# Second-Order Evolution Problems with Time-Dependent Maximal Monotone Operator and Applications

• C. Castaing
• M. D. P. Monteiro Marques
• P. Raynaud de Fitte
Chapter
Part of the Advances in Mathematical Economics book series (MATHECON, volume 22)

## Abstract

We consider at first the existence and uniqueness of solution for a general second-order evolution inclusion in a separable Hilbert space of the form
$$\displaystyle 0\in \ddot u(t) + A(t) \dot u(t) + f(t, u(t)), \hskip 2pt t\in [0, T]$$
where A(t) is a time dependent with Lipschitz variation maximal monotone operator and the perturbation f(t, .) is boundedly Lipschitz. Several new results are presented in the sense that these second-order evolution inclusions deal with time-dependent maximal monotone operators by contrast with the classical case dealing with some special fixed operators. In particular, the existence and uniqueness of solution to
$$\displaystyle 0= \ddot u(t) + A(t) \dot u(t) + \nabla \varphi (u(t)), \hskip 2pt t\in [0, T]$$
where A(t) is a time dependent with Lipschitz variation single-valued maximal monotone operator and ∇φ is the gradient of a smooth Lipschitz function φ are stated. Some more general inclusion of the form
$$\displaystyle 0\in \ddot u(t) + A(t) \dot u(t) + \partial \Phi (u(t)), \hskip 2pt t\in [0, T]$$
where  Φ(u(t)) denotes the subdifferential of a proper lower semicontinuous convex function Φ at the point u(t) is provided via a variational approach. Further results in second-order problems involving both absolutely continuous in variation maximal monotone operator and bounded in variation maximal monotone operator, A(t), with perturbation f : [0, T] × H × H are stated. Second- order evolution inclusion with perturbation f and Young measure control νt
$$\displaystyle \left \{ \begin {array}{lll} 0\in \ddot u_{x, y, \nu }(t) + A(t) \dot u_{x, y, \nu }(t) + f(t, u_{x, y, \nu }(t))+ \operatorname {{\mathrm {bar}}}(\nu _t), \hskip 2pt t \in [0, T] \\ u_{x, y, \nu }(0) = x, \dot u_{x, y, \nu } (0) =y \in D(A(0)) \end {array} \right .$$
where $$\operatorname {{\mathrm {bar}}}(\nu _t)$$ denotes the barycenter of the Young measure νt is considered, and applications to optimal control are presented. Some variational limit theorems related to convex sweeping process are provided.

## Keywords

Bolza control problem Lipschitz mapping Maximal monotone operators Pseudo-distance Subdifferential Viscosity Young measures

