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March  2020, 25(3): 1129-1139. doi: 10.3934/dcdsb.2019212

## Global asymptotic stability of nonconvex sweeping processes

 Department of Mathematical Sciences, University of Texas at Dallas, 75080 Richardson, USA

* Corresponding author: Oleg Makarenkov

Received  November 2018 Revised  May 2019 Published  September 2019

Building upon the technique that we developed earlier for perturbed sweeping processes with convex moving constraints and monotone vector fields (Kamenskii et al, Nonlinear Anal. Hybrid Syst. 30, 2018), the present paper establishes the conditions for global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraints. Our conclusion can be formulated as follows: closer the constraint to a convex one, weaker monotonicity is required to keep the sweeping process globally asymptotically stable. We explain why the proposed technique is not capable to prove global asymptotic stability of a periodic regime in a crowd motion model (Cao-Mordukhovich, DCDS-B 22, 2017). We introduce and analyze a toy model which clarifies the extent of applicability of our result.

Citation: Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
##### References:
 [1] L. Adam and J. Outrata, On optimal control of a sweeping process coupled with an ordinary differential equation, Discrete Contin. Dyn. Syst.–Ser. B, 19 (2014), 2709-2738.  doi: 10.3934/dcdsb.2014.19.2709.  Google Scholar [2] J. Bastien, F. Bernardin and C.-H. Lamarque, Non Smooth Deterministic or Stochastic Discrete, Dynamical Systems: Applications to Models with Friction or Impact, Wiley, 2013,512 pp.  Google Scholar [3] H. Benabdellah, Existence of solutions to the nonconvex sweeping process, Journal of Differential Equations, 164 (2000), 286-295.  doi: 10.1006/jdeq.1999.3756.  Google Scholar [4] B. Brogliato, Absolute stability and the Lagrange–Dirichlet theorem with monotone multivalued mappings, Systems & Control Letters, 51 (2004), 343-353.  doi: 10.1016/j.sysconle.2003.09.007.  Google Scholar [5] B. Brogliato and W. M. H. Heemels, Observer design for Lur'e systems with multivalued mappings: A passivity approach, IEEE Transactions on Automatic Control, 54 (2009), 1996-2001.  doi: 10.1109/TAC.2009.2023968.  Google Scholar [6] T. H. Cao and B. S. Mordukhovich, Optimality conditions for a controlled sweeping process with applications to the crowd motion model, Discrete Cont. Dyn. Syst., Ser B., 22 (2017), 267-306.  doi: 10.3934/dcdsb.2017014.  Google Scholar [7] T. H. Cao and B. Mordukhovich, Optimal control of a nonconvex perturbed sweeping process, Journal of Differential Equations, 266 (2019), 1003-1050.  doi: 10.1016/j.jde.2018.07.066.  Google Scholar [8] C. Castaing and M. D. Monteiro Marques, BV periodic solutions of an evolution problem associated with continuous moving convex sets, Set-Valued Analysis, 3 (1995), 381-399.  doi: 10.1007/BF01026248.  Google Scholar [9] G. Colombo and V. V. Goncharov, The sweeping processes without convexity, Set-Valued Analysis, 7 (1999), 357-374.  doi: 10.1023/A:1008774529556.  Google Scholar [10] G. Colombo and M. D. Monteiro Marques, Sweeping by a continuous prox-regular set, Journal of Differential Equations, 187 (2003), 46-62.  doi: 10.1016/S0022-0396(02)00021-9.  