Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input. (English) Zbl 1533.34025
Summary: We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí’s theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron \(\chi\) of a fixed shape. We provide examples of elastoplastic spring models leading to structurally stable situations where \(\chi\) is a one- or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory.
MSC:
| 34A60 | Ordinary differential inclusions |
| 34D45 | Attractors of solutions to ordinary differential equations |
| 93D40 | Finite-time stability |
| 47J22 | Variational and other types of inclusions |
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
Keywords:
sweeping process; finite-time stability; convex polyhedra; elastoplasticity; differential inclusions; shakedown theoryReferences:
| [1] | V. Acary and B. Brogliato, Numerical Methods for Nonsmooth Dynamical Systems. Lecture Notes in Applied and Computational Mechanics. Springer-Verlag Berlin Heidelberg (2008) xxi+525. · Zbl 1173.74001 |
| [2] | S. Adly, H. Attouch and A. Cabot, Finite time stabilization of nonlinear oscillators subject to dry friction. Nonsmooth Mechanics and Analysis. Adv. Mech. Math., Vol. 12, Springer, New York (2006) 289-304. · doi:10.1007/0-387-29195-4_24 |
| [3] | S. Adly, A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics. Springer Briefs in Mathematics (2017) xv+159. · Zbl 1391.49001 |
| [4] | R. Agarwal and V. Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. WorldScientific (1993) 324. · Zbl 0785.34003 |
| [5] | H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011) xvi+468. · Zbl 1218.47001 |
| [6] | S.P. Bhat and D.S. Bernstein, Lyapunov analysis of finite-time differential equations, in Proceedings of the American Control Conference. IEEE, Washington, USA (1995) 1831-1832. |
| [7] | S.P. Bhat and D.S. Bernstein, Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38 (2000) 751-766. · Zbl 0945.34039 · doi:10.1137/S0363012997321358 |
| [8] | F.J. Bejarano and L.M. Fridman, High order sliding mode observer for linear systems with unbounded unknown inputs. Int. J. Control. 9 (2010) 1920-1929. · Zbl 1213.93019 · doi:10.1080/00207179.2010.501386 |
| [9] | T. Bonnesen and W. Fenchel, Theory of convex Bodies. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. BCS Associates (1987) x+172. · Zbl 0628.52001 |
| [10] | H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer-Verlag, New York (2011) xiv+600. · Zbl 1220.46002 |
| [11] | B. Brogliato, Nonsmooth Mechanics: Models, Dynamics and Control. 3rd edn. Springer (2016). · Zbl 1333.74002 · doi:10.1007/978-3-319-28664-8 |
| [12] | A. Cabot, Stabilization of oscillators subject to dry friction: finite time convergence versus exponential decay results. Trans. Am. Math. Soc. 360 (2008) 103-121. · Zbl 1133.34008 · doi:10.1090/S0002-9947-07-03990-6 |
| [13] | F. Chernousko, I. Ananievskii and S. Reshmin, Control of Nonlinear Dynamical Systems: Methods and Applications. Springer-Verlag (2008) xii+396. · Zbl 1155.93001 |
| [14] | G. Colombo, P. Gidoni and E. Vilches, Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction. Discrete Contin. Dyn. Syst. Ser. A 42 (2022) 737-757. · Zbl 1513.70074 · doi:10.3934/dcds.2021135 |
| [15] | F. Dinuzzo and A. Ferrara, Higher order sliding mode controllers with optimal reaching. IEEE Trans. Automat. Contr. 54 (2009) 2126-2136. · Zbl 1367.93120 · doi:10.1109/TAC.2009.2026940 |
| [16] | H. Du, C. Qian, S. Yang and S. Li, Recursive design of finite-time convergent observers for a class of time-varying nonlinear systems. Automatica 49 (2013) 601-609. · Zbl 1259.93029 · doi:10.1016/j.automatica.2012.11.036 |
| [17] | C.O. Frederick and P.J. Armstrong, Convergent internal stresses and steady cyclic states of stress. J. Strain Anal. 1 (1966) 154-159. · doi:10.1243/03093247V012154 |
| [18] | A. Filippov, Differential Equations with Discontinuous Right-hand Sides. Springer Dordrecht, ser. Mathematics and Its Applications (Soviet Series) (1988) x+304. · Zbl 0664.34001 |
| [19] | P. Gidoni, Rate-independent soft crawlers. Q. J. Mech. Appl. Math. 71 (2018) 369-409. · Zbl 1451.74169 |
| [20] | B. Grünbaum, Convex Polytopes. 2nd edn. Graduate Texts in Mathematics, 221. Springer-Verlag, New York (2003) xvi+468. · Zbl 1024.52001 |
| [21] | I. Gudoshnikov and O. Makarenkov, Stabilization of the response of cyclically loaded lattice spring models with plasticity. ESAIM: COCV 27 (2021) S8. · Zbl 1484.74035 · doi:10.1051/cocv/2020043 |
| [22] | I. Gudoshnikov and O. Makarenkov, Structurally stable families of periodic solutions in sweeping processes of networks of elastoplastic springs. Phys. D 406 (2020) 132443. · Zbl 1495.34033 · doi:10.1016/j.physd.2020.132443 |
| [23] | I. Gudoshnikov, Y. Jiao, O. Makarenkov and D. Chen, Sweeping Process Approach to Stress Analysis in Elastoplastic Lattice Springs Models with Applications to Hyperuniform Network Materials. (2022) submitted, preprint available at arXiv:2204.03015. |
