# American Institute of Mathematical Sciences

June  2020, 13(6): 1791-1801. doi: 10.3934/dcdss.2020105

## On evolution quasi-variational inequalities and implicit state-dependent sweeping processes

 1 Université de Limoges, Laboratoire XLIM UMR CNRS 7252,123, avenue Albert Thomas, 87060 Limoges, France 2 Laboratoire LMPEA, Faculté des Sciences Exactes et Informatique, Université de Jijel, B.P. 98, Jijel 18000, Algeria

Received  April 2018 Revised  December 2018 Published  September 2019

In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint ${C(\cdot, u)}$ depends upon the unknown state $u$, which causes one of the main difficulties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a fixed point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set $C$ does not depend explicitly on the state $u$ (i.e. $C: = C(t)$) given in [4], we prove a new existence result of solutions of the quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.

Citation: Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1791-1801. doi: 10.3934/dcdss.2020105
##### References:
 [1] V. Acary, O. Bonnefon and B. Brogliato, Nonsmooth Modeling and Simulation for Switched Circuits, Springer, Lecture notes in electrical engineering, 2011. doi: 10.1007/978-90-481-9681-4. [2] K. Addi, S. Adly, B. Brogliato and D. Goeleven, A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear Anal. Hybrid Syst., 1 (2007), 30-43.  doi: 10.1016/j.nahs.2006.04.001. [3] S. Adly and T. Haddad, An implicit sweeping process approach to quasistatic evolution variational inequalities, SIAM J. Math. Anal., 50 (2018), 761-778.  doi: 10.1137/17M1120658. [4] 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.  doi: 10.1007/s10107-014-0754-4. [5] S. Adly and B. K. Le, On semicoercive sweeping process with velocity constraint, Optimization letters, 12 (2018), 831-843.  doi: 10.1007/s11590-017-1149-2. [6] D. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. [7] C. Henry, An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl., 41 (1973), 179-186.  doi: 10.1016/0022-247X(73)90192-3. [8] B. Maury and J. Venel, A mathematical framework for a crowd motion model, C. R. Math. Acad. Sci., 346 (2008), 1245-1250.  doi: 10.1016/j.crma.2008.10.014. [9] 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. [10] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooths Mechanical Problems, Shokcks and dry Friction, Progress in Nonlinear Differential Equations an Their Applications, Birkhauser, 1993. doi: 10.1007/978-3-0348-7614-8. [11] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7. [12] J. J. Moreau, Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Engrg., 177 (1999), 329-349.  doi: 10.1016/S0045-7825(98)00387-9. [13] E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed Point Theorems, Springer-Verlag, New York, 1986.

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##### References:
 [1] V. Acary, O. Bonnefon and B. Brogliato, Nonsmooth Modeling and Simulation for Switched Circuits, Springer, Lecture notes in electrical engineering, 2011. doi: 10.1007/978-90-481-9681-4. [2] K. Addi, S. Adly, B. Brogliato and D. Goeleven, A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular circuits in electronics, Nonlinear Anal. Hybrid Syst., 1 (2007), 30-43.  doi: 10.1016/j.nahs.2006.04.001. [3] S. Adly and T. Haddad, An implicit sweeping process approach to quasistatic evolution variational inequalities, SIAM J. Math. Anal., 50 (2018), 761-778.  doi: 10.1137/17M1120658. [4] 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.  doi: 10.1007/s10107-014-0754-4. [5] S. Adly and B. K. Le, On semicoercive sweeping process with velocity constraint, Optimization letters, 12 (2018), 831-843.  doi: 10.1007/s11590-017-1149-2. [6] D. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. [7] C. Henry, An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl., 41 (1973), 179-186.  doi: 10.1016/0022-247X(73)90192-3. [8] B. Maury and J. Venel, A mathematical framework for a crowd motion model, C. R. Math. Acad. Sci., 346 (2008), 1245-1250.  doi: 10.1016/j.crma.2008.10.014. [9] 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. [10] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooths Mechanical Problems, Shokcks and dry Friction, Progress in Nonlinear Differential Equations an Their Applications, Birkhauser, 1993. doi: 10.1007/978-3-0348-7614-8. [11] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7. [12] J. J. Moreau, Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Engrg., 177 (1999), 329-349.  doi: 10.1016/S0045-7825(98)00387-9. [13] E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed Point Theorems, Springer-Verlag, New York, 1986.
The moving set $C(u)$ of Example 1
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