August  2022, 11(4): 1001-1035. doi: 10.3934/eect.2021034

On a second-order functional evolution problem with time and state dependent maximal monotone operators

LMPA Laboratory, Department of Mathematics, Mohammed Seddik Ben Yahia University, Jijel-Algeria

Received  December 2020 Revised  May 2021 Published  August 2022 Early access  July 2021

The present paper proposes, in a real separable Hilbert space, to analyze the existence of solutions for a class of perturbed second-order state-dependent maximal monotone operators with a finite delay. The dependence of the operators is -in some sense- absolutely continuous (or bounded continuous) variation in time, and Lipschitz continuous in state. The approach to solve our problem is based on a discretization scheme. The uniqueness result is applied to optimal control.

Citation: Soumia Saïdi. On a second-order functional evolution problem with time and state dependent maximal monotone operators. Evolution Equations and Control Theory, 2022, 11 (4) : 1001-1035. doi: 10.3934/eect.2021034
References:
[1]

M. S. AbdoA. G. Ibrahim and S. K. Panchal, State-dependent delayed sweeping process with a noncompact perturbation in Banach spaces, Acta Univ. Apulensis, 54 (2018), 139-159.  doi: 10.17114/j.aua.2018.54.10.

[2]

S. Adly and H. Attouch, Finite Convergence of Proximal-Gradient Inertial Algorithms with Dry Friction Damping, preprint, 2019, hal-02388038. doi: 10.1137/19M1307779.

[3]

S. Adly, H. Attouch and A. Cabot, Finite time stabililization of nonlinear oscillators subject to dry friction, Nonsmooth Mechanics and Analysis, Advances in Mechanics and Mathematics, 12 (2006), 289–304. doi: 10.1007/0-387-29195-4_24.

[4]

S. Adly and B. K. Le, Unbounded second-order state-dependent Moreau's sweeping processes in Hilbert spaces, J. Optim. Theory Appl., 169 (2016), 407-423.  doi: 10.1007/s10957-016-0905-2.

[5]

S. Adly and B. K. Le, Second-order state-dependent sweeping process with unbounded and nonconvex constraints, Pure and Applied Functional Analysis, 3 (2018), 271-285.  doi: 10.1007/s10957-018-1427-x.

[6]

S. Adly and F. Nacry, An existence result for discontinuous second-order nonconvex state-dependent sweeping processes, Appl. Math. Optim., 79 (2019), 515-546.  doi: 10.1007/s00245-017-9446-9.

[7]

H. AttouchA. Cabot and M. O. Czarnecki, Asymptotic behavior of nonautonomous monotone and subgradient evolution equations, Trans. Amer. Math. Soc., 370 (2018), 755-790.  doi: 10.1090/tran/6965.

[8]

H. AttouchA. Cabot and P. Redont, The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations, Adv. Math. Sci. Appl., 12 (2002), 273-306. 

[9]

D. Azzam-LaouirF. AliouaneC. Castaing and M. D. P. Monteiro Marques, Second order time and state dependent sweeping process in Hilbert space, J. Optim. Theory Appl., 182 (2019), 153-188.  doi: 10.1007/s10957-018-01455-x.

[10]

D. Azzam-Laouir, W. Belhoula, C. Castaing and M. D. P. Monteiro Marques, Perturbed evolution problems with absolutely continuous variation in time and applications, J. Fixed Point Theory Appl., 21 (2019). doi: 10.1007/s11784-019-0666-2.

[11]

D. Azzam-LaouirW. BelhoulaC. Castaing and M. D. P. Monteiro Marques, Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators, Evol. Equ. Control Theory, 9 (2020), 219-254.  doi: 10.3934/eect.2020004.

[12]

D. Azzam-LaouirC. Castaing and M. D. P. Monteiro Marques, Perturbed evolution problems with continuous bounded variation in time and applications, Set-Valued Var. Anal., 26 (2018), 693-728.  doi: 10.1007/s11228-017-0432-9.

[13]

D. Azzam-Laouir and S. Lounis, Nonconvex perturbations of second order maximal monotone differential inclusions, Topol. Methods Nonlinear Anal., 35 (2010), 305-317. 

[14]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands, 1976.

[15]

M. Bounkhel, General existence results for second order non convex sweeping process with unbounded perturbations, Port. Math., 60 (2003), 269-304. 

[16]

M. Bounkhel, Existence results for first and second order nonconvex sweeping processes with perturbations and with delay: Fixed point approach, Georgian Math. J., 13 (2006), 239-249.  doi: 10.1515/GMJ.2006.239.

[17]

M. Bounkhel and R. Al-Yusof, First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal., 18 (2010), 151-182.  doi: 10.1007/s11228-010-0134-z.

