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March  2018, 8(1): 81-95. doi: 10.3934/naco.2018005

Unbounded state-dependent sweeping processes with perturbations in uniformly convex and q-uniformly smooth Banach spaces

1. 

XLIM UMR-CNRS 7252, Université de Limoges, 87060 Limoges, France

2. 

Centro de Modelamiento Matemático (CMM), Universidad de Chile, Santiago, Chile

* Corresponding author

Received  December 2016 Revised  January 2018 Published  March 2018

Fund Project: The second author is supported by Fondecyt Postdoc Project 3150332.

In this paper, the existence of solutions for a class of first and second order unbounded state-dependent sweeping processes with perturbation in uniformly convex and $q$-uniformly smooth Banach spaces are analyzed by using a discretization method. The sweeping process is a particular differential inclusion with a normal cone to a moving set and is of a great interest in many concrete applications. The boundedness of the moving set, which plays a crucial role for the existence of solutions in many works in the literature, is not necessary in the present paper. The compactness assumption on the moving set is also improved.

Citation: Samir Adly, Ba Khiet Le. Unbounded state-dependent sweeping processes with perturbations in uniformly convex and q-uniformly smooth Banach spaces. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 81-95. doi: 10.3934/naco.2018005
References:
[1]

S. AdlyT. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program. Ser. B, 148 (2014), 5-47.   Google Scholar

[2]

S. Adly and B. K. Le, Unbounded second order state-dependent Moreau's sweeping processes in Hilbert spaces, J. Optim. Theory Appl., 169 (2016), 407-423.   Google Scholar

[3]

Y. Alber, Generalized projection operators in banach spaces: properties and applications, Funct. Different. Equations, 1 (1994), 1-21.   Google Scholar

[4]

Y. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, In: Theory and Applications of Nonlinear Operators of Monotonic and Accretive Type (ed. A. Kartsatos), Marcel Dekker, New York, (1996), 15-50.  Google Scholar

[5]

Y. Alber and I. Ryazantseva, Nonlinear Ill-Posed Problem of Monotone Type, Springer Netherlands, 2006.  Google Scholar

[6]

J. -P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Spinger-Verlag, Berlin, 1984.  Google Scholar

[7]

J. M. Borwein and Q. Z. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control Optim., 34 (1996), 1568-1591.   Google Scholar

[8]

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.   Google Scholar

[9]

M. Bounkhel, Existence results for second order convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces, Electronic Journal of Qualitative Theory of Differential Equations, 27 (2012), 1-10.   Google Scholar

[10]

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.   Google Scholar

[11]

M. Bounkhel and L. Thibault, On various notions of regularity of sets in nonsmooth analysis, Nonlinear Anal.: Theory, Methods and Applications, 48 (2002), 223-246.   Google Scholar

[12]

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

[13]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math Prog. Ser. B, 104 (2005), 347-373.   Google Scholar

[14]

J. Diestel, Geometry of Banach Spaces, Selected Topics, Lecture Notes in Mathematics, Springer, New York, 485 (1975).  Google Scholar

[15]

K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992.  Google Scholar

[16]

J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusions with perturbation, J. Differential Equations, 226 (2006), 135-179.   Google Scholar

[17]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in: Impacts in Mechanical Systems. Analysis and Modelling (ed. B. Brogliato), Springer, Berlin, (2000), 1-60.  Google Scholar

[18]

J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.   Google Scholar

[19]

J. J. Moreau, Sur l'evolution d'un système élastoplastique, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A118-A121.   Google Scholar

[20]

J. J. Moreau, Rafle par un convexe variable Ⅰ, Sém. Anal. Convexe Montpellier, (1971), Exposé 15.  Google Scholar

[21]

J. J. Moreau, Rafle par un convexe variable Ⅱ, Sém. Anal. Convexe Montpellier, (1972), Exposé 3.  Google Scholar

[22]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.   Google Scholar

[23]

L. Thibault, Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable. Applications, Thèse, Université Montpellier, 1976.  Google Scholar

show all references

References:
[1]

S. AdlyT. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program. Ser. B, 148 (2014), 5-47.   Google Scholar

[2]

S. Adly and B. K. Le, Unbounded second order state-dependent Moreau's sweeping processes in Hilbert spaces, J. Optim. Theory Appl., 169 (2016), 407-423.   Google Scholar

[3]

Y. Alber, Generalized projection operators in banach spaces: properties and applications, Funct. Different. Equations, 1 (1994), 1-21.   Google Scholar

[4]

Y. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, In: Theory and Applications of Nonlinear Operators of Monotonic and Accretive Type (ed. A. Kartsatos), Marcel Dekker, New York, (1996), 15-50.  Google Scholar

[5]

Y. Alber and I. Ryazantseva, Nonlinear Ill-Posed Problem of Monotone Type, Springer Netherlands, 2006.  Google Scholar

[6]

J. -P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Spinger-Verlag, Berlin, 1984.  Google Scholar

[7]

J. M. Borwein and Q. Z. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control Optim., 34 (1996), 1568-1591.   Google Scholar

[8]

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.   Google Scholar

[9]

M. Bounkhel, Existence results for second order convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces, Electronic Journal of Qualitative Theory of Differential Equations, 27 (2012), 1-10.   Google Scholar

[10]

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.   Google Scholar

[11]

M. Bounkhel and L. Thibault, On various notions of regularity of sets in nonsmooth analysis, Nonlinear Anal.: Theory, Methods and Applications, 48 (2002), 223-246.   Google Scholar

[12]

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

[13]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math Prog. Ser. B, 104 (2005), 347-373.   Google Scholar

[14]

J. Diestel, Geometry of Banach Spaces, Selected Topics, Lecture Notes in Mathematics, Springer, New York, 485 (1975).  Google Scholar

[15]

K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992.  Google Scholar

[16]

J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusions with perturbation, J. Differential Equations, 226 (2006), 135-179.   Google Scholar

[17]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in: Impacts in Mechanical Systems. Analysis and Modelling (ed. B. Brogliato), Springer, Berlin, (2000), 1-60.  Google Scholar

[18]

J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.   Google Scholar

[19]

J. J. Moreau, Sur l'evolution d'un système élastoplastique, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A118-A121.   Google Scholar

[20]

J. J. Moreau, Rafle par un convexe variable Ⅰ, Sém. Anal. Convexe Montpellier, (1971), Exposé 15.  Google Scholar

[21]

J. J. Moreau, Rafle par un convexe variable Ⅱ, Sém. Anal. Convexe Montpellier, (1972), Exposé 3.  Google Scholar

[22]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.   Google Scholar

[23]

L. Thibault, Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable. Applications, Thèse, Université Montpellier, 1976.  Google Scholar

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