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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. Adly, T. 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. [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. [3] Y. Alber, Generalized projection operators in banach spaces: properties and applications, Funct. Different. Equations, 1 (1994), 1-21. [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. [5] Y. Alber and I. Ryazantseva, Nonlinear Ill-Posed Problem of Monotone Type, Springer Netherlands, 2006. [6] J. -P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Spinger-Verlag, Berlin, 1984. [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. [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. [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. [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. [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. [12] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer, Berlin, 1977. [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. [14] J. Diestel, Geometry of Banach Spaces, Selected Topics, Lecture Notes in Mathematics, Springer, New York, 485 (1975). [15] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992. [16] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusions with perturbation, J. Differential Equations, 226 (2006), 135-179. [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. [18] J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299. [19] J. J. Moreau, Sur l'evolution d'un système élastoplastique, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A118-A121. [20] J. J. Moreau, Rafle par un convexe variable Ⅰ, Sém. Anal. Convexe Montpellier, (1971), Exposé 15. [21] J. J. Moreau, Rafle par un convexe variable Ⅱ, Sém. Anal. Convexe Montpellier, (1972), Exposé 3. [22] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374. [23] L. Thibault, Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable. Applications, Thèse, Université Montpellier, 1976.

show all references

##### References:
 [1] S. Adly, T. 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. [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. [3] Y. Alber, Generalized projection operators in banach spaces: properties and applications, Funct. Different. Equations, 1 (1994), 1-21. [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. [5] Y. Alber and I. Ryazantseva, Nonlinear Ill-Posed Problem of Monotone Type, Springer Netherlands, 2006. [6] J. -P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Spinger-Verlag, Berlin, 1984. [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. [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. [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. [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. [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. [12] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer, Berlin, 1977. [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. [14] J. Diestel, Geometry of Banach Spaces, Selected Topics, Lecture Notes in Mathematics, Springer, New York, 485 (1975). [15] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992. [16] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusions with perturbation, J. Differential Equations, 226 (2006), 135-179. [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. [18] J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299. [19] J. J. Moreau, Sur l'evolution d'un système élastoplastique, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A118-A121. [20] J. J. Moreau, Rafle par un convexe variable Ⅰ, Sém. Anal. Convexe Montpellier, (1971), Exposé 15. [21] J. J. Moreau, Rafle par un convexe variable Ⅱ, Sém. Anal. Convexe Montpellier, (1972), Exposé 3. [22] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374. [23] L. Thibault, Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable. Applications, Thèse, Université Montpellier, 1976.
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