[18]

M. Bounkhel and D. Azzam, Existence results on the second-order nonconvex sweeping processes with perturbation, Set Valued Anal., 12 (2004), 291-318.  doi: 10.1023/B:SVAN.0000031356.03559.91.

[19]

H. Brézis, Opérateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert, Lecture Notes in Math., North-Holland, 1973.

[20]

B. Brogliato and A. Tanwani, Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability, SIAM Rev., 62 (2020), 3-129.  doi: 10.1137/18M1234795.

[21]

C. Castaing, Quelques problèmes d'évolution du second ordre, Sém. d'Ana. Convexe, Montpellier, vol. 18, 1988.

[22]

C. CastaingT. X. Duc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set Valued Anal., 1 (1993), 109-139.  doi: 10.1007/BF01027688.

[23]

C. CastaingA. Faik and A. Salvadori, Evolution equations governed by m-accretive and subdifferential operators with delay, Int. J. Appl. Math., 2 (2000), 1005-1026. 

[24]

C. Castaing, C. Godet-Thobie and L. X. Truong, Fractional order of evolution inclusion coupled with a time and state dependent maximal monotone operator, Mathematics MDPI, (2020), 1–30.

[25]

C. Castaing and A. G. Ibrahim, Functional differential inclusion on closed sets in Banach spaces, Adv. Math. Econ., 2 (2000), 21-39.  doi: 10.1007/978-4-431-67909-7_2.

[26]

C. CastaingA. G. Ibrahim and M. Yarou, Existence problems in second order evolution inclusions: Discretization and variational approach, Taiwanese J. Math., 12 (2008), 1433-1475.  doi: 10.11650/twjm/1500405034.

[27]

C. CastaingA. G. Ibrahim and M. Yarou, Some contributions to nonconvex sweeping process, J. Nonlinear Convex Anal., 10 (2009), 1-20. 

[28]

C. Castaing and M. D. P. Monteiro Marques, Topological properties of solution sets for sweeping processes with delay, Port. Math., 54 (1997), 485-507. 

[29]

C. CastaingM. D. P. Monteiro Marques and P. Raynaud de Fitte, Second-order evolution problems with time-dependent maximal monotone operator and applications, Adv. Math. Econ., 22 (2018), 25-77. 

[30]

C. Castaing, P. Raynaud de Fitte and M. Valadier, Young Measures on Topological Spaces With Applications in Control Theory and Probability Theory, Kluwer Academic Publishers, Dordrecht, 2004. doi: 10.1007/1-4020-1964-5.

[31]

C. CastaingA. Salvadori and L. Thibault, Functional evolution equations governed by nonconvex sweeping process, J. Nonlinear Convex Anal., 2 (2001), 217-241. 

[32]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math, Springer-Verlag Berlin Heidelberg, 1977.

[33]

T. X. Duc Ha and M. D. P. Monteiro Marques, Nonconvex second order differential inclusions with memory, Set-Valued Anal., 3 (1995), 71-86.  doi: 10.1007/BF01033642.

[34]

A. G. Ibrahim and F. A. Aladsani, Second order evolution inclusions governed by sweeping process in Banach spaces, Le Matematiche, LXIV (2009), 17-39. 

[35]

M. Kunze and M. D. P. Monteiro Marques, BV solutions to evolution problems with time-dependent domains, Set-Valued Anal., 5 (1997), 57-72.  doi: 10.1023/A:1008621327851.

[36]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in Impacts in Mechanical Systems, vol. 551, Springer, Berlin, Heidelberg, 2000, 1–60. doi: 10.1007/3-540-45501-9_1.

[37]

B. K. Le, Well-posedeness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions, Optimization, 69 (2020), 1187-1217.  doi: 10.1080/02331934.2019.1686504.

[38]

M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Birkhauser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-7614-8.

[39]

J. J. Moreau, Unilateral Contact and Dry Friction in FiniteFreedom Dynamics. Nonsmooth Mechanics, CISM Courses and Lectures, vol. 302, Springer, Vienna, New York, 1988.

[40]

J. Noel, Second-order general perturbed sweeping process differential inclusion, J. Fixed Point Theory Appl., 20 (2018), 1-21.  doi: 10.1007/s11784-018-0609-3.

[41]

L. Paoli, An existence result for non-smooth vibro-impact problem, J. Differential Equations, 211 (2005), 247-281.  doi: 10.1016/j.jde.2004.11.008.

[42]

M. Schatzman, Problèmes unilatéraux d'évolution du second ordre en temps, Thèse de Doctorat d'Etat es Sciences Mathématiques, Université Pierre et Marie Curie, Paris 6, 1979.

[43]

F. Selamnia, D. Azzam-Laouir and M. D. P. Monteiro Marques, Evolution problems involving state-dependent maximal monotone operators, Appl. Anal., (2020).