Google Scholar [11] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, Journal of Differential Equations, 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.  Google Scholar [12] J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Mathematical Programming, 104 (2005), 347-373.  doi: 10.1007/s10107-005-0619-y.  Google Scholar [13] C. O. Frederick and P. J. Armstrong, Convergent internal stresses and steady cyclic states of stress, The Journal of Strain Analysis for Engineering Design, 1 (1966), 154-159.  doi: 10.1243/03093247V012154.  Google Scholar [14] M. Kamenskii, O. Makarenkov, L. N. Wadippuli and P. R. de Fitte, Global stability of almost periodic solutions of monotone sweeping processes and their response to non-monotone perturbations, Nonlinear Analysis: Hybrid Systems, 30 (2018), 213-224.  doi: 10.1016/j.nahs.2018.05.007.  Google Scholar [15] 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.  Google Scholar [16] P. Krejci, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gattotoscho, 1996.  Google Scholar [17] M. Kunze, Periodic solutions of non-linear kinematic hardening models, Math. Methods Appl. Sci., 22 (1999), 515-529.  doi: 10.1002/(SICI)1099-1476(199904)22:6<515::AID-MMA48>3.0.CO;2-S.  Google Scholar [18] R. I. Leine and N. Van de Wouw, Stability and Convergence of Mechanical Systems with Unilateral Constraints, Lecture Notes in Applied and Computational Mechanics, 36. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-76975-0.  Google Scholar [19] E. H. Lockwood, A Book of Curves, Cambridge University Press, New York, 1961.   Google Scholar [20] B. Maury and J. Venel, A discrete contact model for crowd motion, ESAIM: Mathematical Modelling and Numerical Analysis, 45 (2011), 145-168.  doi: 10.1051/m2an/2010035.  Google Scholar [21] B. S. Mordukhovich, Variational Analysis and Applications, Springer, 2018. doi: 10.1007/978-3-319-92775-6.  Google Scholar [22] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Transactions of American Mathematical Society, 352 (2000), 5231-5249.  doi: 10.1090/S0002-9947-00-02550-2.  Google Scholar [23] C. Polizzotto, Variational methods for the steady state response of elasticplastic solids subjected to cyclic loads, International Journal of Solids and Structures, 40 (2003), 2673-2697.  doi: 10.1016/S0020-7683(03)00093-3.  Google Scholar [24] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar [25] W. Rudin, Principles of Mathematical Analysis, McGraw-hill New York, 1976.  Google Scholar [26] A. Tanwani, B. Brogliato and C. Prieur, Stability and observer design for Lur'e systems with multivalued, nonmonotone, time-varying nonlinearities and state jumps, SIAM Journal on Control and Optimization, 52 (2014), 3639-3672.  doi: 10.1137/120902252.  Google Scholar [27] L. Thibault, Sweeping process with regular and nonregular sets, Journal of Differential Equations, 193 (2003), 1-26.  doi: 10.1016/S0022-0396(03)00129-3.  Google Scholar [28] Y. V. Trubnikov and A. I. Perov, Differential Equations with Monotone Nonlinearities, "Nauka i Tekhnika", Minsk, 1986.  Google Scholar [29] V. A. Zorich, Mathematical Analysis. II, Translated from the 2002 fourth Russian edition by Roger Cooke, Universitext, Springer-Verlag, Berlin, 2004.  Google Scholar [30] Z. Zhu, H. Leung and Z. Ding, Optimal synchronization of chaotic systems in noise, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46 (1999), 1320-1329.   Google Scholar

