# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021012
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## BV solutions of a convex sweeping process with a composed perturbation

 Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Lermontov str., 134, Irkutsk, 664033 Russia

Received  April 2020 Revised  August 2020 Early access March 2021

A measurable sweeping process with a composed perturbation is considered in a separable Hilbert space. The values of the moving set generating the sweeping process are closed, convex sets. The retraction of the sweeping process is bounded by a positive Radon measure. The perturbation is the sum of two multivalued mappings. The values of the first one are closed, bounded, not necessarily convex sets. It is measurable in the time variable, is Lipschitz continuous in the phase variable, and satisfies a conventional growth condition. The values of the second one are closed, convex, not necessarily bounded sets. We assume that this mapping has a closed with respect to the phase variable graph.

The remaining assumptions concern the intersection of the second mapping and the multivalued mapping defined by the growth conditions. We suppose that this intersection has a measurable selector and has certain compactness properties.

We prove the existence of solutions for our inclusion. The proof is based on the author's theorem on continuous with respect to a parameter selectors passing through fixed points of contraction multivalued maps with closed, nonconvex, decomposable values depending on the parameter, and the classical Ky Fan fixed point theorem. The results which we obtain are new.

Citation: Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, doi: 10.3934/eect.2021012
##### References:
 [1] V. Acary, O. Bonnefon and B. Brogliato, Nonsmooth Modeling and Simulation for Switched Circuits, Springer, 2011. doi: 10.1007/978-90-481-9681-4.  Google Scholar [2] S. Adly, A Variational Approach to Nonsmooth Dynamics. Applications in Unilateral Mechanics and Electronics, Springer Int. Publ., 2017 doi: 10.1007/978-3-319-68658-5.  Google Scholar [3] S. Adly, A. Hantoute and B. K. Le, Nonsmooth Lur'e Dynamical Systems in Hilbert Spaces, Set-Valued Var. Anal., 24 (2016), 13-35.  doi: 10.1007/s11228-015-0334-7.  Google Scholar [4] S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program., 148 (2014), 5-47.  doi: 10.1007/s10107-014-0754-4.  Google Scholar [5] S. Adly, F. Nacry and L. Thibault, Discontinuous sweeping process with prox-regular sets, ESAIM; COCV, 23 (2017), 1293-1329.  doi: 10.1051/cocv/2016053.  Google Scholar [6] H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math, 12 (1972), 373-390.  doi: 10.1007/BF02764629.  Google Scholar [7] J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Springer-Verlag, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar [8] D. Azzam-Laouir, A. Makhlouf and L. Thibault, On perturbed sweeping process, Applicable Anal., 95 (2016), 303-322.  doi: 10.1080/00036811.2014.1002482.  Google Scholar [9] N. Bourbaki, Integration, Chapitre V, Hermann, Paris, 1967.  Google Scholar [10] B. Brogliato, Nonsmooth Mechanics, 3$^{rd}$ edition, Springer, 2016. doi: 10.1007/978-3-319-28664-8.  Google Scholar [11] B. Brogliato and D. Goeleven, Existence, uniqueness of solutions and stability of nonsmooth multivalued Lur'e dynamical systems, J. Convex Anal., 20 (2013), 881-900.   