## References

1. 1.
Adly S, Haddad T (2018) An implicit sweeping process approach to quasistatic evolution variational inequalities. Siam J Math Anal 50(1):761–778
2. 2.
Adly S, Haddad T, Thibault L (2014) Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. Math Program 148(1–2, Ser. B):5–47
3. 3.
Aliouane F, Azzam-Laouir D, Castaing C, Monteiro Marques MDP (2018, Preprint) Second order time and state dependent sweeping process in Hilbert spaceGoogle Scholar
4. 4.
Attouch H, Cabot A, Redont P (2002) The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations. Adv Math Sci Appl 12(1):273–306. Gakkotosho, TokyoGoogle Scholar
5. 5.
Azzam-Laouir D, Izza S, Thibault L (2014) Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process. Set Valued Var Anal 22:271–283
6. 6.
Azzam-Laouir D, Makhlouf M, Thibault L (2016) On perturbed sweeping process. Appl Anal 95(2):303–322
7. 7.
Azzam-Laouir D, Castaing C, Monteiro Marques MDP (2017) Perturbed evolution problems with continuous bounded variation in time and applications. Set-Valued Var Anal. https://doi.org/10.1007/s11228-017-0432-9
8. 8.
Azzam-Laouir D, Castaing C, Belhoula W, Monteiro Marques MDP (2017, Preprint) Perturbed evolution problems with absolutely continuous variation in time and applicationsGoogle Scholar
9. 9.
Barbu (1976) Nonlinear semigroups and differential equations in Banach spaces. Noordhoff International Publisher, Leyden
10. 10.
Benabdellah H, Castaing C (1995) BV solutions of multivalued differential equations on closed moving sets in Banach spaces. Banach center publications, vol 32. Institute of Mathematics, Polish Academy of Sciences, Warszawa
11. 11.
Benabdellah H, Castaing C, Salvadori A (1997) Compactness and discretization methods for differential inclusions and evolution problems. Atti Sem Mat Fis Univ Modena XLV:9–51Google Scholar
12. 12.
Brezis H (1972) Opérateurs maximaux monotones dans les espaces de Hilbert et equations dévolution. Lectures notes 5. North Holland Publishing Co, AmsterdamGoogle Scholar
13. 13.
Brezis H (1979) Opérateurs maximaux monotones et semi-groupes de contraction dans un espace de Hilbert. North Holland Publishing Co, AmsterdamGoogle Scholar
14. 14.
Castaing C (1970) Quelques résultats de compacité liés a l’ intégration. C R Acd Sci Paris 270:1732–1735; et Bulletin Soc Math France 31:73–81 (1972)Google Scholar
15. 15.
Castaing C (1980) Topologie de la convergence uniforme sur les parties uniformément intégrables de $$L^{1}_{E}$$ et théorèmes de compacité faible dans certains espaces du type Köthe-Orlicz. Travaux Sém Anal Convexe 10(1):27. exp. no. 5Google Scholar
16. 16.
Castaing C, Marcellin S (2007) Evolution inclusions with pln functions and application to viscosity and control. JNCA 8(2):227–255Google Scholar
17. 17.
Castaing C, Monteiro Marques MDP (1996) Evolution problems associated with nonconvex closed moving sets with bounded variation. Portugaliae Mathematica 53(1):73–87; FascGoogle Scholar
18. 18.
Castaing C, Monteiro Marques MDP (1995) BV Periodic solutions of an evolution problem associated with continuous convex sets. Set Valued Anal 3:381–399
19. 19.
Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions. Lectures notes in mathematics. Springer, Berlin, p 580Google Scholar
20. 20.
Castaing C, Duc Ha TX, Valadier M (1993) Evolution equations governed by the sweeping process. Set-Valued Anal 1:109–139
21. 21.
Castaing C, Salvadori A, Thibault L (2001) Functional evolution equations governed by nonconvex sweeeping process. J Nonlinear Convex Anal 2(2):217–241Google Scholar
22. 22.
Castaing C, Raynaud de Fitte P, Valadier M (2004) Young measures on topological spaces with applications in control theory and probability theory. Kluwer Academic Publishers, Dordrecht
23. 23.
Castaing C, Raynaud de Fitte P, Salvadori A (2006) Some variational convergence results with application to evolution inclusions. Adv Math Econ 8:33–73Google Scholar
24. 24.
Castaing C, Ibrahim AG, Yarou M (2009) Some contributions to nonconvex sweeping process. J Nonlinear Convex Anal 10(1):1–20Google Scholar
25. 25.
Castaing C, Monteiro Marques MDP, Raynaud de Fitte P (2014) Some problems in optimal control governed by the sweeping process. J Nonlinear Convex Anal 15(5):1043–1070Google Scholar
26. 26.
Castaing C, Monteiro Marques MDP, Raynaud de Fitte P (2016) A Skorohod problem governed by a closed convex moving set. J Convex Anal 23(2):387–423Google Scholar
