doi: 10.3934/mcrf.2022019
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Second-order problems involving time-dependent subdifferential operators and application to control

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

* Corresponding author: Soumia Saïdi

Received  November 2021 Revised  March 2022 Early access April 2022

The paper provides a new result concerning the existence of solutions for second-order evolution problems associated with time-dependent subdifferential operators involving both single-valued and mixed semi-continuous set-valued perturbations. Optimal control problems corresponding to such differential inclusions using relaxation theorems with Young measures are investigated. The existence of solutions for a coupled system governed by a second-order differential equation with an evolution problem is also addressed.

Citation: Soumia Saïdi, Fatima Fennour. Second-order problems involving time-dependent subdifferential operators and application to control. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022019
References:
[1]

S. Adly and H. Attouch, Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping, SIAM J. Optim., 30 (2020), 2134-2162.  doi: 10.1137/19M1307779.

[2]

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

[3]

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.

[4]

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.

[5]

S. Aizicovici and N. H. Pavel, Anti-periodic solutions to a class of nonlinear differential equations in Hilbert Space, J. Funct. Anal., 99 (1991), 387-408.  doi: 10.1016/0022-1236(91)90046-8.

[6]

F. AliouaneD. Azzam-LaouirC. 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.

[7]

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. 

[8]

H. Attouch and A. Damlamian, Problèmes d'évolution dans les Hilberts et applications, J. Math. Pures Appl., 54 (1975), 53-74. 

[9]

H. AttouchP.-E. Maingé and P. Redont, A second-order differential system with hessian driven damping; application to non-elastic shock laws, Differ. Equ. Appl., 4 (2012), 27-65.  doi: 10.7153/dea-04-04.

[10]

D. Azzam-Laouir, Mixed semicontinuous perturbation of a second order nonconvex sweeping process, Electron. J. Qual. Theory Differ. Equ., (2008), No. 37, 9 pp. doi: 10.14232/ejqtde.2008.1.37.

[11]

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), Paper No. 40, 32 pp. doi: 10.1007/s11784-019-0666-2.

[12]

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

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D. Azzam-Laouir and I. Boutana-Harid, Mixed semicontinuous perturbation to an evolution problem with time dependent maximal monotone operator, J. Nonlinear Convex Anal., 20 (2019), 39-52. 

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D. Azzam-Laouir and S. Izza, Existence of solutions for second-order perturbed nonconvex sweeping process, Comput. Math. Appl., 62 (2011), 1736-1744.  doi: 10.1016/j.camwa.2011.06.015.

[15]

D. Azzam-LaouirS. Izza and L. Thibault, Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process, Set-Valued Var. Anal., 22 (2014), 271-283.  doi: 10.1007/s11228-013-0248-1.

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

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D. Azzam-Laouir and S. Melit, Existence of solutions for a second order boundary value problem with the Clarke subdifferential, Filomat, 31 (2017), 2763-2771.  doi: 10.2298/FIL1709763A.

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M. Bergounioux and L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 35, 38 pp. doi: 10.1051/cocv/2019021.

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

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M. Bounkhel and T. Haddad, An existence result for a new variant of the nonconvex sweeping process, Port. Math., 65 (2008), 33-47.  doi: 10.4171/PM/1797.

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R. D. Bourgin, Geometric Aspects of Convex Sets with the Radon Nikodym Property, Lecture Notes in Math., no. 993, Springer Verlag, Berlin, 1983. doi: 10.1007/BFb0069321.

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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.

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K. Camlibel, L. Iannelli and A. Tanwani, Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators, Math. Program., 2021. doi: 10.1007/s10107-021-01666-7.

[26]

C. Castaing, Quelques problèmes d'évolution du second ordre, Séminaire d'Analyse Convexe, Montpellier, 18 (1988), exposé n 5.

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

[28]

C. CastaingC. Godet-Thobie and F. Z. Mostefai, On a fractional differential inclusion with boundary conditions and application to subdifferential operators, J. Nonlinear Convex Anal., 18 (2017), 1717-1752. 

[29]

C. CastaingC. Godet-ThobieP. D. Phung and L. X. Truong, On fractional differential inclusions with nonlocal boundary conditions, Fract. Calc. Appl. Anal., 22 (2019), 444-478.  doi: 10.1515/fca-2019-0027.

[30]

C. CastaingC. Godet-Thobie and S. Saïdi, On fractional evolution inclusion coupled with a time and state dependent maximal monotone operator, Set-Valued Var. Anal., (2021).  doi: 10.1007/s11228-021-00611-2.

[31]

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.

[32]

C. CastaingC. Godet-ThobieL. X. Truong and and B. Satco, Optimal control problems governed by a second order ordinary differential equation with $m$-point boundary condition, Adv. Math. Econ., 18 (2014), 1-59.  doi: 10.1007/978-4-431-54834-8_1.

[33]

C. Castaing and T. Haddad, Relaxation and Bolza problem involving a second order evolution inclusion, J. Nonlinear Convex Anal., 9 (2008), 141-159. 

