Jourani, Abderrahim; Vilches, Emilio Galerkin-like method and generalized perturbed sweeping process with nonregular sets. (English) Zbl 1371.34103 SIAM J. Control Optim. 55, No. 4, 2412-2436 (2017). Summary: In this paper we present a new method to solve differential inclusions in Hilbert spaces. This method is a Galerkin-like method where we approach the original problem by projecting the state into a \(n\)-dimensional Hilbert space but not the velocity. We prove that the approached problem always has a solution and that, under some compactness conditions, the approached problems have a subsequence which converges strongly pointwisely to a solution of the original differential inclusion. We apply this method to the generalized perturbed sweeping process governed by nonregular sets (equi-uniformly subsmooth or positively \(\alpha\)-far). This differential inclusion includes Moreau’s sweeping process, the state-dependent sweeping process, and second-order sweeping process for which we give very general existence results. Cited in 10 Documents MSC: 34G25 Evolution inclusions 34A60 Ordinary differential inclusions 49J52 Nonsmooth analysis 49J53 Set-valued and variational analysis 34A45 Theoretical approximation of solutions to ordinary differential equations Keywords:sweeping process; subsmooth sets; positively \(\alpha\)-far sets; differential inclusions; second-order sweeping process; normal cone PDFBibTeX XMLCite \textit{A. Jourani} and \textit{E. Vilches}, SIAM J. Control Optim. 55, No. 4, 2412--2436 (2017; Zbl 1371.34103) Full Text: DOI References: [1] S. Adly and B. K. Le, {\it Unbounded second-order state-dependent Moreau’s sweeping processes in Hilbert spaces}, J. Optim. Theory Appl., 169 (2016), pp. 407-423. · Zbl 1345.34116 [2] S. Aizicovici and V. Staicu, {\it Multivalued evolution equations with nonlocal initial conditions in Banach spaces}, NoDEA, Nonlinear Differ. Equ. Appl., 14 (2007), pp. 361-376. · Zbl 1145.35076 [3] F. Aliouane and D. Azzam-Laouir, {\it A second order differential inclusion with proximal normal cone in Banach spaces}, Topol. Fixed Point Theory Appl., 44 (2014), pp. 143-160. · Zbl 1360.34132 [4] C. Aliprantis and K. Border, {\it Infinite Dimensional Analysis}, 3rd ed., Springer, Berlin, 2006. · Zbl 1156.46001 [5] D. Azzam-Laouir, {\it Mixed semicontinuous perturbation of a second order nonconvex sweeping process}, Electron. J. Qual. Theory Differ. Equ., 37 (2008), pp. 1-9. · Zbl 1186.34023 [6] D. Azzam-Laouir and S. Izza, {\it Existence of solutions for second-order perturbed nonconvex sweeping process}, Comput. Math. Appl., 62 (2011), pp. 1736-1744. · Zbl 1231.34023 [7] D. Azzam-Laouir, S. Izza, and L. Thibault, {\it Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process}, Set-Valued Var. Anal., 22 (2014), pp. 271-283. · Zbl 1307.34033 [8] P. Ballard, {\it A counterexample to uniqueness of quasistatic elastic contact problems with small friction}, Int. J. Eng. Sci., 37 (1999), pp. 163-178. · Zbl 1210.74018 [9] H. Benabdellah, {\it Existence of solutions to the nonconvex sweeping process}, J. Differ. Equ., 164 (2000), pp. 286-295. · Zbl 0957.34061 [10] M. Bounkhel, {\it General existence results for second order nonconvex sweeping process with unbounded perturbations}, Port. Math., 60 (2003), pp. 269-304. · Zbl 1055.34116 [11] M. Bounkhel, {\it Regularity concepts in nonsmooth