27. 27.
Castaing C, Le Xuan T, Raynaud de Fitte P, Salvadori A (2017) Some problems in second order evolution inclusions with boundary condition: a variational approach. Adv Math Econ 21:1–46Google Scholar
28. 28.
Colombo G, Goncharov VV (1999) The sweeping processes without convexity. Set Valued Anal 7:357–374Google Scholar
29. 29.
Cornet B (1983) Existence of slow solutions for a class of differential inclusions. J Math Anal Appl 96:130–147
30. 30.
Edmond JF, Thibault L (2005) Relaxation and optimal control problem involving a perturbed sweeping process. Math Program Ser B 104:347–373Google Scholar
31. 31.
Flam S, Hiriart-Urruty J-B, Jourani A (2009) Feasibility in finite time. J Dyn Control Syst 15:537–555
32. 32.
Florescu LC, Godet-Thobie C (2012) Young measures and compactness in measure spaces. De Gruyter, BerlinGoogle Scholar
33. 33.
Grothendieck A (1964) Espaces vectoriels topologiques Mat, 3rd edn. Sociedade de matematica, Saõ PauloGoogle Scholar
34. 34.
Haddad T, Noel J, Thibault L (2016) Perturbed Sweeping process with subsmooth set depending on the state. Linear Nonlinear Anal 2(1):155–174Google Scholar
35. 35.
Henry C (1973) An existence theorem for a class of differential equations with multivalued right-hand side. J Math Anal Appl 41:179–186
36. 36.
Idzik A (1988) Almost fixed points theorems. Proc Am Math Soc 104:779–784
37. 37.
Kenmochi N (1981) Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull Fac Educ Chiba Univ 30:1–87Google Scholar
38. 38.
Kunze M, Monteiro Marques MDP (1997) BV solutions to evolution problems with time-dependent domains. Set Valued Anal 5:57–72Google Scholar
39. 39.
Monteiro Marques MDP (1984) Perturbations convexes semi-continues supérieurement de problèmes d’évolution dans les espaces de Hilbert, vol 14. Séminaire d’Analyse Convexe, Montpellier, exposé n 2Google Scholar
40. 40.
Monteiro Marques MDP (1993) Differential inclusions nonsmooth mechanical problems, shocks and dry friction. Progress in nonlinear differential equations and their applications, vol 9. Birkhauser, BaselGoogle Scholar
41. 41.
Moreau JJ (1977) Evolution problem associated with a moving convex set in a Hilbert Space. J Differ Equ 26:347–374
42. 42.
Moreau JJ, Valadier M (1987) A chain rule involving vector functions of bounded variations. J Funct Anal 74(2):333–345
43. 43.
Paoli L (2005) An existence result for non-smooth vibro-impact problem. J Differ Equ 211(2):247–281
44. 44.
Park S (2006) Fixed points of approximable or Kakutani maps. J Nonlinear Convex Anal 7(1):1–17Google Scholar
45. 45.
Recupero V (2016) Sweeping processes and rate independence. J Convex Anal 23:921–946Google Scholar
46. 46.
Rockafellar RT (1971) Integrals which are convex functionals, II. Pac J Math 39(2):439–369
47. 47.
Saidi S, Thibault L, Yarou M (2013) Relaxation of optimal control problems involving time dependent subdifferential operators. Numer Funct Anal Optim 34(10):1156–1186
48. 48.
Schatzman M (1979) Problèmes unilatéraux d’ évolution du second ordre en temps. Thèse de Doctorat d’ Etates Sciences Mathématiques, Université Pierre et Marie Curie, Paris 6Google Scholar
49. 49.
Thibault L (1976) Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable. Applications. Thèse, Université Montpellier IIGoogle Scholar
50. 50.
Thibault L (2003) Sweeping process with regular and nonregular sets. J Differ Equ 193:1–26
51. 51.
Valadier M (1988) Quelques résultats de base concernant le processus de la rafle. Sém. Anal. Convexe, Montpellier, vol 3Google Scholar
52. 52.
Valadier M (1990) Lipschitz approximations of the sweeping process (or Moreau) process. J Differ Equ 88(2):248–264Google Scholar
53. 53.
Vladimirov AA (1991) Nonstationary dissipative evolution equation in Hilbert space. Nonlinear Anal 17:499–518Google Scholar
54. 54.
Vrabie IL (1987) Compactness methods for Nonlinear evolutions equations. Pitman monographs and surveys in pure and applied mathematics, vol 32. Longman Scientific and Technical, Wiley/New YorkGoogle Scholar

© Springer Nature Singapore Pte Ltd. 2018

## Authors and Affiliations

• C. Castaing
• 1
• M. D. P. Monteiro Marques
• 2
• P. Raynaud de Fitte
• 3
1. 1.IMAGUniversité de MontpellierMontpellierFrance
2. 2.CMAF-CIO, Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisbonPortugal
3. 3.Laboratoire de Mathématiques Raphaël SalemNormandie UniversitéRouenFrance