[34]

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

[35]

C. CastaingM. D. P. Monteiro Marques and P. Raynaud de Fitte, Some problems in optimal control governed by the sweeping process, J. Nonlinear Convex Anal., 15 (2014), 1043-1070. 

[36]

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. 

[37]

C. CastaingM. D. P. Monteiro Marques and S. Saïdi, Evolution problems with time-dependent subdifferential operators, Adv. Math. Econ., 23 (2020), 1-39. 

[38]

C. CastaingP. Raynaud de Fitte and A. Salvadori, Some variational convergence results with application to evolution inclusions, Adv. Math. Econ., 8 (2006), 33-73.  doi: 10.1007/4-431-30899-7_2.

[39]

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.

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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math, 580, Springer-Verlag Berlin Heidelberg, 1977.

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A. Cernea, On a fractional differential inclusion with boundary condition, Stud. Univ. Babeş-Bolyai Math., 55 (2010), 105-113.

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J. F. Edmond and L. Thibault, Relaxation and optimal control problem involving a perturbed sweeping process, Math. Program., 104 (2005), 347-373.  doi: 10.1007/s10107-005-0619-y.

[43]

A. Fryszkowski and L. Górniewicz, Mixed semi-continuous mappings and their applications to differential inclusions, Set-Valued Anal., 8 (2000), 203-217.  doi: 10.1023/A:1008763724495.

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T. HaddadA. Jourani and L. Thibault, Reduction of sweeping process to unconstrained differential inclusion, Pac. J. Optim., 4 (2008), 493-512. 

[45]

T. Haddad and L. Thibault, Mixed semi-continuous perturbation of nonconvex sweeping process, Math. Program., 123 (2010), 225-240.  doi: 10.1007/s10107-009-0315-4.

[46]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. â…¡, volume 500 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4615-4665-8_17.

[47]

A. Jourani and E. Vilches, Galerkin-like method and generalized perturbed sweeping process with nonregular sets, SIAM J. Control Optim., 55 (2017), 2412-2436.  doi: 10.1137/16M1078288.

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N. Kenmochi, Some nonlinear parabolic inequalities, Israel J. Math., 22 (1975), 304-331.  doi: 10.1007/BF02761596.

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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.

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S. LounisT. Haddad and M. Sene, Non-convex second-order Moreau's sweeping processes in Hilbert spaces, J. Fixed Point Theory Appl., 19 (2017), 2895-2908.  doi: 10.1007/s11784-017-0461-x.

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J. J. Moreau, Rafle par un convexe variable, I, Sém. Anal. Convexe, Montpellier, Vol. 1 (1971), Exposé No. 15.

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show all references

References:
[1]

S. Adly and H. Attouch, Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping, SIAM J. Optim., 30 (2020), 2134-2162.  doi: 10.1137/19M1307779.

[2]

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

[3]

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.

[4]

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.

[5]

S. Aizicovici and N. H. Pavel, Anti-periodic solutions to a class of nonlinear differential equations in Hilbert Space, J. Funct. Anal., 99 (1991), 387-408.  doi: 10.1016/0022-1236(91)90046-8.

[6]

F. AliouaneD. Azzam-LaouirC. 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.

[7]

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. 

[8]

H. Attouch and A. Damlamian, Problèmes d'évolution dans les Hilberts et applications, J. Math. Pures Appl., 54 (1975), 53-74. 

[9]

H. AttouchP.-E. Maingé and P. Redont, A second-order differential system with hessian driven damping; application to non-elastic shock laws, Differ. Equ. Appl., 4 (2012), 27-65.  doi: 10.7153/dea-04-04.

[10]

D. Azzam-Laouir, Mixed semicontinuous perturbation of a second order nonconvex sweeping process, Electron. J. Qual. Theory Differ. Equ., (2008), No. 37, 9 pp. doi: 10.14232/ejqtde.2008.1.37.

[11]

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), Paper No. 40, 32 pp. doi: 10.1007/s11784-019-0666-2.

[12]

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

[13]

D. Azzam-Laouir and I. Boutana-Harid, Mixed semicontinuous perturbation to an evolution problem with time dependent maximal monotone operator, J. Nonlinear Convex Anal., 20 (2019), 39-52. 

[14]

D. Azzam-Laouir and S. Izza, Existence of solutions for second-order perturbed nonconvex sweeping process, Comput. Math. Appl., 62 (2011), 1736-1744.  doi: 10.1016/j.camwa.2011.06.015.

[15]

D. Azzam-LaouirS. Izza and L. Thibault, Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process, Set-Valued Var. Anal., 22 (2014), 271-283.  doi: 10.1007/s11228-013-0248-1.

[16]

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

[17]

D. Azzam-Laouir and S. Melit, Existence of solutions for a second order boundary value problem with the Clarke subdifferential, Filomat, 31 (2017), 2763-2771.  doi: 10.2298/FIL1709763A.

[18]

E. J. Balder, New Fundamentals of Young Measure Convergence, Calculus of Variations and Optimal Control, (Haifa, 1998) (Boca Raton, FL) Chapman and Hall, 2000, 24-48.