[44]

A. A. Vladimirov, Nonstationary dissipative evolution equations in Hilbert space, Nonlinear Anal., 17 (1991), 499-518.  doi: 10.1016/0362-546X(91)90061-5.

[45]

I. I. Vrabie, Compactness Methods for Nonlinear Evolution Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 32, Longman Scientific and Technical, John Wiley and Sons, Inc., New York, 1987.

show all references

References:
[1]

M. S. AbdoA. G. Ibrahim and S. K. Panchal, State-dependent delayed sweeping process with a noncompact perturbation in Banach spaces, Acta Univ. Apulensis, 54 (2018), 139-159.  doi: 10.17114/j.aua.2018.54.10.

[2]

S. Adly and H. Attouch, Finite Convergence of Proximal-Gradient Inertial Algorithms with Dry Friction Damping, preprint, 2019, hal-02388038. doi: 10.1137/19M1307779.

[3]

S. Adly, H. Attouch and A. Cabot, Finite time stabililization of nonlinear oscillators subject to dry friction, Nonsmooth Mechanics and Analysis, Advances in Mechanics and Mathematics, 12 (2006), 289–304. doi: 10.1007/0-387-29195-4_24.

[4]

S. Adly and B. K. Le, Unbounded second-order state-dependent Moreau's sweeping processes in Hilbert spaces, J. Optim. Theory Appl., 169 (2016), 407-423.  doi: 10.1007/s10957-016-0905-2.

[5]

S. Adly and B. K. Le, Second-order state-dependent sweeping process with unbounded and nonconvex constraints, Pure and Applied Functional Analysis, 3 (2018), 271-285.  doi: 10.1007/s10957-018-1427-x.

[6]

S. Adly and F. Nacry, An existence result for discontinuous second-order nonconvex state-dependent sweeping processes, Appl. Math. Optim., 79 (2019), 515-546.  doi: 10.1007/s00245-017-9446-9.

[7]

H. AttouchA. Cabot and M. O. Czarnecki, Asymptotic behavior of nonautonomous monotone and subgradient evolution equations, Trans. Amer. Math. Soc., 370 (2018), 755-790.  doi: 10.1090/tran/6965.

[8]

H. AttouchA. Cabot and P. Redont, The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations, Adv. Math. Sci. Appl., 12 (2002), 273-306. 

[9]

D. Azzam-LaouirF. AliouaneC. Castaing and M. D. P. Monteiro Marques, Second order time and state dependent sweeping process in Hilbert space, J. Optim. Theory Appl., 182 (2019), 153-188.  doi: 10.1007/s10957-018-01455-x.

[10]

D. Azzam-Laouir, W. Belhoula, C. Castaing and M. D. P. Monteiro Marques, Perturbed evolution problems with absolutely continuous variation in time and applications, J. Fixed Point Theory Appl., 21 (2019). doi: 10.1007/s11784-019-0666-2.

[11]

D. Azzam-LaouirW. BelhoulaC. Castaing and M. D. P. Monteiro Marques, Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators, Evol. Equ. Control Theory, 9 (2020), 219-254.  doi: 10.3934/eect.2020004.

[12]

D. Azzam-LaouirC. Castaing and M. D. P. Monteiro Marques, Perturbed evolution problems with continuous bounded variation in time and applications, Set-Valued Var. Anal., 26 (2018), 693-728.  doi: 10.1007/s11228-017-0432-9.

[13]

D. Azzam-Laouir and S. Lounis, Nonconvex perturbations of second order maximal monotone differential inclusions, Topol. Methods Nonlinear Anal., 35 (2010), 305-317. 

[14]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, The Netherlands, 1976.

[15]

M. Bounkhel, General existence results for second order non convex sweeping process with unbounded perturbations, Port. Math., 60 (2003), 269-304. 

[16]

M. Bounkhel, Existence results for first and second order nonconvex sweeping processes with perturbations and with delay: Fixed point approach, Georgian Math. J., 13 (2006), 239-249.  doi: 10.1515/GMJ.2006.239.

[17]

M. Bounkhel and R. Al-Yusof, First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal., 18 (2010), 151-182.  doi: 10.1007/s11228-010-0134-z.

[18]

M. Bounkhel and D. Azzam, Existence results on the second-order nonconvex sweeping processes with perturbation, Set Valued Anal., 12 (2004), 291-318.  doi: 10.1023/B:SVAN.0000031356.03559.91.

[19]

H. Brézis, Opérateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert, Lecture Notes in Math., North-Holland, 1973.

[20]

B. Brogliato and A. Tanwani, Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability, SIAM Rev., 62 (2020), 3-129.  doi: 10.1137/18M1234795.

