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March  2020, 9(1): 27-38. doi: 10.3934/eect.2020015

## Almost mixed semi-continuous perturbation of Moreau's sweeping process

 LMPA Laboratory, Department of Mathematics, Jijel University, 18000, Algeria

* Corresponding author: Doria Affane

Received  May 2018 Revised  August 2019 Published  March 2020 Early access  October 2019

In this work, we introduce a new concept of semi-continuous set-valued mappings, called almost mixed semi-continuity, by taking maps that are upper semi-continuous with almost convex values in some points and lower semi-continuous in remaining points. We generalize earlier results obtained for both mixed semi-continuous maps and almost convex sets. We discuss the existence of solution for evolution problems driven by the so-called sweeping process subject to external forces, known as perturbation to the system, by this type of set-valued mappings. Finally, we give some topological properties of the attainable and solution sets in order to solve an optimal time problem.

Citation: Doria Affane, Meriem Aissous, Mustapha Fateh Yarou. Almost mixed semi-continuous perturbation of Moreau's sweeping process. Evolution Equations and Control Theory, 2020, 9 (1) : 27-38. doi: 10.3934/eect.2020015
##### References:
 [1] D. Affane, M. Aissous and M. F. Yarou, Existence results for sweeping process with almost convex perturbation, Bull. Math. Soc. Sci. Math. Roumanie, 61 (2018), 119-134. [2] D. Affane and D. Azzam-Laouir, Almost convex valued perturbation to time optimal control sweeping processes, Essaim: Control, Optim. Calcul. Variat., 23 (2017), 1-12.  doi: 10.1051/cocv/2015036. [3] H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math., 12 (1972), 373-390.  doi: 10.1007/BF02764629. [4] M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlin. Convex Anal., 6 (2005), 359-374. [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977. [6] A. Cellina and A. Ornelas, Existence of solution to differential inclusion and optimal control problems in the autonomous case, SIAM J. Control Optim., 42 (2003), 260-265.  doi: 10.1137/S0363012902408046. [7] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998. [8] 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. [9] B. Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl., 96 (1983), 130-147.  doi: 10.1016/0022-247X(83)90032-X. [10] A. Fryszkowski and L. Gorniewicz, Mixed semi-continuous mappings and their applications to differential inclusions, Set-Valued Anal., 8 (2000), 203-217.  doi: 10.1023/A:1008763724495. [11] T. Haddad and L. Thibault, Mixed semi-continuous perturbation of nonconvex sweeping process, Math. Program. Ser. B, 123 (2010), 225-240.  doi: 10.1007/s10107-009-0315-4. [12] C. Henry, An existence theorem for a class of differential equation with multi-valued right hand side, J. Math. Anal. Appl., 41 (1973), 179-186.  doi: 10.1016/0022-247X(73)90192-3. [13] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Math. Soc., 352 (2000), 5231-5249.  doi: 10.1090/S0002-9947-00-02550-2. [14] L. Thibault, Sweeping process with regular and nonregular sets, J. Diff. Eqs., 193 (2003), 1-26.  doi: 10.1016/S0022-0396(03)00129-3. [15] A. A. Tolstonogov, Solutions of a differential inclusion with unbounded right-hand side (Russian), Sib. Math. Zh., 29 (1988), 212-225.  doi: 10.1007/BF00970283.

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##### References:
 [1] D. Affane, M. Aissous and M. F. Yarou, Existence results for sweeping process with almost convex perturbation, Bull. Math. Soc. Sci. Math. Roumanie, 61 (2018), 119-134. [2] D. Affane and D. Azzam-Laouir, Almost convex valued perturbation to time optimal control sweeping processes, Essaim: Control, Optim. Calcul. Variat., 23 (2017), 1-12.  doi: 10.1051/cocv/2015036. [3] H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math., 12 (1972), 373-390.  doi: 10.1007/BF02764629. [4] M. Bounkhel and L. Thibault, Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlin. Convex Anal., 6 (2005), 359-374. [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977. [6] A. Cellina and A. Ornelas, Existence of solution to differential inclusion and optimal control problems in the autonomous case, SIAM J. Control Optim., 42 (2003), 260-265.  doi: 10.1137/S0363012902408046. [7] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998. [8] 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. [9] B. Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl., 96 (1983), 130-147.  doi: 10.1016/0022-247X(83)90032-X. [10] A. Fryszkowski and L. Gorniewicz, Mixed semi-continuous mappings and their applications to differential inclusions, Set-Valued Anal., 8 (2000), 203-217.  doi: 10.1023/A:1008763724495. [11] T. Haddad and L. Thibault, Mixed semi-continuous perturbation of nonconvex sweeping process, Math. Program. Ser. B, 123 (2010), 225-240.  doi: 10.1007/s10107-009-0315-4. [12] C. Henry, An existence theorem for a class of differential equation with multi-valued right hand side, J. Math. Anal. Appl., 41 (1973), 179-186.  doi: 10.1016/0022-247X(73)90192-3. [13] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Math. Soc., 352 (2000), 5231-5249.  doi: 10.1090/S0002-9947-00-02550-2. [14] L. Thibault, Sweeping process with regular and nonregular sets, J. Diff. Eqs., 193 (2003), 1-26.  doi: 10.1016/S0022-0396(03)00129-3. [15] A. A. Tolstonogov, Solutions of a differential inclusion with unbounded right-hand side (Russian), Sib. Math. Zh., 29 (1988), 212-225.  doi: 10.1007/BF00970283.
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