On state-dependent sweeping process in Banach spaces

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June 2018, 7(2): 183-196. doi: 10.3934/eect.2018009

On state-dependent sweeping process in Banach spaces

Dalila Azzam-Laouir ^1,, and Fatiha Selamnia ^2,

1.	Laboratoire de Physique Théorique, FSEI, Université Mohammed Seddik Benyahia-Jijel, Algérie
2.	Département de Mathématiques, FSEI, Université Mohammed Seddik Benyahia-Jijel, Algérie

* Corresponding author: Dalila Azzam-Laouir

Received June 2017 Revised January 2018 Published May 2018

In this paper we prove, in a separable reflexive uniformly smooth Banach space, the existence of solutions of a perturbed first order differential inclusion governed by the proximal normal cone to a moving set depending on the time and on the state. The perturbation is assumed to be separately upper semicontinuous.

Keywords: Differential inclusion, perturbation, proximal normal cone, state-dependent sweeping process, uniformly smooth Banach space.

Mathematics Subject Classification: Primary: 34A60, 49J52; Secondary: 58C20.

Citation: Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations & Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009

References:

[1]	S. Adly and B. K. Le, Unbounded state-dependent sweeping process with perturbations in uniformly convex and q-uniformly smooth Banach spaces, Numerical Algebra, Control & Optimization, 8 (2018), 81-95. doi: 10.3934/naco.2018005.
[2]	D. Azzam-Laouir, S. Izza and L. Thibault, Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process, Set-Valued Var. Anal, 22 (2014), 271-283. doi: 10.1007/s11228-013-0248-1.
[3]	D. Azzam-Laouir, A. Makhlouf and L. Thibault, On perturbed sweeping process, Applicable. Anal., 95 (2016), 303-322. doi: 10.1080/00036811.2014.1002482.
[4]	H. Benabdellah, Existence of solutions to the nonconvex sweeping process, J. Differ. Equ, 164 (2000), 286-295. doi: 10.1006/jdeq.1999.3756.
[5]	F. Bernard, L. Thibault and N. Zlateva, Characterizations of Prox-Regular sets in uniformly convex Banach spaces, J. Convex Anal, 13 (2006), 525-559.
[6]	F. Bernicot and J. Venel, Existence of sweeping process in Banach spaces under directional prox-regularity, J. Convex Anal, 17 (2010), 451-484.
[7]	M. Bounkhel and R. AL-Yusof, First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal, 18 (2010), 151-182. doi: 10.1007/s11228-010-0134-z.
[8]	M. Bounkhel and C. Castaing, State dependent sweeping process in $p$-uniformly smooth and $q$-uniformly convex Banach spaces, Set-Valued Var. Anal, 20 (2012), 187-201. doi: 10.1007/s11228-011-0186-8.
[9]	M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal, 6 (2005), 359-374.
[10]	C. Castaing and M. D. P. Monteiro Marques, Perturbations convexes semi-continues supérieurement de problèmes d'évolution dans les espaces de Hilbert, Sém. Anal. Convexe Montp, 14 (1984), Exp. 2, 23pp.
[11]	C. Castaing, T. X. Dúc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Anal, 1 (1993), 109-139. doi: 10.1007/BF01027688.
[12]	C. Castaing, A. G. Ibrahim and M. Yarou, Some contributions to nonconvex sweeping process, J. Nonlinear Convex Anal., 10 (2009), 1-20.
[13]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580, Springer Verlag, Berlin, 1977.
[14]	N. Chemetov and M. D. P. Monteiro Marques, Non-convex quasi-variational differential inclusions, Set-Valued Anal, 15 (2007), 209-221. doi: 10.1007/s11228-007-0045-9.