| [24] | I. Gudoshnikov, O. Makarenkov and D. Rachinskiy, Finite-time stability of polyhedral sweeping processes with application to elastoplastic systems. SIAM J. Control Optim. 60 (2022) 1320-1346. · Zbl 07535618 · doi:10.1137/20M1388796 |
| [25] | W.M. Haddad and A. L’Afflitto, Finite-Time Stabilization and Optimal Feedback Control. IEEE Trans. Automat. Contr. 61 (2016) 1069-1074. · Zbl 1359.93366 · doi:10.1109/TAC.2015.2454891 |
| [26] | W. Han and B.D. Reddy, Plasticity. Mathematical Theory and Numerical Analysis. 2nd edn. Interdisciplinary Applied Mathematics, 9. Springer, New York (2013). · Zbl 1258.74002 |
| [27] | W.P.M.H. Heemels, J.M. Schumacher and S. Weiland, Linear complementarity systems. SIAM J. Appl. Math. 60 (2000) 1234-1269. · Zbl 0954.34007 · doi:10.1137/S0036139997325199 |
| [28] | J.B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis. Grundlehren Text Editions, Springer-Verlag, Berlin (2001) x+259. · Zbl 0998.49001 |
| [29] | Y. Hong, Finite-time stabilization and stabilizability of a class of controllable systems. Syst. Control Lett. 46 (2002) 231-236. · Zbl 0994.93049 · doi:10.1016/S0167-6911(02)00119-6 |
| [30] | Y. Hong, J. Wang and D. Cheng, Adaptive finite-time control of nonlinear systems with parametric uncertainty. IEEE Trans. Automat. Contr. 51 (2006) 858-862. · Zbl 1366.93290 · doi:10.1109/TAC.2006.875006 |
| [31] | P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations. GAKUTO International Series. Mathematical Sciences and Applications, 8. Gakkōtosho Co. Ltd., Tokyo (1996) viii+211. · Zbl 1187.35003 |
| [32] | M. Kunze and M.D.M. Marques, An introduction to Moreau’s sweeping process, in Impacts in Mechanical Systems. Lecture Notes in Physics, edited by Brogliato, Vol. 551. Springer, Berlin, Heidelberg (2000). · doi:10.1007/BFb0103843 |
| [33] | J. Matoušek and B. Gärtner, Understanding and Using Linear Programming. Berlin, Springer (2007) viii+226. · Zbl 1133.90001 |
| [34] | B. Mordukhovich, Variational Analysis and Applications. Springer Monographs in Mathematics. Springer Cham (2018) xix+622. · Zbl 1402.49003 |
| [35] | J.-J. Moreau, On Unilateral Constraints, Friction and Plasticity. New Variational Techniques in Mathematical Physics (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1973). Edizioni Cremonese, Rome (1974) 171-322. |
| [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: Joint Symposium IUTAM/IMU held in Marseille, September 1-6, 1975, edited by P. Germain and B. Nayroles. Springer (1976) 56-89. · Zbl 0337.73004 |
| [37] | J.-J. Moreau, Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177 (1999) 329-349. · Zbl 0968.70006 · doi:10.1016/S0045-7825(98)00387-9 |
| [38] | J.-J. Moreau, An introduction to unilateral dynamics, in Novel Approaches in Civil Engineering. Lecture Notes in Applied and Computational Mechanics, edited by M. Frémond and F. Maceri, Vol. 14. Springer, Berlin, Heidelberg (2004). · Zbl 1123.70014 |
| [39] | J. Nocedal and S.J. Wright, Numerical optimization. 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York, NY, (2006) xxii+664. · Zbl 1104.65059 |
| [40] | H. Pan, W. Sun and H. Gao, Finite-time vibration control for active suspension systems, in Vibration Control and Actuation of Large-Scale Systems, Emerging Methodologies and Applications in Modelling. Academic Press (2020) 185-223. |
| [41] | C. Qian and W. Lin, Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization. Syst. Control. Lett. 42 (2001) 185-200. · Zbl 0974.93050 · doi:10.1016/S0167-6911(00)00089-X |
| [42] | R.T. Rockafellar, Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, NJ (1970) xviii+451. · Zbl 0193.18401 |
| [43] | A. Schrijver, Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons Ltd. (1987) 484. |
| [44] | S. Seo, H. Shim and J.H. Seo, Global finite-time stabilization of a nonlinear system using dynamic exponent scaling. 47th IEEE Conference, Decision and Control, Cancun, Mexico, Dec. 9-11 (2008) 3805-3810. |
| [45] | A. Shapiro, Differentiability properties of metric projections onto convex sets. J. Optim. Theory Appl. 169 (2016) 953-964. · Zbl 1342.90192 · doi:10.1007/s10957-016-0871-8 |
| [46] | V. Utkin, J. Guldner and J. Shi, Sliding Mode Control in Electro-mechanical Systems. CRC Press (2009) 503. |
| [47] | S.T. Venkataraman and S. Gulati, Terminal sliding modes: a new approach to nonlinear control systems. Proc. 5th Int. Conf. Advanced Robotics, Pisa, Italy (1991) 443-448. |
| [48] | S. Yua, X. Yub, B. Shirinzadehc and Z. Mand, Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 41 (2005) 1957-1964. · Zbl 1125.93423 |
| [49] | X. Zhang, G. Feng and Y. Sun, Finite-time stabilization by state feedback control for a class of time-varying nonlinear systems. Automatica 48 (2012) 499-504. · Zbl 1244.93142 · doi:10.1016/j.automatica.2011.07.014 |
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