Abstract
We consider a control problem in a finite-dimensional setting, which consists in finding a minimizer for a standard functional defined by way of two continuous and bounded below functions and a convex function, where the control functions take values in a closed convex set and the state functions solve a differential system made up of a differential inclusion governed by a maximal monotone operator; and an ordinary differential equation with a Lipschitz mapping in the right-hand side. We first show the existence of a unique absolutely continuous solution of our system, by transforming it to a sole evolution differential inclusion, and then use a result from the literature. Secondly, we prove the existence of an optimal solution to our problem. The main novelties are: the presence of the time-dependent maximal monotone operators, which may depend as well as their domains on the time variable; and the discretization scheme for the approximation of the solution.
Similar content being viewed by others
References
Adam, L., Outrata, J.: On optimal control of a sweeping process coupled with an ordinary differential equation. Disc. Contin. Dyn. Syst. Ser B. 19(9), 2709–2738 (2014)
Adly, S., Haddad, T., Thibault, L.: Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. Math. Progr. Ser. B. 148(1–2), 5–47 (2014)
Adly, S., Zakaryan, T.: Sensitivity properties of parametric nonconvex evolution inclusions with application to optimal control problems. Set-Valued Var. Anal. 27, 549–568 (2019)
Azzam-Laouir, D., Belhoula, W., Castaing, C., Monteiro Marques, M.D.P.: Perturbed evolution problems with absolutely continuous variation in time and applications. J. Fixed Point Theory Appl. 21, 40 (2019)
Azzam-Laouir, D., Belhoula, W., Castaing, C., Monteiro Marques, M.D.P.: Multi-valued perturbation to evolution problems involving time-dependent maximal monotone operators. Evolut. Equ. Control Theory 9(1), 219–254 (2020)
Azzam-Laouir, D., Boutana-Harid, I.: Mixed semicontinuous perturbation to an evolution problem with time-dependent maximal monotone operator. J. Nonlinear Convex Anal. 20(1), 39–52 (2018)
Azzam-Laouir, D., Castaing, C., Monteiro Marques, M.D.P.: Perturbed evolution proplems with continuous bounded variation in time and applications. Set-Valued Var. Anal. 26(3), 693–728 (2018)
Azzam-Laouir, D., Makhlouf, M., Thibault, L.: On perturbed sweeping process. Appl. Anal. 95(2), 303–322 (2016)
Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff. Int. Publ, Leyden (1976)
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York (2010)
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)
Bounkhel, M., Thibault, L.: Non convex sweeping process and prox-regularity in Hilbert spaces. J. Nonlinear Convex Anal. 6, 359–374 (2005)
Brezis, H.: Opérateurs Maximaux Monotones. North Holland Publ. Company, Amsterdam-London (1973)
Briceño-Arias, L.M., Hoang, N.D., Peypouquet, J.: Existence, stability and optimality for optimal control problems governed by maximal monotone operators. J. Differ. Equ. 260, 733–757 (2016)
Brogliato, B.: Nonsmooth Impact Mechanics: Models, Dynamics and Control. Springer, 3rd edition (2016)
Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Disc. Contin. Dyn. Syst. Ser. B. 18(2), 331–348 (2013)
Cao, T.H., Mordukovich, B.S.: Optimal control of nonconvex perturbed sweeping process. J. Differ. Equ. 266, 1003–1050 (2019)
Castaing, C., Duc Ha, T.X., Valadier, M.: Evolution equations governed by the sweeping process. Set-Valued Anal. 1, 109–139 (1993)
Castaing, C., Monteiro Marques, M.D.P.: Evolution problems associated with nonconvex closed moving sets with bounded variation. Portugal. Math. 53, 73–87 (1996)
Castaing, C., Salvadori, A., Thibault, L.: Functional evolution equations governed by nonconvex sweeeping process. J. Nonlinear Convex Anal. 2(2), 217–241 (2001)
Colombo, G., Goncharov, V.V.: The sweeping processes without convexity. Set-Valued Anal. 7, 357–374 (1999)
Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Discrete approximations of a controlled sweeping process. Set-Valued Var. Anal. 23, 69–86 (2015)
Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. J. Differ. Equ. 260(4), 3397–3447 (2016)
Cortes, J.: Discontinuous dynamical systems: a tutorial on solutions, nonsmooth analysis, and stability. Control Syst. Mag. 28(3), 36–73 (2008)
Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences. Springer, 78 (2008)
De Pinho, M.D.R., Ferreira, M.M.A., Smirnov, G.V.: Optimal control involving sweeping processes. Set-Valued Var. Anal. 27(2), 523–548 (2019)
