On a class of evolution problems driven by maximal monotone operators with integral perturbation. (English) Zbl 1536.34022
Summary: The present paper is dedicated to the study of a first-order differential inclusion driven by time and state-dependent maximal monotone operators with integral perturbation, in the context of Hilbert spaces. Based on a fixed point method, we derive a new existence theorem for this class of differential inclusions. Then, we investigate an optimal control problem subject to such a class, by considering control maps acting in the state of the operators and the integral perturbation.
MSC:
| 34A60 | Ordinary differential inclusions |
| 34G25 | Evolution inclusions |
| 47H10 | Fixed-point theorems |
| 47J35 | Nonlinear evolution equations |
| 49J52 | Nonsmooth analysis |
| 49J53 | Set-valued and variational analysis |
| 45J05 | Integro-ordinary differential equations |
Keywords:
integro-differential inclusion; maximal monotone operator; integral perturbation; optimal solutionReferences:
| [1] | F. Amiour, M. Sene, and T. Haddad, “Existence results for state-dependent maximal monotone differential inclusions: fixed point approach,” Numer. Funct. Anal. Optim., vol. 43, no. 7, pp. 838-859, 2022, doi: 10.1080/01630563.2022.2059675. · Zbl 1502.34071 |
| [2] | 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., vol. 21, no. 2, 2019, Art. ID. 40, doi: 10.1007/s11784-019-0666-2. · Zbl 1418.34122 |
| [3] | V. Barbu, Nonlinear semigroups and differential equations in Banach spaces. Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Lei- den, 1976, translated from the Romanian. · Zbl 0328.47035 |
| [4] | A. Bouabsa and S. Saïdi, “Coupled systems of subdifferential type with integral perturbation and fractional differential equations,” Advances in the Theory of Nonlinear Analysis and its Applications, vol. 7, no. 1, pp. 253-271, 2023, doi: 10.31197/atnaa.1149751. |
| [5] | A. Bouach, T. Haddad, and B. S. Mordukhovich, “Optimal control of nonconvex integro- differential sweeping processes,” J. Differential Equations, vol. 329, pp. 255-317, 2022, doi: 10.1016/j.jde.2022.05.004. · Zbl 1489.49017 |
| [6] | A. Bouach, T. Haddad, and L. Thibault, “Nonconvex integro-differential sweeping process with applications,” SIAM J. Control Optim., vol. 60, no. 5, pp. 2971-2995, 2022, doi: 10.1137/21M1397635. · Zbl 07600095 |
| [7] | A. Bouach, T. Haddad, and L. Thibault, “On the discretization of truncated integro-differential sweeping process and optimal control,” J. Optim. Theory Appl., vol. 193, no. 1-3, pp. 785-830, 2022, doi: 10.1007/s10957-021-01991-z. · Zbl 1489.49008 |
| [8] | Y. Brenier, W. Gangbo, G. Savaré, and M. Westdickenberg, “Sticky particle dynamics with interactions,” J. Math. Pures Appl., vol. 99, no. 5, pp. 577-617, 2013, doi: 10.1016/j.matpur.2012.09.013. · Zbl 1282.35236 |
| [9] | H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, ser. North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973, vol. 5. · Zbl 0252.47055 |
| [10] | L. M. Briceño Arias, N. D. Hoang, and J. Peypouquet, “Existence, stability and optimality for optimal control problems governed by maximal monotone operators,” J. Differential Equations, vol. 260, no. 1, pp. 733-757, 2016, doi: 10.1016/j.jde.2015.09.006. · Zbl 1326.49025 |
| [11] | M. Brokate and P. Krejčí, “Optimal control of ODE systems involving a rate independent variational inequality,” Discrete Contin. Dyn. Syst. Ser. B, vol. 18, no. 2, pp. 331-348, 2013, doi: 10.3934/dcdsb.2013.18.331. · Zbl 1260.49002 |
| [12] | T. H. Cao and B. S. Mordukhovich, “Optimal control of a nonconvex perturbed sweeping process,” J. Differential Equations, vol. 266, no. 2-3, pp. 1003-1050, 2019, doi: 10.1016/j.jde.2018.07.066. · Zbl 1404.49014 |
| [13] | C. Castaing, C. Godet-Thobie, M. D. P. Monteiro Marques, and A. Salvadori, “Evolution problems with m-accretive operators and perturbations,” Mathematics, vol. 10, no. 3, 2022, Art. ID 317, doi: 10.3390/math10030317. |
| [14] | C. Castaing, C. Godet-Thobie, S. Saïdi, and M. D. P. Monteiro Marques, “Various perturbations of time dependent maximal monotone/accretive operators in evolution inclusions with applications,” Appl. Math. Optim., vol. 87, no. 2, 2023, Art. ID 24, doi: 10.1007/s00245-022- 09898-5. · Zbl 1516.34092 |
| [15] | 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, vol. 8, no. 9, 2020, Art. ID 1395, doi: 10.3390/math8091395. |
| [16] | C. Castaing, P. Raynaud de Fitte, and M. Valadier, Young measures on topological spaces, ser. Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2004, vol. 571, doi: 10.1007/1-4020-1964-5. · Zbl 1067.28001 |
