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On state-dependent sweeping process in Banach spaces
1. | Laboratoire de Physique Théorique, FSEI, Université Mohammed Seddik Benyahia-Jijel, Algérie |
2. | Département de Mathématiques, FSEI, Université Mohammed Seddik Benyahia-Jijel, Algérie |
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.
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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