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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. AdlyA. 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. AdlyT. 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. AdlyF. 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-LaouirA. 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. AdlyA. 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. AdlyT. 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. AdlyF. 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-LaouirA. 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.

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