analysis}, Springer, 2012. · Zbl 1259.49022 [12] M. Bounkhel and R. Al-Yusof, {\it First and second order convex sweeping processes in reflexive smooth Banach spaces}, Set-Valued Var. Anal., 18 (2010), pp. 151-182. · Zbl 1200.34067 [13] M. Bounkhel and D. Azzam-Laouir, {\it Existence results on the second-order nonconvex sweeping processes with perturbations}, Set-Valued Anal., 12 (2004), pp. 291-318. · Zbl 1048.49002 [14] M. Bounkhel and C. Castaing, {\it State dependent sweeping process in \(p\)-uniformly smooth and \(q\)-uniformly convex Banach spaces}, Set-Valued Var. Anal., 20 (2012), pp. 187-201. · Zbl 1262.34072 [15] M. Bounkhel and T. Haddad, {\it An existence result for a new variant of the nonconvex sweeping process}, Port. Math., 65 (2008), pp. 33-47. · Zbl 1135.49014 [16] M. Bounkhel and L. Thibault, {\it Nonconvex sweeping process and prox-regularity in Hilbert space}, J. Nonlinear Convex Anal., 6 (2005), pp. 359-374. · Zbl 1086.49016 [17] M. Brokate, P. Krejčí, and H. Schnabel, {\it On uniqueness in evolution quasivariational inequalities}, J. Convex Anal., 11 (2004), pp. 111-130. · Zbl 1061.49006 [18] C. Castaing, {\it Quelques problèmes d’évolution du second ordre. Exposé} 5, Sém. Anal. Convexe Montpellier (1988). · Zbl 0676.47029 [19] C. Castaing, T. Duc Ha, and M. Valadier, {\it Evolution equations governed by the sweeping process}, Set-Valued Anal., 1 (1993), pp. 109-139. · Zbl 0813.34018 [20] C. Castaing, A. G. Ibrahim, and M. Yarou, {\it Some contributions to nonconvex sweeping process}, J. Nonlinear Convex Anal., 10 (2009), pp. 1-20. · Zbl 1185.34017 [21] N. Chemetov and M. D. P. Monteiro-Marques, {\it Nonconvex quasi-variational differential inclusions}, Set-Valued Anal., 15 (2007), pp. 209-221. · Zbl 1134.34036 [22] N. Chemetov, M. D. P. Monteiro-Marques, and U. Stefanelli, {\it Ordered non-convex quasi-variational sweeping process}, J. Convex Anal., 14 (2008), pp. 201-214. · Zbl 1148.34004 [23] M. Chraibi Kaadoud, {\it Etude théorique et numérique de problèmes d’évolution en présence de liaisons unilaterales et de frottement}, Ph.D. thesis, USTL, Montpellier, 1987. [24] F. Clarke, Y. Ledyaev, R. Stern, and P. Wolenski, {\it Nonsmooth Analysis and Control Theory}, Springer, Berlin, 1998. · Zbl 1047.49500 [25] G. Colombo and V. V. Goncharov, {\it The sweeping process without convexity}, Set-Valued Anal., 7 (1999), pp. 357-374. · Zbl 0957.34060 [26] K. Deimling, {\it Multivalued Differential Equations}, De Gruyten, Berlin, 1992. · Zbl 0760.34002 [27] N. Dunford and J. T. Schwartz, {\it Linear Operators. Part I: General Theory}, Interscience, New York, 1957. [28] J. F. Edmond and L. Thibault, {\it BV solutions of nonconvex sweeping process differential inclusion with perturbation}, J. Differ. Equ., 226 (2006), pp. 135-179. · Zbl 1110.34038 [29] L. A. Faik and A. Syam, {\it Differential inclusions governed by a nonconvex sweeping process}, J. Nonlinear Convex Anal., 2 (2001), pp. 381-392. · Zbl 1036.49013 [30] L. Gasinski and N. Papageorgiou, {\it Nonlinear Analysis}, Chapman & Hall/CRC, Boca Raton, FL, 2005. · Zbl 1144.35453 [31] T. Haddad, {\it Differential inclusion governed by a state dependent sweeping process}, Int. J. Difference Equ., 8 (2013), pp. 63-70. [32] T. Haddad and T. Haddad, {\it State-dependent sweeping process with perturbation}, in Advances in Applied Mathematics and Approximation