hysteresis and backlash, IEEE Trans. Industrial Electronics, 56 (2009), 3840-3847.
doi: 10.1109/TIE.2009.2015752. |
[44] |
I. Rychlik,
A new definition of the rainflow cycle counting method, Int. J. Fatigue, 9 (1987), 119-121.
doi: 10.1016/0142-1123(87)90054-5. |
[45] |
H. Sayyaadi and M. R. Zakerzadeh,
Position control of shape memory alloy actuator based on the generalized Prandtl-Ishlinskii inverse model, Mechatronics, 22 (2012), 945-957.
doi: 10.1016/j.mechatronics.2012.06.003. |
[46] |
J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts and J. D. Shore,
Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations, Phys. Rev. Lett., 70 (1993), 3347-3350.
doi: 10.1103/PhysRevLett.70.3347. |
[47] |
A. Visintin,
Differential Models of Hysteresis, Springer, 1994.
doi: 10.1007/978-3-662-11557-2. |
show all references
References:
[1] |
S. Adly, T. Haddad and L. Thibault,
Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program Ser. B, 148 (2014), 5-47.
doi: 10.1007/s10107-014-0754-4. |
[2] |
F. Al-Bender, W. Symens, J. Swevers and H. Van Brussel,
Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems, Int. J. Non-Linear Mechanics, 39 (2004), 1721-1735.
doi: 10.1016/j.ijnonlinmec.2004.04.005. |
[3] |
M. Brokate and J. Sprekels,
Hysteresis and Phase Transitions, Applied Mathematical Sciences, 121. Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[4] |
M. Brokate, P. Krejci and D. Rachinskii,
Some analytical properties of the multidimensional continuous Mroz model of plasticity, Control Cybernetics, 27 (1998), 199-215.
|
[5] |
G. Colombo, R. Henrion, N. D. Hoang and B. S. Mordukhovich,
Optimal control of the sweeping process: The polyhedral case, J. Differ. Equ., 260 (2016), 3397-3447.
doi: 10.1016/j.jde.2015.10.039. |
[6] |
G. Colombo and M. D. P. Monteiro Marques,
Sweeping by a continuous prox-regular set, J. Diff. Equ., 187 (2003), 46-62.
doi: 10.1016/S0022-0396(02)00021-9. |
[7] |
G. Colombo and M. Palladino,
The minimum time function for the controlled Moreau's sweeping process, SIAM J. Control Optimization, 54 (2016), 2036-2062.
doi: 10.1137/15M1043364. |
[8] |
D. Davino, P. Krejci and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility,
Smart Mater. Struct. , 22 (2013), 095009. |
[9] |
D. Davino, P. Krejci, A. Pimenov, D. Rachinskii and C. Visone,
Analysis of an operator-differential model for magnetostrictive energy harvesting, Communications Nonlinear Science and Numerical Simulation, 39 (2016), 504-519.
doi: 10.1016/j.cnsns.2016.04.004. |
[10] |
W. Desch and J. Turi,
The Stop operator related to a convex polyhedron, J. Diff. Equ., 157 (1999), 329-347.
doi: 10.1006/jdeq.1998.3601. |
[11] |
S. Di Marino, B. Maury and F. Santambrogio,
Measure sweeping processes, J. Convex Anal., 23 (2016), 567-601.
|
[12] |
M. Eleuteri, J. Kopfova and P. Krejčı,
A new phase field model for material fatigue in an oscillating elastoplastic beam, Discrete Continuous Dynam. Systems - A, 35 (2015), 2465-2495.
doi: 10.3934/dcds.2015.35.2465. |
[13] |
G.-Y. Gu, L.-M. Zhu and C.-Y. Su,
Modeling and compensation of asymmetric hysteresis nonlinearity for piezoceramic actuators with a modified Prandtl-Ishlinskii model, IEEE Trans Industrial Electronics, 61 (2014), 1583-1595.
doi: 10.1109/TIE.2013.2257153. |
[14] |
A. Yu. Ishlinskii,
Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR Techn. Ser., 9 (1944), 583-590.
|
[15] |
M. A. Janaideh and P. Krejci,
Inverse rate-dependent Prandtl-Ishlinskii model for feedforward compensation of hysteresis in a piezomicropositioning actuator, IEEE/ASME Trans. Mechatronics, 18 (2013), 1498-1507.
doi: 10.1109/TMECH.2012.2205265. |
[16] |
B. Kaltenbacher and P. Krejci,
A thermodynamically consistent phenomenological model for ferroelectric and ferroelastic hysteresis, ZAMM - Z. Angew. Math. Mech., 96 (2016), 874-891.
doi: 10.1002/zamm.201400292. |
[17] |
M. Kamenskii and O. Makarenkov,
On the response of autonomous sweeping processes to periodic perturbations, Set-Valued and Variational Analysis, 24 (2016), 551-563.
doi: 10.1007/s11228-015-0348-1. |
[18] |
J. Kopfova and V. Recupero,
BV-norm continuity of sweeping processes driven by a set with constant shape, J. Differ. Equ., 261 (2016), 5875-5899.
doi: 10.1016/j.jde.2016.08.025. |
[19] |
M. Krasnosel'skii and A. Pokrovskii,
Systems with Hysteresis, Springer, 1989.
