# American Institute of Mathematical Sciences

• Previous Article
Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations
• EECT Home
• This Issue
• Next Article
Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass
April  2022, 11(2): 537-557. doi: 10.3934/eect.2021012

## 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 Published  April 2022 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 and Control Theory, 2022, 11 (2) : 537-557. 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. [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. [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. [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. [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. [6] H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math, 12 (1972), 373-390.  doi: 10.1007/BF02764629. [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. [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. [9] N. Bourbaki, Integration, Chapitre V, Hermann, Paris, 1967. [10] B. Brogliato, Nonsmooth Mechanics, 3$^{rd}$ edition, Springer, 2016. doi: 10.1007/978-3-319-28664-8. [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. [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. [13] N. Dinculeanu, Vector Measures, Veb Deitscher Verlag der Wissenschaften, Berlin, 1966. [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. [15] C. J. Himmelberg, Measurable relations, Fundamenta Math., 87 (1975), 53-72.  doi: 10.4064/fm-87-1-53-72. [16] P. Krejči, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Intern. Ser. Math. Sci., 8, Appl. Gakkõtosho Co., Ltd., Tokyo, 1966. [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. [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. [19] B. Maury and J. Venel, Un modéle de mouvement de foule, ESAIM Proc., 18 (2007), 143-152.  doi: 10.1051/proc:071812. [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. [21] J. J. Moreau, On unilateral constraints, friction and plasticity, in New Variational Techniques in Mathematical Physics, Edizioni Cremonese, Rome, 1974,173–222. [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. [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. [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. [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. [26] L. Schwartz, Analysis, 1, Mir, Moscow, 1972. [27] L. Thibault, Moreau sweeping process with bounded truncated retraction, J. Convex Anal., 23 (2016), 1051-1098. [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. [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. [30] A. A. Tolstonogov, Compactness of BV solutions of a convex sweeping process of measurable differential inclusion, J. Convex Anal., 27 (2020), 675-697. [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.

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. [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. [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. [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. [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. [6] H. Attouch and A. Damlamian, On multivalued evolution equations in Hilbert spaces, Israel J. Math, 12 (1972), 373-390.  doi: 10.1007/BF02764629. [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. [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. [9] N. Bourbaki, Integration, Chapitre V, Hermann, Paris, 1967. [10] B. Brogliato, Nonsmooth Mechanics, 3$^{rd}$ edition, Springer, 2016. doi: 10.1007/978-3-319-28664-8. [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. [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. [13] N. Dinculeanu, Vector Measures, Veb Deitscher Verlag der Wissenschaften, Berlin, 1966. [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. [15] C. J. Himmelberg, Measurable relations, Fundamenta Math., 87 (1975), 53-72.  doi: 10.4064/fm-87-1-53-72. [16] P. Krejči, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Intern. Ser. Math. Sci., 8, Appl. Gakkõtosho Co., Ltd., Tokyo, 1966. [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. [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. [19] B. Maury and J. Venel, Un modéle de mouvement de foule, ESAIM Proc., 18 (2007), 143-152.  doi: 10.1051/proc:071812. [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. [21] J. J. Moreau, On unilateral constraints, friction and plasticity, in New Variational Techniques in Mathematical Physics, Edizioni Cremonese, Rome, 1974,173–222. [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. [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. [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. [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. [26] L. Schwartz, Analysis, 1, Mir, Moscow, 1972. [27] L. Thibault, Moreau sweeping process with bounded truncated retraction, J. Convex Anal., 23 (2016), 1051-1098. [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. [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. [30] A. A. Tolstonogov, Compactness of BV solutions of a convex sweeping process of measurable differential inclusion, J. Convex Anal., 27 (2020), 675-697. [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.
 [1] 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 [2] Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations and Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009 [3] Dmitrii Rachinskii. On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3361-3386. doi: 10.3934/dcdsb.2018246 [4] Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100 [5] Tan H. Cao, Boris S. Mordukhovich. Optimality conditions for a controlled sweeping process with applications to the crowd motion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 267-306. doi: 10.3934/dcdsb.2017014 [6] Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709 [7] Kenneth Kuttler. Measurable solutions for elliptic and evolution inclusions. Evolution Equations and Control Theory, 2020, 9 (4) : 1041-1055. doi: 10.3934/eect.2020041 [8] Kevin T. Andrews, Kenneth L. Kuttler, Ji Li, Meir Shillor. Measurable solutions to general evolution inclusions. Evolution Equations and Control Theory, 2020, 9 (4) : 935-960. doi: 10.3934/eect.2020055 [9] Tan H. Cao, Boris S. Mordukhovich. Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4191-4216. doi: 10.3934/dcdsb.2019078 [10] Yinuo Wang, Chuandong Li, Hongjuan Wu, Hao Deng. Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1767-1776. doi: 10.3934/dcdss.2022005 [11] N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119 [12] Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005 [13] Alexander Vladimirov. Equicontinuous sweeping processes. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 565-573. doi: 10.3934/dcdsb.2013.18.565 [14] Zhaoli Liu, Jiabao Su. Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 617-634. doi: 10.3934/dcds.2004.10.617 [15] Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure and Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49 [16] Piotr Gwiazda, Sander C. Hille, Kamila Łyczek, Agnieszka Świerczewska-Gwiazda. Differentiability in perturbation parameter of measure solutions to perturbed transport equation. Kinetic and Related Models, 2019, 12 (5) : 1093-1108. doi: 10.3934/krm.2019041 [17] Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93 [18] Michael Blank. Recurrence for measurable semigroup actions. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1649-1665. doi: 10.3934/dcds.2020335 [19] Domingo Gomez-Perez, Ana-Isabel Gomez, Andrew Tirkel. Arrays composed from the extended rational cycle. Advances in Mathematics of Communications, 2017, 11 (2) : 313-327. doi: 10.3934/amc.2017024 [20] John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501

2021 Impact Factor: 1.169