A minimization problem subject to a coupled system by maximal monotone operators. (English) Zbl 1542.49007
Summary: We study in the current paper an optimization problem subject to a controlled system with maximal monotone operators and integral perturbation, in Hilbert spaces. First, we establish a new existence and uniqueness theorem for a coupled system by two first-order differential inclusions governed by maximal monotone operators and single-valued perturbations. Then, we minimize an integral functional over the controls acting in the state of the operators into the coupled system under consideration.
MSC:
| 49J27 | Existence theories for problems in abstract spaces |
| 49K27 | Optimality conditions for problems in abstract spaces |
| 47H05 | Monotone operators and generalizations |
| 34A60 | Ordinary differential inclusions |
| 34G25 | Evolution inclusions |
| 49J52 | Nonsmooth analysis |
| 49J53 | Set-valued and variational analysis |
Keywords:
coupled system; differential inclusion; maximal monotone operator; integral perturbation; optimal solutionReferences:
| [1] | Adam, L.; Outrata, J., On optimal control of a sweeping process coupled with an ordinary differential equation, Discrete Contin. Dyn. Syst. Ser. B, 19, 9, 2709-2738 (2014) · Zbl 1304.49052 |
| [2] | Adly, S.; Haddad, T.; Thibault, L., Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program Ser. B., 148, 5-47 (2014) · Zbl 1308.49013 · doi:10.1007/s10107-014-0754-4 |
| [3] | Aissous, M.; Nacry, F.; Thuong Nguyen, VA, First and second order state-dependent bounded subsmooth sweeping processes, Linear Nonlinear Anal., 6, 3, 447-472 (2020) · Zbl 1473.34013 |
| [4] | Azzam-Laouir, D.; Belhoula, W.; Castaing, C.; Monteiro Marques, MDP, Perturbed evolution problems with absolutely continuous variation in time and applications, J. Fixed Point Theory Appl., 21, 2, 1-32 (2019) · Zbl 1418.34122 · doi:10.1007/s11784-019-0666-2 |
| [5] | Azzam-Laouir, D.; Belhoula, W.; Castaing, C.; Monteiro Marques, MDP, Multivalued perturbation to evolution problems involving time dependent maximal monotone operators, Evol. Equ. Control Theory, 9, 1, 219-254 (2020) · Zbl 1469.34084 · doi:10.3934/eect.2020004 |
| [6] | Azzam-Laouir, D.; Castaing, C.; Monteiro Marques, MDP, Perturbed evolution problems with continuous bounded variation in time and applications, Set Valued Var. Anal., 26, 3, 693-728 (2018) · Zbl 1403.34046 · doi:10.1007/s11228-017-0432-9 |
| [7] | Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces (1976), Leyden: Noordhoff International Publishing, Leyden · Zbl 0328.47035 · doi:10.1007/978-94-010-1537-0 |
| [8] | Benguessoum, M.; Azzam-Laouir, D.; Castaing, C., On a time and state dependent maximal monotone operator coupled with a sweeping process with perturbations, Set Valued Var. Anal., 29, 191-219 (2021) · Zbl 1469.34057 · doi:10.1007/s11228-020-00544-2 |
| [9] | Bouabsa, A.; Saïdi, S., Coupled systems of subdifferential type with integral perturbation and fractional differential equations, Adv. Theory Nonlinear Anal. Appl., 7, 1, 253-271 (2023) · Zbl 1512.34039 |
| [10] | Bouach, A.; Haddad, T.; Mordukhovich, BS, Optimal control of nonconvex integro-differential sweeping processes, J. Differ. Equ,, 329, 255-317 (2022) · Zbl 1489.49017 · doi:10.1016/j.jde.2022.05.004 |
| [11] | Bouach, A.; Haddad, T.; Thibault, L., Nonconvex integro-differential sweeping process with applications, SIAM J. Control Optim., 60, 5, 2971-2995 (2022) · Zbl 07600095 · doi:10.1137/21M1397635 |
