## Sweeping processes with prescribed behavior on jumps.(English)Zbl 1435.34063

In the main result of the paper, the authors present a generalized formulation of sweeping process where the behavior of the solution is prescribed at the jump points of the driving moving set; they impose a sort of family of initial conditions which make the concept of solution more general than the one of the classical sweeping processes. As applications, some consequences and particular cases of the main result are finally discussed by the authors.

### MSC:

 34G25 Evolution inclusions 49J52 Nonsmooth analysis 47J20 Variational and other types of inequalities involving nonlinear operators (general) 34A36 Discontinuous ordinary differential equations
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### References:

 [1] Acary, V., Bonnefon, O., Brogliato, B.: Nonsmooth Modeling and Simulation for Switched Circuits. Lecture Notes in Electrical Engineering. Springer, New York (2011) · Zbl 1208.94003 [2] Addy, K; Adly, S; Brogliato, B; Goeleven, D, A method using the approach of Moreau and panagiotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear Anal. Hybrid Syst., 1, 30-43, (2007) · Zbl 1172.94650 [3] Adly, S; Haddad, T; Thibault, L, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program. Ser. B, 148, 5-47, (2014) · Zbl 1308.49013 [4] Ambrosio, L, Metric space valued functions of bounded variation, Ann. Sc. Norm. Sup. Pisa, 17, 439-478, (1990) · Zbl 0724.49027 [5] Benabdellah, H, Existence of solutions to the nonconvex sweeping process, J. Differ. Equ., 164, 286-295, (2000) · Zbl 0957.34061 [6] Bernicot, F; Venel, J, Differential inclusions with proximal normal cones in Banach spaces, J. Convex Anal., 17, 451-484, (2010) · Zbl 1204.34083 [7] Bernicot, F; Venel, J, Stochastic perturbations of sweeping process, J. Differ. Equ., 251, 1195-1224, (2011) · Zbl 1226.34014 [8] Bounkhel, M; Thibault, L, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal., 6, 359-374, (2005) · Zbl 1086.49016 [9] Brezis, H.: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematical Studies. North-Holland Publishing Company, Amsterdam (1973) · Zbl 0252.47055 [10] Brogliato, B; Thibault, L, Existence and uniqueness of solutions for non-autonomous complementarity dynamical systems, J. Convex Anal., 17, 961-990, (2010) · Zbl 1217.34026 [11] Brokate, M; Krejčí, P; Schnabel, H, On uniqueness in evolution quasivariational inequalities, J. Convex Anal., 11, 111-130, (2004) · Zbl 1061.49006 [12] Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions, Applied Mathematical Sciences, 121. Springer, New York (1996) · Zbl 0951.74002 [13] Castaing, C.: Version aléatoire de problème de raflee par un convexe variable. C. R. Acad. Sci. Paris Sér. A 277, 1057-1059 (1973) · Zbl 0268.35013 [14] Castaing,C.: Equations différentielles. Rafle par un convexe aléatoire à variation continue roite, C. R. Acad. Sci. Paris Sér. A 282, 515-518 (1976) · Zbl 0321.35018 [15] Castaing, C; Duc Ha, TX; Valadier, M, Evolution equations governed by the sweeping process, Set-Valued Anal., 1, 109-139, (1993) · Zbl 0813.34018 [16] Castaing, C., Monteiro Marques, M.D.P.: BV periodic solutions of an evolution problem associated with continuous moving convex sets. Set- Valued Anal. 3, 381-399 (1995) · Zbl 0845.35142 [17] Castaing, C; Monteiro Marques, MDP, Evolution problems associated with non-convex closed moving sets with bounded variation, Portugaliae Math., 53, 73-87, (1996) · Zbl 0848.35052 [18] Castaing, C; Monteiro Marques, MDP; Raynaud de Fitte, P, A Skorokhod problem governed by a closed convex moving set, J. Convex Anal., 23, 387-423, (2016) · Zbl 1350.34048 [19] Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Springer, Berlin, Heidelberg, New York (1977) · Zbl 0346.46038 [20] Colombo, G; Goncharov, VV, The sweeping process without convexity, Set-Valued Anal., 7, 357-374, (1999) · Zbl 0957.34060 [21] Colombo, G; Henrion, R; Hoang, ND; Mordukhovich, BS, Disctrete approximations of a controlled sweeping