[19]

H. BenabdellahC. Castaing and A. Salvadori, Compactness and discretization methods for differential inclusions and evolution problems, Atti. Sem. Math. Fis. Univ. Modena, 45 (1997), 9-51. 

[20]

M. Bergounioux and L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 35, 38 pp. doi: 10.1051/cocv/2019021.

[21]

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

[22]

M. Bounkhel and T. Haddad, An existence result for a new variant of the nonconvex sweeping process, Port. Math., 65 (2008), 33-47.  doi: 10.4171/PM/1797.

[23]

R. D. Bourgin, Geometric Aspects of Convex Sets with the Radon Nikodym Property, Lecture Notes in Math., no. 993, Springer Verlag, Berlin, 1983. doi: 10.1007/BFb0069321.

[24]

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.

[25]

K. Camlibel, L. Iannelli and A. Tanwani, Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators, Math. Program., 2021. doi: 10.1007/s10107-021-01666-7.

[26]

C. Castaing, Quelques problèmes d'évolution du second ordre, Séminaire d'Analyse Convexe, Montpellier, 18 (1988), exposé n 5.

[27]

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

[28]

C. CastaingC. Godet-Thobie and F. Z. Mostefai, On a fractional differential inclusion with boundary conditions and application to subdifferential operators, J. Nonlinear Convex Anal., 18 (2017), 1717-1752. 

[29]

C. CastaingC. Godet-ThobieP. D. Phung and L. X. Truong, On fractional differential inclusions with nonlocal boundary conditions, Fract. Calc. Appl. Anal., 22 (2019), 444-478.  doi: 10.1515/fca-2019-0027.

[30]

C. CastaingC. Godet-Thobie and S. Saïdi, On fractional evolution inclusion coupled with a time and state dependent maximal monotone operator, Set-Valued Var. Anal., (2021).  doi: 10.1007/s11228-021-00611-2.

[31]

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.

[32]

C. CastaingC. Godet-ThobieL. X. Truong and and B. Satco, Optimal control problems governed by a second order ordinary differential equation with $m$-point boundary condition, Adv. Math. Econ., 18 (2014), 1-59.  doi: 10.1007/978-4-431-54834-8_1.

[33]

C. Castaing and T. Haddad, Relaxation and Bolza problem involving a second order evolution inclusion, J. Nonlinear Convex Anal., 9 (2008), 141-159. 

[34]

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

[35]

C. CastaingM. D. P. Monteiro Marques and P. Raynaud de Fitte, Some problems in optimal control governed by the sweeping process, J. Nonlinear Convex Anal., 15 (2014), 1043-1070. 

[36]

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. 

[37]

C. CastaingM. D. P. Monteiro Marques and S. Saïdi, Evolution problems with time-dependent subdifferential operators, Adv. Math. Econ., 23 (2020), 1-39. 

[38]

C. CastaingP. Raynaud de Fitte and A. Salvadori, Some variational convergence results with application to evolution inclusions, Adv. Math. Econ., 8 (2006), 33-73.  doi: 10.1007/4-431-30899-7_2.

[39]

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.

[40]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math, 580, Springer-Verlag Berlin Heidelberg, 1977.

[41]

A. Cernea, On a fractional differential inclusion with boundary condition, Stud. Univ. Babeş-Bolyai Math., 55 (2010), 105-113.

[42]

J. F. Edmond and L. Thibault, Relaxation and optimal control problem involving a perturbed sweeping process, Math. Program., 104 (2005), 347-373.  doi: 10.1007/s10107-005-0619-y.

[43]

A. Fryszkowski and L. Górniewicz, Mixed semi-continuous mappings and their applications to differential inclusions, Set-Valued Anal., 8 (2000), 203-217.  doi: 10.1023/A:1008763724495.

[44]

T. HaddadA. Jourani and L. Thibault, Reduction of sweeping process to unconstrained differential inclusion, Pac. J. Optim., 4 (2008), 493-512. 

[45]

T. Haddad and L. Thibault, Mixed semi-continuous perturbation of nonconvex sweeping process, Math. Program., 123 (2010), 225-240.  doi: 10.1007/s10107-009-0315-4.

[46]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. â…¡, volume 500 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4615-4665-8_17.

[47]

A. Jourani and E. Vilches, Galerkin-like method and generalized perturbed sweeping process with nonregular sets, SIAM J. Control Optim., 55 (2017), 2412-2436.  doi: 10.1137/16M1078288.

[48]

N. Kenmochi, Some nonlinear parabolic inequalities, Israel J. Math., 22 (1975), 304-331.  doi: 10.1007/BF02761596.

[49]

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.

[50]

S. LounisT. Haddad and M. Sene, Non-convex second-order Moreau's sweeping processes in Hilbert spaces, J. Fixed Point Theory Appl., 19 (2017), 2895-2908.  doi: 10.1007/s11784-017-0461-x.

[51]

K. Maruo, On certain nonlinear differential equations of second order in time, Osaka J. Math., 23 (1986), 1-53. 

[52]

J. J. Moreau, Rafle par un convexe variable, I, Sém. Anal. Convexe, Montpellier, Vol. 1 (1971), Exposé No. 15.

[53]

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