| [12] | Bouach, A.; Haddad, T.; Thibault, L., On the discretization of truncated integro-differential sweeping process and optimal control, J. Optim. Theory Appl., 193, 1, 785-830 (2022) · Zbl 1489.49008 · doi:10.1007/s10957-021-01991-z |
| [13] | Bouhali, N.; Azzam-Laouir, D.; Monteiro Marques, MDP, Optimal control of an evolution problem involving time-dependent maximal monotone operators, J. Optim. Theory Appl., 194, 1, 59-91 (2022) · Zbl 1490.49006 · doi:10.1007/s10957-022-02009-y |
| [14] | Brenier, Y.; Gangbo, W.; Savare, G.; Westdickenberg, M., Sticky particle dynamics with interactions, J. Math. Pures Appl., 99, 577-617 (2013) · Zbl 1282.35236 · doi:10.1016/j.matpur.2012.09.013 |
| [15] | Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Lecture Notes in Mathematics (1973), Amsterdam: North-Holland, Amsterdam · Zbl 0252.47055 |
| [16] | Briceño-Arias, LM; Hoang, ND; Peypouquet, J., Existence, stability and optimality for optimal control problems governed by maximal monotone operators, J. Differ. Equ., 260, 733-757 (2016) · Zbl 1326.49025 · doi:10.1016/j.jde.2015.09.006 |
| [17] | Brogliato, B.; Tanwani, A., Dynamical systems coupled with monotone set-valued operators: formalisms, applications, well-posedness and stability, SIAM Rev., 62, 1, 3-129 (2020) · Zbl 1450.34002 · doi:10.1137/18M1234795 |
| [18] | Brokate, M.; Krejčí, P., Optimal control of ODE systems involving a rate independent variational inequality, Discrete Contin. Dyn. Syst. Ser. B, 18, 2, 331-348 (2013) · Zbl 1260.49002 |
| [19] | Cao, TH; Mordukhovich, BS, Optimal control of a nonconvex perturbed sweeping process, J. Differ. Equ., 266, 1003-1050 (2019) · Zbl 1404.49014 · doi:10.1016/j.jde.2018.07.066 |
| [20] | Castaing, C.; Godet-Thobie, C.; Monteiro Marques, MDP; Salvadori, A., Evolution problems with \(m\)-accretive operators and perturbations, Mathematics, 10, 3, 1-32 (2022) · doi:10.3390/math10030317 |
| [21] | Castaing, C.; Godet-Thobie, C.; Saïdi, S.; Monteiro Marques, MDP, Various perturbations of time dependent maximal monotone/accretive operators in evolution inclusions with applications, Appl. Math. Optim., 87, 24 (2023) · Zbl 1516.34092 · doi:10.1007/s00245-022-09898-5 |
| [22] | Castaing, C.; Godet-Thobie, C.; Truong, LX, Fractional order of evolution inclusion coupled with a time and state dependent maximal monotone operator, Math. MDPI, 8, 9, 1-30 (2020) |
| [23] | Castaing, C.; Raynaud de Fitte, P.; Valadier, M., Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory (2004), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 1067.28001 |
| [24] | Castaing, C.; Saïdi, S., Lipschitz perturbation to evolution inclusions driven by time-dependent maximal monotone operators, Topol. Methods Nonlinear Anal., 58, 2, 677-712 (2021) · Zbl 1494.34135 · doi:10.12775/TMNA.2021.012 |
| [25] | Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics (1977), Berlin: Springer, Berlin · Zbl 0346.46038 · doi:10.1007/BFb0087685 |
| [26] | Colombo, G.; Henrion, R.; Hoang, ND; Mordukhovich, BS, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst. Ser. B, 19, 117-159 (2012) · Zbl 1264.49013 |
| [27] | Colombo, G.; Henrion, R.; Hoang, ND; Mordukhovich, BS, Discrete approximations of a controlled sweeping process, Set Valued Var. Anal., 23, 69-86 (2015) · Zbl 1312.49015 · doi:10.1007/s11228-014-0299-y |
| [28] | Colombo, G.; Henrion, R.; Hoang, ND; Mordukhovich, BS, Optimal control of the sweeping process over polyhedral controlled sets, J. Differ. Equ., 260, 3397-3447 (2016) · Zbl 1334.49070 · doi:10.1016/j.jde.2015.10.039 |