Google Scholar [12] B. Brogliato and A. Tanwani, Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability, SIAM Review, 62 (2020), 3-129.  doi: 10.1137/18M1234795.  Google Scholar [13] N. Dinculeanu, Vector Measures, Veb Deitscher Verlag der Wissenschaften, Berlin, 1966.  Google Scholar [14] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Diff. Equat., 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.  Google Scholar [15] C. J. Himmelberg, Measurable relations, Fundamenta Math., 87 (1975), 53-72.  doi: 10.4064/fm-87-1-53-72.  Google Scholar [16] P. Krejči, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Intern. Ser. Math. Sci., 8, Appl. Gakkõtosho Co., Ltd., Tokyo, 1966.  Google Scholar [17] M. Kunze and M. Monteiro-Marques, An introduction to Moreau's sweeping process, in Lecture Notes in Phys., 551, Springer, Berlin, 2000, 1–60. doi: 10.1007/3-540-45501-9_1.  Google Scholar [18] F. Ky, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acod Sci. USA, 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.  Google Scholar [19] B. Maury and J. Venel, Un modéle de mouvement de foule, ESAIM Proc., 18 (2007), 143-152.  doi: 10.1051/proc:071812.  Google Scholar [20] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooths Mechanical Problems. Shocks and Dry Friction, Birkhäuser, Basel-Boston-Berlin, 1993. doi: 10.1007/978-3-0348-7614-8.  Google Scholar [21] J. J. Moreau, On unilateral constraints, friction and plasticity, in New Variational Techniques in Mathematical Physics, Edizioni Cremonese, Rome, 1974,173–222.  Google Scholar [22] J. J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints, in Unilateral Problems in Structural Analysis, Springer, Vienna, (1985), 173–222. doi: 10.1007/978-3-7091-2632-5_9.  Google Scholar [23] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Equat., 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar [24] F. Nacry, Perturbed BV sweeping process involving prox-regular sets, J. Nonlinear and Convex Anal., 18 (2017), 1619-1651.  doi: 10.1631/jzus.a1600416.  Google Scholar [25] F. Nacry and L. Thibault, BV prox-regular sweeping process with bounded truncated variation, Optimization, 69 (2020), 1391-1437.  doi: 10.1080/02331934.2018.1514039.  Google Scholar [26] L. Schwartz, Analysis, 1, Mir, Moscow, 1972. Google Scholar [27] L. Thibault, Moreau sweeping process with bounded truncated retraction, J. Convex Anal., 23 (2016), 1051-1098.   Google Scholar [28] S. A. Timoshin and A. A. Tolstonogov, Existence and relaxation of BV solutions for a sweeping process with a nonconvex-valued perturbation, J. Convex Anal., 27 (2020), 647-674.   Google Scholar [29] A. A. Tolstonogov, Continuous selectors of fixed point sets of multifunctions with decomposable valued, Set-valued Anal., 6 (1998), 129-147.  doi: 10.1023/A:1008690526107.  Google Scholar [30] A. A. Tolstonogov, Compactness of BV solutions of a convex sweeping process of measurable differential inclusion, J. Convex Anal., 27 (2020), 675-697.   Google Scholar [31] A. A. Tolstonogov and D. A. Tolstonogov, $L_p$-continuous extreme selectors of multifunctions with decomposable values. Existence theorems, Set-valued Anal., 4 (1996), 173-203.  doi: 10.1007/BF00425964.  Google Scholar