[15]	K. Chraibi, Etude Théorique et Numérique de $ Probl\grave{m}es $ D'évolution en Présence de Liaisons Unilatérales et de Frottements, Ph. D. Thesis, Université de Montpellier, 1987.
[16]	F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal, 2 (1995), 117-144.
[17]	G. Colombo and V. V. Goncharov, The sweeping processes without convexity, Set-Valued Anal, 7 (1999), 357-374. doi: 10.1023/A:1008774529556.
[18]	G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set, J. Diff. Equations, 187 (2003), 46-62. doi: 10.1016/S0022-0396(02)00021-9.
[19]	J. Diestel, Geometry of Banach Spaces: Selected Topics, Springer-Verlag, New-York, 1975.
[20]	J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program, Ser. B, 104 (2005), 347-373. doi: 10.1007/s10107-005-0619-y.
[21]	J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Diff. Equations, 226 (2006), 135-179. doi: 10.1016/j.jde.2005.12.005.
[22]	T. Haddad, Nonconvex differential inequality and state dependent sweeping process, J. Optim. Theory Appl, 159 (2013), 386-398. doi: 10.1007/s10957-013-0353-1.
[23]	T. Haddad, I. Kecis and L. Thibault, Reduction of state dependent sweeping process to unconstrained differential inclusion, J. Global Optim, 62 (2015), 167-182. doi: 10.1007/s10898-014-0220-0.
[24]	T. Haddad, J. Noel and L. Thibault, Perturbed Sweeping process with a subsmooth set depending on the state, Linear and Nonlinear Analysis, 2 (2016), 155-274.
[25]	S. Izza, Contibution à L'étude de Certaines Classes D'inclusions Différentielles Gouvernées par le Processus de la Rafle, Thése de doctorat en Sciences, Université Mohammed Seddik Benyahia-Jijel, 2016.
[26]	A. Jourani and E. Vilches, Moreau-Yosida regularization of state-dependent sweeping processes with nonregular sets, J Optim Theory Appl, 173 (2017), 91-116. doi: 10.1007/s10957-017-1083-6.
[27]	M. Kunze and M. D. P. Monteiro Marques, On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Methods Nonlinear Anal, 12 (1998), 179-191. doi: 10.12775/TMNA.1998.036.
[28]	M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems-shocks and Dry Friction, Birkhauser, Basel-Boston-Berlin, 1993. doi: 10.1007/978-3-0348-7614-8.
[29]	J. J. Moreau, Rafle par un convexe variable Ⅰ, Sém. Anal. Convexe Montpellier, 1 (1971), Exp. No. 15, 43 pp.
[30]	J. J. Moreau. Rafle par un convexe variable Ⅱ, Sém. Anal. Convexe Montpellier, 2 (1972), Exp. No. 3, 36 pp.
[31]	J. J. Moreau, Evolution problem associated with a moving convex set in Hilbert space, J. Differential Equations, 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.
[32]	J. Noel and L. Thibault, Nonconvex sweeping process with a moving set depending on the state, Vietnam J. Math., 42 (2014), 595-612. doi: 10.1007/s10013-014-0109-8.
[33]	R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. doi: 10.1090/S0002-9947-00-02550-2.
[34]	L. Thibault, Sweeping process with regular and nonregular sets, J. Diff. Equations, 193 (2003), 1-26. doi: 10.1016/S0022-0396(03)00129-3.
[35]	M. Valadier, Quelques problèmes d'entrainement unilatéral en dimension finie, Sém. Anal. Convexe Montp., 18 (1988), Exp. No. 8, 21 pp.
[36]	M. Valadier, Entrainement unilatéral, lignes de descente, fonctions lipschitziennes non pathologiques, C.R. Acad. Sci. Paris Sér. 1 Math, 308 (1989), 241-244.