| [17] | G. Colombo, R. Henrion, N. D. Hoang, and B. S. Mordukhovich, “Optimal control of the sweeping process,” Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, vol. 19, no. 1-2, pp. 117-159, 2012. · Zbl 1264.49013 |
| [18] | G. Colombo, R. Henrion, N. D. Hoang, and B. S. Mordukhovich, “Discrete approximations of a controlled sweeping process,” Set-Valued Var. Anal., vol. 23, no. 1, pp. 69-86, 2015, doi: 10.1007/s11228-014-0299-y. · Zbl 1312.49015 |
| [19] | G. Colombo, R. Henrion, N. D. Hoang, and B. S. Mordukhovich, “Optimal control of the sweeping process over polyhedral controlled sets,” J. Differential Equations, vol. 260, no. 4, pp. 3397-3447, 2016, doi: 10.1016/j.jde.2015.10.039. · Zbl 1334.49070 |
| [20] | G. Colombo and C. Kozaily, “Existence and uniqueness of solutions for an integral perturbation of Moreau’s sweeping process,” J. Convex Anal., vol. 27, no. 1, pp. 229-238, 2020. |
| [21] | M. d. R. de Pinho, M. M. A. Ferreira, and G. V. Smirnov, “Optimal control involving sweeping processes,” Set-Valued Var. Anal., vol. 27, no. 2, pp. 523-548, 2019, doi: 10.1007/s11228-018- 0501-8. · Zbl 1419.49030 |
| [22] | M. d. R. de Pinho, M. M. A. Ferreira, and G. Smirnov, “Necessary conditions for optimal control problems with sweeping systems and end point constraints,” Optimization, vol. 71, no. 11, pp. 3363-3381, 2022, doi: 10.1080/02331934.2022.2101111. · Zbl 1506.49013 |
| [23] | M. de Pinho, M. M. A. Ferreira, and G. Smirnov, “Optimal control with sweeping processes: numerical method,” J. Optim. Theory Appl., vol. 185, no. 3, pp. 845-858, 2020, doi: 10.1007/s10957-020-01670-5. · Zbl 1450.49011 |
| [24] | K. Deimling, Non Linear Functional Analysis. Springer-Verlag, Berlin, 1985, doi: 10.1007/978-3-662-00547-7. · Zbl 0559.47040 |
| [25] | M. Guessous, “An elementary proof of Komlós-Révész theorem in Hilbert spaces,” J. Convex Anal., vol. 4, no. 2, pp. 321-332, 1997. · Zbl 0898.46027 |
| [26] | E. Klein and A. C. Thompson, Theory of correspondences, ser. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1984. · Zbl 0556.28012 |
| [27] | M. Kunze and M. D. P. Monteiro Marques, “BV solutions to evolution problems with time-dependent domains,” Set-Valued Anal., vol. 5, no. 1, pp. 57-72, 1997, doi: 10.1023/A:1008621327851. · Zbl 0880.34017 |
| [28] | B. K. Le, “Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions,” Optimization, vol. 69, no. 6, pp. 1187-1217, 2020, doi: 10.1080/02331934.2019.1686504. · Zbl 1440.49011 |
| [29] | C. Nour and V. Zeidan, “Numerical solution for a controlled nonconvex sweeping process,” IEEE Control Syst. Lett., vol. 6, pp. 1190-1195, 2022, doi: 10.1109/lcsys.2021.3089977. |
| [30] | C. Nour and V. Zeidan, “Optimal control of nonconvex sweeping processes with separable endpoints: nonsmooth maximum principle for local minimizers,” J. Differential Equations, vol. 318, pp. 113-168, 2022, doi: 10.1016/j.jde.2022.02.021. · Zbl 1539.49016 |
| [31] | C. Nour and V. Zeidan, “A control space ensuring the strong convergence of continuous approximation for a controlled sweeping process,” Set-Valued Var. Anal., vol. 31, no. 3, 2023, Art. ID 23, doi: 10.1007/s11228-023-00686-z. · Zbl 1527.49021 |
| [32] | C. Nour and V. Zeidan, “Nonsmooth optimality criterion for a W1,2-controlled sweeping process: Nonautonomous perturbation,” Appl. Set-Valued Anal. Optim., vol. 5, no. 2, pp. 193-212, 2023, doi: 10.23952/asvao.5.2023.2.06. |
| [33] | S. Saïdi, “A perturbed second-order problem with time and state-dependent maximal mono- tone operators,” Discuss. Math. Differ. Incl. Control Optim., vol. 41, no. 1, pp. 61-86, 2021, doi: 10.7151/dmdico. · Zbl 1513.34236 |
| [34] | F. Selamnia, D. Azzam-Laouir, and M. D. P. Monteiro Marques, “Evolution problems involving state-dependent maximal monotone operators,” Appl. Anal., vol. 101, no. 1, pp. 297-313, 2022, doi: 10.1080/00036811.2020.1738401. · Zbl 1497.34092 |
| [35] | A. A. Vladimirov, “Nonstationary dissipative evolution equations in a Hilbert space,” Nonlinear Anal., vol. 17, no. 6, pp. 499-518, 1991, doi: 10.1016/0362-546X(91)90061-5. · Zbl 0756.34064 |
| [36] | I. I. Vrabie, Compactness methods for nonlinear evolutions, ser. Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987, vol. 32. · Zbl 0721.47050 |
| [37] | V. Zeidan, C. Nour, and H. Saoud, “A nonsmooth maximum principle for a controlled non-convex sweeping process,” J. Differential Equations, vol. 269, no. 11, pp. 9531-9582, 2020, doi: 10.1016/j.jde.2020.06.053. · Zbl 1450.49007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
