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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

H. Benabdellah, Existence of solutions to the nonconvex sweeping process, J. Differ. Equ, 164 (2000), 286-295. doi: 10.1006/jdeq.1999.3756. Google Scholar

[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. Google Scholar

[6]

F. Bernicot and J. Venel, Existence of sweeping process in Banach spaces under directional prox-regularity, J. Convex Anal, 17 (2010), 451-484. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal, 6 (2005), 359-374. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

C. Castaing, A. G. Ibrahim and M. Yarou, Some contributions to nonconvex sweeping process, J. Nonlinear Convex Anal., 10 (2009), 1-20. Google Scholar

[13]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580, Springer Verlag, Berlin, 1977. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[17]

G. Colombo and V. V. Goncharov, The sweeping processes without convexity, Set-Valued Anal, 7 (1999), 357-374. doi: 10.1023/A:1008774529556. Google Scholar

[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. Google Scholar

[19]

J. Diestel, Geometry of Banach Spaces: Selected Topics, Springer-Verlag, New-York, 1975. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

J. J. Moreau, Rafle par un convexe variable Ⅰ, Sém. Anal. Convexe Montpellier, 1 (1971), Exp. No. 15, 43 pp. Google Scholar

[30]

J. J. Moreau. Rafle par un convexe variable Ⅱ, Sém. Anal. Convexe Montpellier, 2 (1972), Exp. No. 3, 36 pp. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

M. Valadier, Quelques problèmes d'entrainement unilatéral en dimension finie, Sém. Anal. Convexe Montp., 18 (1988), Exp. No. 8, 21 pp. Google Scholar

[36]

M. Valadier, Entrainement unilatéral, lignes de descente, fonctions lipschitziennes non pathologiques, C.R. Acad. Sci. Paris Sér. 1 Math, 308 (1989), 241-244. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

H. Benabdellah, Existence of solutions to the nonconvex sweeping process, J. Differ. Equ, 164 (2000), 286-295. doi: 10.1006/jdeq.1999.3756. Google Scholar

[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. Google Scholar

[6]

F. Bernicot and J. Venel, Existence of sweeping process in Banach spaces under directional prox-regularity, J. Convex Anal, 17 (2010), 451-484. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal, 6 (2005), 359-374. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

C. Castaing, A. G. Ibrahim and M. Yarou, Some contributions to nonconvex sweeping process, J. Nonlinear Convex Anal., 10 (2009), 1-20. Google Scholar

[13]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580, Springer Verlag, Berlin, 1977. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[17]

G. Colombo and V. V. Goncharov, The sweeping processes without convexity, Set-Valued Anal, 7 (1999), 357-374. doi: 10.1023/A:1008774529556. Google Scholar

[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. Google Scholar

[19]

J. Diestel, Geometry of Banach Spaces: Selected Topics, Springer-Verlag, New-York, 1975. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

J. J. Moreau, Rafle par un convexe variable Ⅰ, Sém. Anal. Convexe Montpellier, 1 (1971), Exp. No. 15, 43 pp. Google Scholar

[30]

J. J. Moreau. Rafle par un convexe variable Ⅱ, Sém. Anal. Convexe Montpellier, 2 (1972), Exp. No. 3, 36 pp. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

M. Valadier, Quelques problèmes d'entrainement unilatéral en dimension finie, Sém. Anal. Convexe Montp., 18 (1988), Exp. No. 8, 21 pp. Google Scholar

[36]

M. Valadier, Entrainement unilatéral, lignes de descente, fonctions lipschitziennes non pathologiques, C.R. Acad. Sci. Paris Sér. 1 Math, 308 (1989), 241-244. Google Scholar

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