| [29] | Colombo, G.; Kozaily, C., Existence and uniqueness of solutions for an integral perturbation of Moreau’s sweeping process, J. Convex Anal., 27, 227-236 (2020) · Zbl 1445.34092 |
| [30] | Dib, K.; Azzam-Laouir, D., Existence of solutions for a couple of differential inclusions involving maximal monotone operators, Appl. Anal., 102, 9, 2628-2650 (2020) · Zbl 1516.34043 · doi:10.1080/00036811.2022.2124974 |
| [31] | Guessous, M., An elementary proof of Komlós-Révész Theorem in Hilbert spaces, J. Convex Anal., 4, 2, 321-332 (1997) · Zbl 0898.46027 |
| [32] | Krasnosel’skimi, A.; Pokrovskiia, V., Systems with Hysteresis (1988), Berlin: Springer, Berlin |
| [33] | Kunze, M.; Monteiro Marques, MDP, BV solutions to evolution problems with time-dependent domains, Set Valued Anal., 5, 57-72 (1997) · Zbl 0880.34017 · doi:10.1023/A:1008621327851 |
| [34] | Le, BK, Well-posedeness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions, Optimization, 69, 6, 1187-1217 (2020) · Zbl 1440.49011 · doi:10.1080/02331934.2019.1686504 |
| [35] | Moreau, J.J.: Unilateral Contact and Dry Friction in Finite Freedom Dynamics. Nonsmooth Mechanics, CISM Courses and Lectures, No 302. Springer, Vienna (1988) · Zbl 0703.73070 |
| [36] | Nacry, F.; Noel, J.; Thibault, L., On first and second order state-dependent prox-regular sweeping processes, Pure Appl. Funct. Anal., 6, 6, 1453-1493 (2021) · Zbl 1478.34022 |
| [37] | Saïdi, S., Coupled problems driven by time and state-dependent maximal monotone operators, Numer. Algebra Control Optim. (2022) · Zbl 07923333 · doi:10.3934/naco.2023006 |
| [38] | Saïdi, S., On a second-order functional evolution problem with time and state dependent maximal monotone operators, Evol. Equ. Control Theory, 11, 4, 1001-1035 (2022) · Zbl 1493.34171 · doi:10.3934/eect.2021034 |
| [39] | Saïdi, S.; Bouabsa, A., A coupled problem described by time-dependent subdifferential operator and non-convex perturbed sweeping process, Evol. Equ. Control Theory, 12, 4, 1145-1173 (2023) · Zbl 1512.34039 · doi:10.3934/eect.2023005 |
| [40] | Selamnia, F.; Azzam-Laouir, D.; Monteiro Marques, MDP, Evolution problems involving state-dependent maximal monotone operators, Appl. Anal., 101, 1, 297-313 (2022) · Zbl 1497.34092 · doi:10.1080/00036811.2020.1738401 |
| [41] | Tanaka, H., Stochastic differential equations with reflecting boundary conditions in convex regions, Hiroshima Math. J., 9, 163-177 (1979) · Zbl 0423.60055 · doi:10.32917/hmj/1206135203 |
| [42] | Tanwani, A.; Brogliato, B.; Prieur, C., Well-Posedness and output regulation for implicit time-varying evolution variational inequalities, SIAM J Control Optim., 56, 2, 751-781 (2018) · Zbl 1381.93052 · doi:10.1137/16M1083657 |
| [43] | Tolstonogov, AA, BV continuous solutions of an evolution inclusion with maximal monotone operator and nonconvex-valued perturbation. Existence theorem, Set Valued Var. Anal., 29, 29-60 (2021) · Zbl 1482.34144 · doi:10.1007/s11228-020-00535-3 |
| [44] | Vladimirov, AA, Nonstationary dissipative evolution equations in Hilbert space, Nonlinear Anal., 17, 499-518 (1991) · Zbl 0756.34064 · doi:10.1016/0362-546X(91)90061-5 |
| [45] | Vrabie, II, Compactness Methods for Nonlinear Evolution Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, 32 (1987), New York: Wiley, New York · Zbl 0721.47050 |
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.