process, Set-Valued Var. Anal., 23, 69-86, (2015) · Zbl 1312.49015 [22] Colombo, G; Henrion, R; Hoang, Nguyen D; Mordukhovich, BS, Optimal control of the sweeping process: the polyhedral case, J. Differ. Equ., 260, 3397-3447, (2016) · Zbl 1334.49070 [23] Colombo, G; Monteiro Marques, MDP, Sweeping by a continuous prox-regular set, J. Differ. Equ., 187, 46-62, (2003) · Zbl 1029.34052 [24] Colombo, G; Palladino, M, The minimum time function for the controlled moreau’s sweeping process, SIAM J. Control, 54, 2036-2062, (2016) · Zbl 1346.49032 [25] Cornet, B, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl., 96, 130-147, (1983) · Zbl 0558.34011 [26] Marino, S; Maury, B; Santambrogio, F, Measure sweeping processes, J. Convex Anal., 23, 567-601, (2016) · Zbl 1344.34032 [27] Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977) · Zbl 0369.46039 [28] Dinculeanu, N.: Vector Measures, International Series of Monographs in Pure and Applied Mathematics. Pergamon Press, Berlin (1967) [29] Edmond, JF; Thibault, L, BV solutions of nonconvex sweeping process differential inclusions with perturbation, J. Differ. Equ., 226, 135-179, (2006) · Zbl 1110.34038 [30] Federer, H.: Geometric Measure Theory. Springer, Berlin, Heidelberg (1969) · Zbl 0176.00801 [31] Flam, S; Hiriart-Urruty, J-B; Jourani, A, Feasibility in finite time, J. Dyn. Control Syst., 15, 537-555, (2009) · Zbl 1203.49012 [32] Haddad, T; Jourani, A; Thibault, L, Reduction of sweeping process to unconstrained differential inclusion, Pac. J. Optim., 4, 493-512, (2008) · Zbl 1185.34018 [33] Henry, C, An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl., 41, 179-186, (1973) · Zbl 0262.49019 [34] Klein, O; Recupero, V, Hausdorff metric BV discontinuity of sweeping processes, J. Phys: Conf. Ser., 727, 012006, (2016) · Zbl 1366.46062 [35] Kopfová, J; Recupero, V, $$BV$$-norm continuity of sweeping processes driven by a set with constant shape, J. Differ. Equ., 261, 5875-5899, (2016) · Zbl 1354.34112 [36] Krasnosel’skiǐ, M.A., Pokrovskiǐ, A.V.: Systems with Hysteresis. Springer, Berlin, Heidelberg (1989) · Zbl 1023.34035 [37] Krejčí, P, Vector hysteresis models, Eur. J. Appl. Math., 2, 281-292, (1991) · Zbl 0754.73015 [38] Krejčí, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto International Series Mathematical Sciences and Applications. Gakkōtosho, Tokyo (1997) [39] Krejčí, P; Laurençot, P, Generalized variational inequalities, J. Convex Anal., 9, 159-183, (2002) · Zbl 1001.49014 [40] Krejčí, P; Recupero, V, Comparing $$BV$$ solutions of rate independent processes, J. Convex Anal., 21, 121-146, (2014) · Zbl 1305.47042 [41] Krejčí, P; Roche, T, Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Contin. Dyn. Syst. Ser. B, 15, 637-650, (2011) · Zbl 1214.49022 [42] Krejčí, P; Vladimirov, A, Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Anal., 11, 91-110, (2003) · Zbl 1035.34010 [43] Kunze, M; Monteiro Marques, MDP; Brogliato, B (ed.), An introduction to moreau’s sweeping processes, impact in mechanical systems—analysis and modelling, 1-60, (2000), Berlin [44] Lang, S.: Real and Functional Analysis, 3rd edn. Springer, New York (1993) · Zbl 0831.46001 [45] Maury, B; Roudneff-Chupin, A; Santambrogio, F, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl., 20, 1787-1821, (2010) · Zbl 1223.35116 [46] Maury, B; Roudneff-Chupin, A; Santambrogio, F, Congestion-driven dendritic growth, Discrete Contin. Dyn. Syst., 34, 1575-1604, (2014) · Zbl 1280.35065 [47] Maury, B; Roudneff-Chupin, A; Santambrogio, F; Venel, J, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6, 485-519, (2011) · Zbl 1260.49039 [48] Maury, B; Venel, J, A mathematical framework for a crowd motion model, C. R. Math. Acad. Sci. Paris, 346, 1245-1250, (2008) · Zbl 1168.34333 [49] Maury, B; Venel, J, A discrete contact model for crowd motion, ESAIM: M2AN, 45, 145-168, (2011) · Zbl 1271.34020 [50] Mielke, A; Dafermos, C (ed.); Feireisl, E (ed.), Evolution in rate-independent systems, 461-559, (2005), Amsterdam · Zbl 1120.47062 [51] Mielke, A., Roubíček, T.: Rate Independent Systems, Theory and Applications. Springer, New York (2015) · Zbl 1339.35006 [52] Monteiro Marques, M.D.P.: Perturbations convexes semi-continues supérieurement dans les espaces de Hilbert. Sem. Anal. Convexe Montpellier , exposé 2 (1984) · Zbl 1344.34032 [53] Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems—Shocks and Dry Friction. Birkhauser Verlag, Basel (1993) · Zbl 0802.73003 [54] Moreau, J. J.: Rafle par un convexe variable, I. Sem. d’Anal. Convexe, Montpellier, 1, Exposé No. 15 (1971) · Zbl 0343.49019 [55] Moreau, J. J.: Rafle par un convexe variable, II. Sem. d’Anal. Convexe, Montpellier, 2, Exposé No. 3 (1972) · Zbl 0343.49020 [56] Moreau, JJ, Problème d’ évolution associé à un convexe mobile dun espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276, a791-a794, (1973) [57] Moreau, J. J.: On unilateral constraints, friction and plasticity. In: Capriz, G., Stampacchia, G. (eds.), New Variational Techniques in Mathematical Physics, pp. 173-322. C.I.M.E. II Ciclo 1973, Ediz. Cremonese, Roma (1974) · Zbl 1312.49015 [58] Moreau, J.J.: Sur les mesures différentielles de fonctions vectorielles et certains problémes d’évolution. C. R. Math. Acad. Sci. Paris Sér. A 282, 837-840 (1976) · Zbl 0329.34050 [59] Moreau, J.J.: Application of convex analysis to the treatment of elastoplastic systems. In: Germain, P., Nayroles, B., (eds.), Application of Methods of Functional Analysis to Problems in Mechanics. Lecture Notes in Mathematics, vol. 503. Springer, Berlin, Heidelberg, New York, pp. 56-89 (1976) [60] Moreau, JJ, Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equ., 26, 347-374, (1977) · Zbl 0356.34067 [61] Moreau, JJ; Frémond, M (ed.); Maceri, F (ed.), An introduction to unilateral dynamics, (2002), Berlin [62] Paoli, L, Multibody dynamics with unilateral constraints and dry friction: how the contact dynamics approach may handle coulomb’s law indeterminacies?, J. Convex Anal., 23, 849-876, (2016) · Zbl 1348.49027 [63] Recupero, V, The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci., 31, 1283-1295, (2008) · Zbl 1140.74021 [64] Recupero, V, Sobolev and strict continuity of general hysteresis operators, Math. Methods Appl. Sci., 32, 2003-2018, (2009) · Zbl 1214.47081 [65] Recupero, V.: $$BV$$ solutions of rate independent variational inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sc. (5) 10, 269-315 (2011). · Zbl 1229.49012 [66] Recupero, V, A continuity method for sweeping processes, J. Differ. Equ., 251, 2125-2142, (2011) · Zbl 1237.34116 [67] Recupero, V, $$BV$$ continuous sweeping processes, J. Differ. Equ., 259, 4253-4272, (2015) · Zbl 1322.49014 [68] Recupero, V, Sweeping processes and rate independence, J. Convex Anal., 23, 921-946, (2016) · Zbl 1357.34103 [69] Sene, M; Thibault, L, Regularization of dynamical systems associated with prox-regular moving sets, J. Nonlinear Convex Anal., 15, 647-663, (2014) · Zbl 1296.34140 [70] Thibault, L, Sweeping process with regular and nonregular sets, J. Differ. Equ., 193, 1-26, (2003) · Zbl 1037.34007 [71] Thibault, L, Regularization of nonconvex sweeping process in Hilbert space, Set-Valued Anal., 16, 319-333, (2008) · Zbl 1162.34010 [72] Thibault, L, Moreau sweeping process with bounded truncated retraction, J. Convex Anal., 23, 1051-1098, (2016) · Zbl 1360.34032 [73] Valadier, M.: Quelques problémes d’entrainement unilatéral en dimension finie. Sem. Anal. Convexe Montpellier, Exposé 8 (1988) · Zbl 0672.49014 [74] Valadier, M, Lipschitz approximation of the sweeping (or Moreau) process, J. Differ. Equ., 88, 248-264, (1990) · Zbl 0716.34059 [75] Valadier, M.: Rafle et viabilité. Sem. Anal. Convexe Montpellier, Exposé 17 (1992) · Zbl 1267.52008 [76] Visintin, A.: Differential Models of Hysteresis, Applied Mathematical Sciences. Springer, Berlin, Heidelberg (1994) · Zbl 0820.35004 [77] Ziemer, W.: Weakly Differentiable Functions. Springer, New York (1989) · Zbl 0692.46022
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