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Topological properties for a perturbed first order sweeping process. (English) Zbl 1476.34064

Summary: In this paper, we consider a perturbed sweeping process for a class of subsmooth moving sets. The perturbation is general and takes the form of a sum of a single-valued mapping and a set-valued mapping. In the first result, we study some topological proprieties of the attainable set, the set-valued mapping considered here is upper semi-continuous with convex values. In the second result, we treat the autonomous problem under assumptions that do not require the convexity of the values and that weaken the assumption on the upper semi-continuity. Then, we deduce a solution of the time optimality problem.

MSC:

34A60 Ordinary differential inclusions
28A25 Integration with respect to measures and other set functions
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[1] S. Adly, F. Nacry and L. Thibault, Discontinuous sweeping process with prox-regular sets, ESAIM Control Optim. Calc. Var., 23 (2017), 1293-1329. · Zbl 1379.49023
[2] K. Addi, B. Brogliato and D. Goeleven, Aqualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems, Appl. Electron. Math. Program., 126 (2011), 31-67. · Zbl 1229.90224
[3] D. Affane, M. Aissous and M. F. Yarou, Existence results for sweeping process with almost convex perturbation, Bull. Math. Soc. Sci. Math. Roumanie, 2 (2018), 119-134. · Zbl 1413.34206
[4] D. Affane, M. Aissous and M. F. Yarou, Almost mixed semi-continuous perturbation of Moreau’s sweeping process, Evol. Equ. Control Theory, 1 (2020), 27-38. · Zbl 1431.34026
[5] D. Affane and D. Azzam-Laouir, Almost convex valued perturbation to time optimal control sweeping processes, ESAIM Control Optim. Calc. Var, 23 (2017), 1-12. · Zbl 1366.34029
[6] D. Aussel, A Danilis and L. Thibault, Subsmooth sets: functional characterizations and related concepts, Trans. Amer. Math. Soc.357 (2004), 1275-1301. · Zbl 1094.49016
[7] H. Benabdellah, Existence of solutions to the nonconvex sweeping process, J. Differential Equations, 164 (2000), 286-295. · Zbl 0957.34061
[8] M. Bounkhel and M. F. Yarou, Existence results for first and second order nonconvex sweeping processs with delay, Port. Math.,61 (2004), 207-230. · Zbl 1098.49016
[9] C. Castaing, A. G. Ibrahim and M. F. Yarou, Some contributions to nonconvex sweeping process, J. Nonlinear Convex Anal., 10 (2009), 1-20. · Zbl 1185.34017
[10] N. Chemetov and M. D. P. Monteiro Marques, Non-convex Quasi-variational Differential Inclusions, Set-Valued Var. Anal., 5(2007), 209-221. · Zbl 1134.34036
[11] K. Chraibi, Resolution du problème de rafle et application a un problème de frottement, Topol. Methods Nonlinear Anal., 18 (2001), 89-102. · Zbl 1009.34008
[12] A. Cellina and A. Ornelas, Existence of solution to differential inclusion and the time optimal control problems in the autonomous case, SIAM J. Control Optim., 42 (2003), 260-265. · Zbl 1058.49003
[13] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998. · Zbl 1047.49500
[14] G. Colombo and V. V. Goncharov, The sweeping processes without convexity, Set-Valued Anal., 18 (1999), 357-374. · Zbl 0957.34060
[15] G. Colombo, R. Henrion, N. D Hoang and B. Sh. Mordukhovich, Discrete approximations of a controlled sweeping process, Set-Valued Var. Anal., 23 (2015), 69-86. · Zbl 1312.49015
[16] G. Colombo, R. Henrion, N. D Hoang and B. Sh. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets, J. Differ. Equ., 260 (2016), 3397-3447. · Zbl 1334.49070
[17] G. Colombo and L. Thibault, Prox-regular sets and applications, in Handbook of Nonconvex Analysis and Applications, International Press-Somerville, MA, (2010), 99-182. · Zbl 1221.49001
[18] S. Di Marino, F. Santambrogio and B. Maury, Measure sweeping processes, J. Conv. An., 23 (2015), 567-601. · Zbl 1344.34032
[19] D. Goeleven, Complementarity and Variational Inequalities in Electronics, Mathematical Analysis and its Applications, Academic Press, London, 2017. · Zbl 1377.94001
[20] A. F. Filippov, On certain questions in the theory of optimal control, Vestnik. Univ., Ser. Mat. Mech., 2 (1959), 25-32; Translated in SIAM J. Control, 1 (1962), 76-84. · Zbl 0139.05102
[21] T. Haddad, J. Noel and L. Thibault, Perturbation sweeping process with a subsmooth set depending on the state, Linear Nonlinear Anal., 2 (2016), 155-174. · Zbl 1350.34049
[22] M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau’s sweeping process, in B. Brogliato, (ed.) Impacts in Mechanical Systems. Analysis and Modelling, Springer, Berlin, 2000 1-60. · Zbl 1047.34012
[23] B. Sh. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in asplund space, Trans. Amer. Math. Soc., 4 (1996), 1235-1279. · Zbl 0881.49009
[24] J. J. Moreau, Rafle par un convexe variable I, Sem. Anal. Convexe Mont-pellier, exposé No. 15 (1971). · Zbl 0343.49019
[25] J. J. Moreau, Rafle par un convexe variable II, Sem. Anal. Convexe Mont-pellier, exposé No. 3 (1972). · Zbl 0343.49020
[26] J. J. Moreau, Evolution problem associated with amoving convex set in a Hilbert Space, J. Differ. Equ., 26 (1977), 347-374. · Zbl 0356.34067
[27] J. J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, Nonsmooth Mechanics and Applications 302 in CISM, Courses and Lectures, Springer Verlag, 1988. · Zbl 0703.73070
[28] V. Recupero, A continuity method for sweeping processes, J. Differ. Equ., 251 (2011), 2125-2142. · Zbl 1237.34116
[29] V. Recupero, BV continuous sweeping processes, J. Differ. Equ., 259 (2015), 4253-4272. · Zbl 1322.49014
[30] V. Recupero, Sweeping processes and rate independence, J. Convex Anal., 23 (2016), 921-946. · Zbl 1357.34103
[31] V. Recupero, F. Santambrogio, Sweeping processes with prescribed behavior on jumps, Ann. Mat. Pura Appl., 197 (2018), 1311-1332. · Zbl 1435.34063
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