show all references

##### References:
 [1] L. Adam and J. Outrata, On optimal control of a sweeping process coupled with an ordinary differential equation, Discrete Contin. Dyn. Syst.–Ser. B, 19 (2014), 2709-2738.  doi: 10.3934/dcdsb.2014.19.2709.  Google Scholar [2] J. Bastien, F. Bernardin and C.-H. Lamarque, Non Smooth Deterministic or Stochastic Discrete, Dynamical Systems: Applications to Models with Friction or Impact, Wiley, 2013,512 pp.  Google Scholar [3] H. Benabdellah, Existence of solutions to the nonconvex sweeping process, Journal of Differential Equations, 164 (2000), 286-295.  doi: 10.1006/jdeq.1999.3756.  Google Scholar [4] B. Brogliato, Absolute stability and the Lagrange–Dirichlet theorem with monotone multivalued mappings, Systems & Control Letters, 51 (2004), 343-353.  doi: 10.1016/j.sysconle.2003.09.007.  Google Scholar [5] B. Brogliato and W. M. H. Heemels, Observer design for Lur'e systems with multivalued mappings: A passivity approach, IEEE Transactions on Automatic Control, 54 (2009), 1996-2001.  doi: 10.1109/TAC.2009.2023968.  Google Scholar [6] T. H. Cao and B. S. Mordukhovich, Optimality conditions for a controlled sweeping process with applications to the crowd motion model, Discrete Cont. Dyn. Syst., Ser B., 22 (2017), 267-306.  doi: 10.3934/dcdsb.2017014.  Google Scholar [7] T. H. Cao and B. Mordukhovich, Optimal control of a nonconvex perturbed sweeping process, Journal of Differential Equations, 266 (2019), 1003-1050.  doi: 10.1016/j.jde.2018.07.066.  Google Scholar [8] C. Castaing and M. D. Monteiro Marques, BV periodic solutions of an evolution problem associated with continuous moving convex sets, Set-Valued Analysis, 3 (1995), 381-399.  doi: 10.1007/BF01026248.  Google Scholar [9] G. Colombo and V. V. Goncharov, The sweeping processes without convexity, Set-Valued Analysis, 7 (1999), 357-374.  doi: 10.1023/A:1008774529556.  Google Scholar [10] G. Colombo and M. D. Monteiro Marques, Sweeping by a continuous prox-regular set, Journal of Differential Equations, 187 (2003), 46-62.  doi: 10.1016/S0022-0396(02)00021-9.  Google Scholar [11] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, Journal of Differential Equations, 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.  Google Scholar [12] J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Mathematical Programming, 104 (2005), 347-373.  doi: 10.1007/s10107-005-0619-y.  Google Scholar [13] C. O. Frederick and P. J. Armstrong, Convergent internal stresses and steady cyclic states of stress, The Journal of Strain Analysis for Engineering Design, 1 (1966), 154-159.  doi: 10.1243/03093247V012154.  Google Scholar [14] M. Kamenskii, O. Makarenkov, L. N. Wadippuli and P. R. de Fitte, Global stability of almost periodic solutions of monotone sweeping processes and their response to non-monotone perturbations, Nonlinear Analysis: Hybrid Systems, 30 (2018), 213-224.  doi: 10.1016/j.nahs.2018.05.007.  Google Scholar [15] 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.  Google Scholar [16] P. Krejci, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gattotoscho, 1996.  Google Scholar [17] M. Kunze, Periodic solutions of non-linear kinematic hardening models, Math. Methods Appl. Sci., 22 (1999), 515-529.  doi: 10.1002/(SICI)1099-1476(199904)22:6<515::AID-MMA48>3.0.CO;2-S.  Google Scholar [18] R. I. Leine and N. Van de Wouw, Stability and Convergence of Mechanical Systems with Unilateral Constraints, Lecture Notes in Applied and Computational Mechanics, 36. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-76975-0.  Google Scholar [19] E. H. Lockwood, A Book of Curves, Cambridge University Press, New York, 1961.   Google Scholar [20] B. Maury and J. Venel, A discrete contact model for crowd motion, ESAIM: Mathematical Modelling and Numerical Analysis, 45 (2011), 145-168.  doi: 10.1051/m2an/2010035.  Google Scholar [21] B. S. Mordukhovich, Variational Analysis and Applications, Springer, 2018. doi: 10.1007/978-3-319-92775-6.  Google Scholar [22] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Transactions of American Mathematical Society, 352 (2000), 5231-5249.  doi: 10.1090/S0002-9947-00-02550-2.  Google Scholar [23] C. Polizzotto, Variational methods for the steady state response of elasticplastic solids subjected to cyclic loads, International Journal of Solids and Structures, 40 (2003), 2673-2697.  doi: 10.1016/S0020-7683(03)00093-3.  Google Scholar [24] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar [25] W. Rudin, Principles of Mathematical Analysis, McGraw-hill New York, 1976.  Google Scholar [26] A. Tanwani, B. Brogliato and C. Prieur, Stability and observer design for Lur'e systems with multivalued, nonmonotone, time-varying nonlinearities and state jumps, SIAM Journal on Control and Optimization, 52 (2014), 3639-3672.  doi: 10.1137/120902252.  Google Scholar [27] L. Thibault, Sweeping process with regular and nonregular sets, Journal of Differential Equations, 193 (2003), 1-26.  doi: 10.1016/S0022-0396(03)00129-3.  Google Scholar [28] Y. V. Trubnikov and A. I. Perov, Differential Equations with Monotone Nonlinearities, "Nauka i Tekhnika", Minsk, 1986.  Google Scholar [29] V. A. Zorich, Mathematical Analysis. II, Translated from the 2002 fourth Russian edition by Roger Cooke, Universitext, Springer-Verlag, Berlin, 2004.  Google Scholar [30] Z. Zhu, H. Leung and Z. Ding, Optimal synchronization of chaotic systems in noise, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 46 (1999), 1320-1329.   Google Scholar
Illustrations of the notations of the example. The closed ball centered at $(-1.5, 0)$ is $\bar B_1$ and the white ellipses are the graphs of $S(t)$ for different values of the argument. The arrows is the vector field of $\dot{x} = -\alpha x$
The parameters $\phi_0$ and $\phi_*.$
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