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

* Corresponding author: Samir Adly

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 & Continuous Dynamical Systems - S, 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.  Google Scholar

[2]

K. AddiS. AdlyB. 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.  Google Scholar

[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.  Google Scholar

[4]

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.  doi: 10.1007/s10107-014-0754-4.  Google Scholar

[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.  Google Scholar

[6]

D. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed Point Theorems, Springer-Verlag, New York, 1986.  Google Scholar

show all references

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.  Google Scholar

[2]

K. AddiS. AdlyB. 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.  Google Scholar

[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.  Google Scholar

[4]

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.  doi: 10.1007/s10107-014-0754-4.  Google Scholar

[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.  Google Scholar

[6]

D. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed Point Theorems, Springer-Verlag, New York, 1986.  Google Scholar

Figure 1.  The moving set $ C(u) $ of Example 1
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