show all references

References:

[1]	S. Adly and B. K. Le, Unbounded state-dependent sweeping process with perturbations in uniformly convex and q-uniformly smooth Banach spaces, Numerical Algebra, Control & Optimization, 8 (2018), 81-95. doi: 10.3934/naco.2018005.
[2]	D. Azzam-Laouir, S. Izza and L. Thibault, Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process, Set-Valued Var. Anal, 22 (2014), 271-283. doi: 10.1007/s11228-013-0248-1.
[3]	D. Azzam-Laouir, A. Makhlouf and L. Thibault, On perturbed sweeping process, Applicable. Anal., 95 (2016), 303-322. doi: 10.1080/00036811.2014.1002482.
[4]	H. Benabdellah, Existence of solutions to the nonconvex sweeping process, J. Differ. Equ, 164 (2000), 286-295. doi: 10.1006/jdeq.1999.3756.
[5]	F. Bernard, L. Thibault and N. Zlateva, Characterizations of Prox-Regular sets in uniformly convex Banach spaces, J. Convex Anal, 13 (2006), 525-559.
[6]	F. Bernicot and J. Venel, Existence of sweeping process in Banach spaces under directional prox-regularity, J. Convex Anal, 17 (2010), 451-484.
[7]	M. Bounkhel and R. AL-Yusof, First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal, 18 (2010), 151-182. doi: 10.1007/s11228-010-0134-z.
[8]	M. Bounkhel and C. Castaing, State dependent sweeping process in $p$-uniformly smooth and $q$-uniformly convex Banach spaces, Set-Valued Var. Anal, 20 (2012), 187-201. doi: 10.1007/s11228-011-0186-8.
[9]	M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal, 6 (2005), 359-374.
[10]	C. Castaing and M. D. P. Monteiro Marques, Perturbations convexes semi-continues supérieurement de problèmes d'évolution dans les espaces de Hilbert, Sém. Anal. Convexe Montp, 14 (1984), Exp. 2, 23pp.
[11]	C. Castaing, T. X. Dúc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Anal, 1 (1993), 109-139. doi: 10.1007/BF01027688.
[12]	C. Castaing, A. G. Ibrahim and M. Yarou, Some contributions to nonconvex sweeping process, J. Nonlinear Convex Anal., 10 (2009), 1-20.
[13]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580, Springer Verlag, Berlin, 1977.
[14]	N. Chemetov and M. D. P. Monteiro Marques, Non-convex quasi-variational differential inclusions, Set-Valued Anal, 15 (2007), 209-221. doi: 10.1007/s11228-007-0045-9.
[15]	K. Chraibi, Etude Théorique et Numérique de $ Probl\grave{m}es $ D'évolution en Présence de Liaisons Unilatérales et de Frottements, Ph. D. Thesis, Université de Montpellier, 1987.
[16]	F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal, 2 (1995), 117-144.
[17]	G. Colombo and V. V. Goncharov, The sweeping processes without convexity, Set-Valued Anal, 7 (1999), 357-374. doi: 10.1023/A:1008774529556.
[18]	G. Colombo and M. D. P. Monteiro Marques, Sweeping by a continuous prox-regular set, J. Diff. Equations, 187 (2003), 46-62. doi: 10.1016/S0022-0396(02)00021-9.
[19]	J. Diestel, Geometry of Banach Spaces: Selected Topics, Springer-Verlag, New-York, 1975.
[20]	J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program, Ser. B, 104 (2005), 347-373. doi: 10.1007/s10107-005-0619-y.
[21]	J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Diff. Equations, 226 (2006), 135-179. doi: 10.1016/j.jde.2005.12.005.
[22]	T. Haddad, Nonconvex differential inequality and state dependent sweeping process, J. Optim. Theory Appl, 159 (2013), 386-398. doi: 10.1007/s10957-013-0353-1.
[23]	T. Haddad, I. Kecis and L. Thibault, Reduction of state dependent sweeping process to unconstrained differential inclusion, J. Global Optim, 62 (2015), 167-182. doi: 10.1007/s10898-014-0220-0.
[24]	T. Haddad, J. Noel and L. Thibault, Perturbed Sweeping process with a subsmooth set depending on the state, Linear and Nonlinear Analysis, 2 (2016), 155-274.
[25]	S. Izza, Contibution à L'étude de Certaines Classes D'inclusions Différentielles Gouvernées par le Processus de la Rafle, Thése de doctorat en Sciences, Université Mohammed Seddik Benyahia-Jijel, 2016.
[26]	A. Jourani and E. Vilches, Moreau-Yosida regularization of state-dependent sweeping processes with nonregular sets, J Optim Theory Appl, 173 (2017), 91-116. doi: 10.1007/s10957-017-1083-6.
[27]	M. Kunze and M. D. P. Monteiro Marques, On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Methods Nonlinear Anal, 12 (1998), 179-191. doi: 10.12775/TMNA.1998.036.
[28]	M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems-shocks and Dry Friction, Birkhauser, Basel-Boston-Berlin, 1993. doi: 10.1007/978-3-0348-7614-8.
[29]	J. J. Moreau, Rafle par un convexe variable Ⅰ, Sém. Anal. Convexe Montpellier, 1 (1971), Exp. No. 15, 43 pp.
[30]	J. J. Moreau. Rafle par un convexe variable Ⅱ, Sém. Anal. Convexe Montpellier, 2 (1972), Exp. No. 3, 36 pp.
[31]	J. J. Moreau, Evolution problem associated with a moving convex set in Hilbert space, J. Differential Equations, 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.
[32]	J. Noel and L. Thibault, Nonconvex sweeping process with a moving set depending on the state, Vietnam J. Math., 42 (2014), 595-612. doi: 10.1007/s10013-014-0109-8.
[33]	R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. doi: 10.1090/S0002-9947-00-02550-2.
[34]	L. Thibault, Sweeping process with regular and nonregular sets, J. Diff. Equations, 193 (2003), 1-26. doi: 10.1016/S0022-0396(03)00129-3.
[35]	M. Valadier, Quelques problèmes d'entrainement unilatéral en dimension finie, Sém. Anal. Convexe Montp., 18 (1988), Exp. No. 8, 21 pp.
[36]	M. Valadier, Entrainement unilatéral, lignes de descente, fonctions lipschitziennes non pathologiques, C.R. Acad. Sci. Paris Sér. 1 Math, 308 (1989), 241-244.

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