doi: 10.1007/978-3-642-61302-9. |
[20] |
P. Krejci,
Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkotosho Co. Ltd, Tokyo, 1996. |
[21] |
P. Krejci and J. Sprekels,
Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Math. Meth. Appl. Sci., 30 (2007), 2371-2393.
doi: 10.1002/mma.892. |
[22] |
P. Krejci, H. Lamba, S. Melnik and D. Rachinskii, Analytical solution for a class of network dynamics with mechanical and financial applications,
Phys. Rev. E, 90 (2014), 032822. |
[23] |
P. Krejci, H. Lamba, S. Melnik and D. Rachinskii,
Kurzweil integral representation of interacting Prandtl-Ishlinskii operators, Discrete Continuous Dynam. Systems - B, 20 (2015), 2949-2965.
doi: 10.3934/dcdsb.2015.20.2949. |
[24] |
P. Krejci, H. Lamba, G. A. Monteiro and D. Rachinskii,
The Kurzweil integral in financial market modeling, Mathematica Bohemica, 141 (2016), 261-286.
doi: 10.21136/MB.2016.18. |
[25] |
P. Krejci and V. Recupero,
Comparing BV solutions of rate independent processes, J. Convex Analysis, 21 (2014), 121-146.
|
[26] |
P. Krejci and T. Roche,
Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Continuous Dynam. Systems - B, 15 (2011), 637-650.
doi: 10.3934/dcdsb.2011.15.637. |
[27] |
P. Krejci and A. Vladimirov,
Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Analysis, 11 (2003), 91-110.
doi: 10.1023/A:1021980201718. |
[28] |
K. Kuhnen,
Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418.
doi: 10.3166/ejc.9.407-418. |
[29] |
M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in Impacts in Mechanical Systems (ed. B. Brogliato), Lecture Notes in Physics, 551, Springer, (2000), 1-60.
doi: 10.1007/3-540-45501-9_1. |
[30] |
B. Maury, A. Roudneff-Chupin and F. Santambrogio,
Congestion-driven dendritic growth, Discrete Continuous Dynam. Systems, 34 (2014), 1575-1604.
doi: 10.3934/dcds.2014.34.1575. |
[31] |
I. Mayergoyz,
Mathematical Models of Hysteresis and Their Applications, Elsevier, 2003. |
[32] |
A. Mielke,
Generalized Prandtl-Ishlinskii operators arising from homogenization and dimension reduction, Physica B, 407 (2012), 1330-1335.
doi: 10.1016/j.physb.2011.10.013. |
[33] |
A. Mielke and T. Roubiček,
Rate Independent Systems, Theory and Applications, Springer, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[34] |
J. J. Moreau, On unilateral constraints, friction and plasticity, in New Variational Techniques in Mathematical Physics (eds. G. Capriz and G. Stampacchia), Springer, (1974), 171-322. |
[35] |
J. J. Moreau,
Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equ., 26 (1977), 347-374.
doi: 10.1016/0022-0396(77)90085-7. |
[36] |
J. J. Moreau, Application of convex analysis to the treatment of elastoplastic systems, in Applications of Methods of Functional Analysis to Problems in Mechanics (eds. P. Germain and B. Nayroles), Springer, (1976), 56-89.
doi: 10.1007/BFb0088746. |
[37] |
L. Prandtl,
Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. Ang. Math. Mech., 8 (1928), 85-106.
|
[38] |
D. I. Rachinskii,
Equivalent combinations of stops, Automat. Remote Control, 59 (1998), 1370-1378.
|
[39] |
V. Recupero,
The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci., 31 (2008), 1283-1295.
doi: 10.1002/mma.968. |
[40] |
V. Recupero,
BV continuous sweeping processes, J. Differ. Equ., 259 (2015), 4253-4272.
doi: 10.1016/j.jde.2015.05.019. |
[41] |
D. D. Rizos and S. D. Fassois,
Friction identification based upon the LuGre and Maxwell slip models, IEEE Trans Control Systems Technology, 17 (2009), 153-160.
|
[42] |
M. Ruderman,
Presliding hysteresis damping of LuGre and Maxwell-slip friction models, Mechatronics, 30 (2015), 225-230.
doi: 10.1016/j.mechatronics.2015.07.007. |
[43] |
M. Ruderman, F. Hoffmann and T. Bertram,
Modeling and identification of elastic robot joints with hysteresis and backlash, IEEE Trans. Industrial Electronics, 56 (2009), 3840-3847.
doi: 10.1109/TIE.2009.2015752. |
[44] |
I. Rychlik,
A new definition of the rainflow cycle counting method, Int. J. Fatigue, 9 (1987), 119-121.
doi: 10.1016/0142-1123(87)90054-5. |
[45] |
H. Sayyaadi and M. R. Zakerzadeh,
Position control of shape memory alloy actuator based on the generalized Prandtl-Ishlinskii inverse model, Mechatronics, 22 (2012), 945-957.
doi: 10.1016/j.mechatronics.2012.06.003. |
[46] |
J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts and J. D. Shore,
Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations, Phys. Rev. Lett., 70 (1993), 3347-3350.
doi: 10.1103/PhysRevLett.70.3347. |
[47] |
A. Visintin,
Differential Models of Hysteresis, Springer, 1994.
doi: 10.1007/978-3-662-11557-2. |









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