Theory, G. A. Anastassiou and O. Duman, eds., Springer Proc. Math. Stat. 41, Springer, New York, 2013, pp. 273-281. · Zbl 1357.49034 [33] T. Haddad, A. Jourani, and L. Thibault, {\it Reduction of sweeping process to unconstrained differential inclusion}, Pac. J. Optim., 4 (2008), pp. 493-512. · Zbl 1185.34018 [34] T. Haddad, I. Kecis, and L. Thibault, {\it Reduction of state dependent sweeping process to unconstrained differential inclusion}, J. Global Optim., 62 (2015), pp. 167-182. · Zbl 1323.34028 [35] P. Hájek and M. Johanis, {\it On Peano’s theorem in Banach spaces}, J. Differ. Equ., 249 (2010), pp. 3342-3351. · Zbl 1216.34057 [36] S. Hu and N. Papageorgiou, {\it Handbook of multivalued analysis. Volume I: Theory}, Kluwer Academic, Dordrecht, 1997. · Zbl 0887.47001 [37] A. Jourani and E. Vilches, {\it Positively \(α\)-far sets and existence results for generalized perturbed sweeping processes}, J. Convex Anal., 23 (2016), pp. 775-821. · Zbl 1350.49015 [38] A. Jourani and E. Vilches, {\it Moreau-Yosida regularization of state-dependent sweeping processes with nonregular sets}, J. Optim. Theory Appl., (2017), pp. 1-26. · Zbl 1376.34059 [39] M. Kunze and M. D. P. Monteiro-Marques, {\it On parabolic quasi-variational inequalities and state-dependent sweeping processes}, Topol. Methods Nonlinear Anal., 12 (1998), pp. 179-191. · Zbl 0923.34018 [40] M. Kunze and M. D. P. Monteiro-Marques, {\it An introduction to Moreau’s sweeping process}, in Impacts in Mechanical Systems, B. Brogliato, ed., Lect. Notes Phys., 551, Springer, Berlin, 2000, pp. 1-60. · Zbl 1047.34012 [41] J. J. Moreau, {\it Rafle par un convexe variable I. Exposé} 15, Sém. Anal. Convexe Montpellier (1971). · Zbl 0343.49019 [42] J. J. Moreau, {\it Rafle par un convexe variable II. Exposé} 3, Sém. Anal. Convexe Montpellier (1972). · Zbl 0343.49020 [43] J. J. Moreau, {\it Multi-applications à rétraction finie}, Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (1974), pp. 169-203. · Zbl 0306.54024 [44] J. J. Moreau, {\it Evolution problem associated with a moving convex set in a Hilbert space}, J. Differ. Equ., 26 (1977), pp. 347-374. · Zbl 0356.34067 [45] J. J. Moreau, {\it Numerical aspects of the sweeping process}, Comput. Methods Appl. Mech. Engrg, 177 (1999), pp. 329-349. · Zbl 0968.70006 [46] J. Noel, {\it Inclusions différentielles d’évolution associées à des ensembles sous lisses}, Ph.D. thesis, Université Montpellier II, 2013. [47] J. Noel and L. Thibault, {\it Nonconvex sweeping process with a moving set depending on the state}, Vietnam J. Math., 42 (2014), pp. 595-612. · Zbl 1315.34068 [48] A. Siddiqi, P. Manchanda, and M. Brokate, {\it On some recent developments concerning Moreauâs sweeping process}, in Trends in Industrial and Applied Mathematics, A. H. Siddiqi and M. Kočvara, eds., Appl. Optim. 72, Springer, New York, Differ. Equ., 2002, pp. 339-354. [49] A. H. Siddiqi and P. Manchanda, {\it Certain remarks on a class of evolution quasi-variational inequalities}, Int. J. Math. Sci., 24 (2000), pp. 851-855. · Zbl 0972.49010 [50] L. Thibault, {\it Sweeping process with regular and nonregular sets}, J. Differ. Equ., 193 (2003), pp. 1-26. · Zbl 1037.34007 [51] L. Thibault, {\it Regularization of nonconvex sweeping process in \textnormalHilbert space}, Set-Valued Anal., 16 (2008), pp. 319-333. · Zbl 1162.34010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.