Edmond, J.F., Thibault, L.: Relaxation and optimal control problem involving a perturbed sweeping process. Math. Program. Ser. B. 104, 347–373 (2005)
Edmond, J.F., Thibault, L.: BV solutions of nonconvex sweeping process differential inclusion with perturbation. J. Differ. Equ. 226, 135–179 (2006)
Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics. Siam Edition, 28 (1991)
Haraux, A.: Nonlinear Evolution Equations - Global behavior of solutions. Lecture Notes in Mathematics. 841, Springer-Verlag (1981)
Jourani, A., Vilches, E.: Positively \(\alpha \)-far sets and existence results for generalized perturbed sweeping processes. J. Convex Anal. 23(3), 775–821 (2016)
Kazufumi, I., Kunisch, K.: Optimal control of parabolic variational inequalities. J. Math. Pures Appl. 93, 329–360 (2010)
Kenmochi, N.: Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Educ. Chiba Univ. 30, 1–81 (1981)
Kunze, M., Monteiro Marques, M.D.P.: BV solutions to evolution problems with time-dependent domains. Set-Valued. Anal. 5, 57–72 (1997)
Kunze, M., Monteiro Marques, M.D.P.: On parabolic quasi-variational inequalities and state dependent sweeping processes. Topol. Methods Nonlinear Anal. 12(16), 179–191 (1998)
Kunze, M., Monteiro Marques, M.D.P.: An introduction to Moreau’s sweeping process. In: Brogliato, B. (ed.) Impacts in Mechanical Systems. Lecture Notes in Physics. 551. Springer-Verlag, Berlin, 17 (2000)
Le, B.K.: Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions. Optimization 69(6), 1187–1217 (2020)
Mahmudov, E.N.: Optimization of Mayer problem with Sturm-Liouville-type differential inclusions. J. Optim. Theory Appl. 177, 345–375 (2018)
Mahmudov, E.N.: Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolut. Equ. Control Theory 8(3), 603–619 (2019)
Monteiro Marques, M.D.P.: Differential inclusions in nonsmooth mechanical problems, shocks and dry friction. Progress in Nonlinear Differential Equations and Their Applications, Birkhauser, 9 (1993)
Mordukovich, B.S.: Variational analysis and optimization of sweeping processes with controlled moving sets. Revista Investigación Operacional. 39(3), 283–302 (2018)
Moreau, J.J.: Rafle par un convexe variable I. Sem. Anal. Convexe. Montpellier. exposé No. 15. (1971)
Pavel, N.H.: Nonlinear Evolution Operators and Semigroups. Lecture Notes in Mathematics, vol. 1260. Springer, New York (1987)
Selamnia, F., Azzam-Laouir, D., Monteiro Marques, M.D.P.: Evolution problems involving state-dependent maximal monotone operators. Appl. Anal. (2020). https://doi.org/10.1080/00036811.2020.1738401
Stewart, D.E.: Dynamics with Inequalities: Impacts and Hard Constraints. SIAM, Philadelphia, PA (2011)
Tanwani, A., Brogliato, B., Prieur, C.: Well-posedness and output regulation for implicit time-varying evolution variational inequalities. SIAM J. Control Optim. 56, 751–781 (2018)
Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193, 1–26 (2003)
Tolstonogov, A.A.: Control sweeping process. J. Convex Anal. 23, 1099–1123 (2016)
Tolstonogov, A.A.: 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)
Vilches, E., Nguyen, B.T.: Evolution inclusions governed by time-dependent maximal monotone operators with a full domain. Set-Valued. Var, Anal (2020)
Vladimirov, A.A.: Nonstationary dissipative evolution equations in a Hilbert space. Nonlinear Anal. 17, 499–518 (1991)
Vrabie, I.I.: Compactness Methods for Nonlinear Evolution Equations. Pitman Monographs and Surveys in Pure and Applied mathematics, Longman Scientific and Technical, John Wiley and Sons, Inc. New York. 32 (1987)
Acknowledgements
The authors thank a referee for pointing out important remarks and suggestions that helped in the improvement of the last version of the paper. The first and second authors were supported by the DGRSDT, PRFU project number 811 C00L03UN180120180005. Manuel D.P. Monteiro Marques was partially supported by National Funding from FCT-Fundação para a Ciência e a Technologia, under the project: UID/MAT/04561/2019.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Lionel Thibault.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bouhali, N., Azzam-Laouir, D. & Monteiro Marques, M.D.P. Optimal Control of an Evolution Problem Involving Time-Dependent Maximal Monotone Operators. J Optim Theory Appl 194, 59–91 (2022). https://doi.org/10.1007/s10957-022-02009-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-022-02009-y
Keywords
- Absolutely continuous variation
- Maximal monotone operator
- Objective function
- Optimal solution
- Pseudo-distance