[21]

C. Castaing, Quelques problèmes d'évolution du second ordre, Sém. d'Ana. Convexe, Montpellier, vol. 18, 1988.

[22]

C. CastaingT. X. Duc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set Valued Anal., 1 (1993), 109-139.  doi: 10.1007/BF01027688.

[23]

C. CastaingA. Faik and A. Salvadori, Evolution equations governed by m-accretive and subdifferential operators with delay, Int. J. Appl. Math., 2 (2000), 1005-1026. 

[24]

C. Castaing, C. Godet-Thobie and L. X. Truong, Fractional order of evolution inclusion coupled with a time and state dependent maximal monotone operator, Mathematics MDPI, (2020), 1–30.

[25]

C. Castaing and A. G. Ibrahim, Functional differential inclusion on closed sets in Banach spaces, Adv. Math. Econ., 2 (2000), 21-39.  doi: 10.1007/978-4-431-67909-7_2.

[26]

C. CastaingA. G. Ibrahim and M. Yarou, Existence problems in second order evolution inclusions: Discretization and variational approach, Taiwanese J. Math., 12 (2008), 1433-1475.  doi: 10.11650/twjm/1500405034.

[27]

C. CastaingA. G. Ibrahim and M. Yarou, Some contributions to nonconvex sweeping process, J. Nonlinear Convex Anal., 10 (2009), 1-20. 

[28]

C. Castaing and M. D. P. Monteiro Marques, Topological properties of solution sets for sweeping processes with delay, Port. Math., 54 (1997), 485-507. 

[29]

C. CastaingM. D. P. Monteiro Marques and P. Raynaud de Fitte, Second-order evolution problems with time-dependent maximal monotone operator and applications, Adv. Math. Econ., 22 (2018), 25-77. 

[30]

C. Castaing, P. Raynaud de Fitte and M. Valadier, Young Measures on Topological Spaces With Applications in Control Theory and Probability Theory, Kluwer Academic Publishers, Dordrecht, 2004. doi: 10.1007/1-4020-1964-5.

[31]

C. CastaingA. Salvadori and L. Thibault, Functional evolution equations governed by nonconvex sweeping process, J. Nonlinear Convex Anal., 2 (2001), 217-241. 

[32]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math, Springer-Verlag Berlin Heidelberg, 1977.

[33]

T. X. Duc Ha and M. D. P. Monteiro Marques, Nonconvex second order differential inclusions with memory, Set-Valued Anal., 3 (1995), 71-86.  doi: 10.1007/BF01033642.

[34]

A. G. Ibrahim and F. A. Aladsani, Second order evolution inclusions governed by sweeping process in Banach spaces, Le Matematiche, LXIV (2009), 17-39. 

[35]

M. Kunze and M. D. P. Monteiro Marques, BV solutions to evolution problems with time-dependent domains, Set-Valued Anal., 5 (1997), 57-72.  doi: 10.1023/A:1008621327851.

[36]

M. Kunze and M. D. P. Monteiro Marques, An introduction to Moreau's sweeping process, in Impacts in Mechanical Systems, vol. 551, Springer, Berlin, Heidelberg, 2000, 1–60. doi: 10.1007/3-540-45501-9_1.

[37]

B. K. Le, Well-posedeness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions, Optimization, 69 (2020), 1187-1217.  doi: 10.1080/02331934.2019.1686504.

[38]

M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Birkhauser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-7614-8.

[39]

J. J. Moreau, Unilateral Contact and Dry Friction in FiniteFreedom Dynamics. Nonsmooth Mechanics, CISM Courses and Lectures, vol. 302, Springer, Vienna, New York, 1988.

[40]

J. Noel, Second-order general perturbed sweeping process differential inclusion, J. Fixed Point Theory Appl., 20 (2018), 1-21.  doi: 10.1007/s11784-018-0609-3.

[41]

L. Paoli, An existence result for non-smooth vibro-impact problem, J. Differential Equations, 211 (2005), 247-281.  doi: 10.1016/j.jde.2004.11.008.

[42]

M. Schatzman, Problèmes unilatéraux d'évolution du second ordre en temps, Thèse de Doctorat d'Etat es Sciences Mathématiques, Université Pierre et Marie Curie, Paris 6, 1979.

[43]

F. Selamnia, D. Azzam-Laouir and M. D. P. Monteiro Marques, Evolution problems involving state-dependent maximal monotone operators, Appl. Anal., (2020).

[44]

A. A. Vladimirov, Nonstationary dissipative evolution equations in Hilbert space, Nonlinear Anal., 17 (1991), 499-518.  doi: 10.1016/0362-546X(91)90061-5.

[45]

I. I. Vrabie, Compactness Methods for Nonlinear Evolution Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 32, Longman Scientific and Technical, John Wiley and Sons, Inc., New York, 1987.

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