show all references

##### References:
 [1] V. Acary, O. Bonnefon and B. Brogliato, Nonsmooth Modeling and Simulation for Switched Circuits, Springer, 2011. doi: 10.1007/978-90-481-9681-4.  Google Scholar [2] S. Adly, A Variational Approach to Nonsmooth Dynamics. Applications in Unilateral Mechanics and Electronics, Springer Int. Publ., 2017 doi: 10.1007/978-3-319-68658-5.  Google Scholar [3] S. Adly, A. Hantoute and B. K. Le, Nonsmooth Lur'e Dynamical Systems in Hilbert Spaces, Set-Valued Var. Anal., 24 (2016), 13-35.  doi: 10.1007/s11228-015-0334-7.  Google Scholar [4] S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program., 148 (2014), 5-47.  doi: 10.1007/s10107-014-0754-4.  Google Scholar [5] S. Adly, F. Nacry and L. Thibault, Discontinuous sweeping process with prox-regular sets, ESAIM; COCV, 23 (2017), 1293-1329.  doi: 10.1051/cocv/2016053.  Google Scholar [6] H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math, 12 (1972), 373-390.  doi: 10.1007/BF02764629.  Google Scholar [7] J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory, Springer-Verlag, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar [8] D. Azzam-Laouir, A. Makhlouf and L. Thibault, On perturbed sweeping process, Applicable Anal., 95 (2016), 303-322.  doi: 10.1080/00036811.2014.1002482.  Google Scholar [9] N. Bourbaki, Integration, Chapitre V, Hermann, Paris, 1967.  Google Scholar [10] B. Brogliato, Nonsmooth Mechanics, 3$^{rd}$ edition, Springer, 2016. doi: 10.1007/978-3-319-28664-8.  Google Scholar [11] B. Brogliato and D. Goeleven, Existence, uniqueness of solutions and stability of nonsmooth multivalued Lur'e dynamical systems, J. Convex Anal., 20 (2013), 881-900.   Google Scholar [12] B. Brogliato and A. Tanwani, Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability, SIAM Review, 62 (2020), 3-129.  doi: 10.1137/18M1234795.  Google Scholar [13] N. Dinculeanu, Vector Measures, Veb Deitscher Verlag der Wissenschaften, Berlin, 1966.  Google Scholar [14] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Diff. Equat., 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.  Google Scholar [15] C. J. Himmelberg, Measurable relations, Fundamenta Math., 87 (1975), 53-72.  doi: 10.4064/fm-87-1-53-72.  Google Scholar [16] P. Krejči, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Intern. Ser. Math. Sci., 8, Appl. Gakkõtosho Co., Ltd., Tokyo, 1966.  Google Scholar [17] M. Kunze and M. Monteiro-Marques, An introduction to Moreau's sweeping process, in Lecture Notes in Phys., 551, Springer, Berlin, 2000, 1–60. doi: 10.1007/3-540-45501-9_1.  Google Scholar [18] F. Ky, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acod Sci. USA, 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.  Google Scholar [19] B. Maury and J. Venel, Un modéle de mouvement de foule, ESAIM Proc., 18 (2007), 143-152.  doi: 10.1051/proc:071812.  Google Scholar [20] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooths Mechanical Problems. Shocks and Dry Friction, Birkhäuser, Basel-Boston-Berlin, 1993. doi: 10.1007/978-3-0348-7614-8.  Google Scholar [21] J. J. Moreau, On unilateral constraints, friction and plasticity, in New Variational Techniques in Mathematical Physics, Edizioni Cremonese, Rome, 1974,173–222.  Google Scholar [22] J. J. Moreau, Standard inelastic shocks and the dynamics of unilateral constraints, in Unilateral Problems in Structural Analysis, Springer, Vienna, (1985), 173–222. doi: 10.1007/978-3-7091-2632-5_9.  Google Scholar [23] J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Equat., 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar [24] F. Nacry, Perturbed BV sweeping process involving prox-regular sets, J. Nonlinear and Convex Anal., 18 (2017), 1619-1651.  doi: 10.1631/jzus.a1600416.  Google Scholar [25] F. Nacry and L. Thibault, BV prox-regular sweeping process with bounded truncated variation, Optimization, 69 (2020), 1391-1437.  doi: 10.1080/02331934.2018.1514039.  Google Scholar [26] L. Schwartz, Analysis, 1, Mir, Moscow, 1972. Google Scholar [27] L. Thibault, Moreau sweeping process with bounded truncated retraction, J. Convex Anal., 23 (2016), 1051-1098.   Google Scholar [28] S. A. Timoshin and A. A. Tolstonogov, Existence and relaxation of BV solutions for a sweeping process with a nonconvex-valued perturbation, J. Convex Anal., 27 (2020), 647-674.   Google Scholar [29] A. A. Tolstonogov, Continuous selectors of fixed point sets of multifunctions with decomposable valued, Set-valued Anal., 6 (1998), 129-147.  doi: 10.1023/A:1008690526107.  Google Scholar [30] A. A. Tolstonogov, Compactness of BV solutions of a convex sweeping process of measurable differential inclusion, J. Convex Anal., 27 (2020), 675-697.   Google Scholar [31] A. A. Tolstonogov and D. A. Tolstonogov, $L_p$-continuous extreme selectors of multifunctions with decomposable values. Existence theorems, Set-valued Anal., 4 (1996), 173-203.  doi: 10.1007/BF00425964.  Google Scholar
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