On a Non-Convex Lagrange Optimal Control Problem

Sara Attab; Doria Affane; Mustapha Fateh Yarou

doi:10.2478/tmmp-2024-0008

.blurhash-client-img { display: none !important; }

On a Non-Convex Lagrange Optimal Control Problem

Tatra Mountains Mathematical Publications

Volume 88 (2024): Issue 3 (December 2024)

By: Sara Attab, Doria Affane and Mustapha Fateh Yarou

Open Access

|May 2024

Abstract

In this paper, we are concerned with an iterative differential inclusion governed by the time-dependent maximal monotone operator with perturbation. The approach to solve our problem is based on the Yosida approximation technique. The theoretical result is applied to prove an existence result for a Lagrange optimal control problem without assumptions concerning convexity.

References

AFFANE, D.—GHALIA, S.: First-order iterative differential inclusion, Electron. J. Math. Anal. Appl. 10 (2022), 1–10.
Search in Google Scholar Back to article
AUBIN, J.-P.—CELLINA, A.: Differential Inclusions. Set-valued Maps and Viability Theory. Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences], Vol. 264 Springer-Verlag, Berlin, 1984.
Search in Google Scholar Back to article
BERINDE, V.: Existence and approximation of solutions of some first order iterative differential equations, Miskolc Math. Notes 11 (2010), 13–26.
Search in Google Scholar Back to article
BERTSEKAS, D. P.: Convex Optimization Theory. Athena Scientific, Nashua, NH, 2009.
Search in Google Scholar Back to article
BRÉZIS, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert. North-Holland Math. Stud. Vol. 5, Elsevier, Amsterdam, 1973.
Search in Google Scholar Back to article
BUICĂ, A.: Existence and continuous dependence of solutions of some functional--differential equations, Babes,-Bolyai Univ., Fac. Math. Comput. Sci., Res. Semin., Preprint 1995 (1995), 1–13.
Search in Google Scholar Back to article
CARLOTA, C.—CHÁ, S.: An existence result for non-convex optimal control problems, WSEAS Trans. Syst. Control. 9 (2014), 687–697.
Search in Google Scholar Back to article
CARLOTA, C.—ORNELAS, A.: Existence of vector minimizers for nonconvex 1-dim integrals with almost convex Lagrangian,J.Differ. Equations 243 (2007), 414–426.
Search in Google Scholar Back to article
CELLINA, A.—ORNELAS, A.: Existence of solutions to differential inclusions and to time optimal control problems in the autonomous case, SIAM J. Control Optim. 42 (2003), 260–265.
Search in Google Scholar Back to article
CESARI, L.: Optimization—Theory and Applications. Problems with Ordinary Differential Equations. Appl. Math. (New York). Vol. 17, Springer, New York, 1983.
Search in Google Scholar Back to article
CHENG, S. S.: Smooth solutions of iterative functional differential equations, In: Proceedings of the international conference: Dynamical systems and applications 2024, Antalya, Turkey, July 5–10, 2004.
Search in Google Scholar Back to article
CLARKE, F. H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley, New York, NY, 1983.
Search in Google Scholar Back to article
EDER, E.: The functional differential equation x (t)= x(x(t)), J. Differ. Equations 54 (1984), 390–400.
Search in Google Scholar Back to article
EL-SAYED, A. M. A.—EBEAD, H.: On an initial value problem of delay-refereed differential equation, Int. J. Math. Trends Technol. 66 (2020), 32–37.
Search in Google Scholar Back to article
EL-SAYED, A. M. A.—HAMDALLAH, E.—EBEAD, H.: Positive solutions of an initial value problem of a delay-self-reference nonlinear differential equation, Malaya J. Matematik. 8 (2020), 1001–1006.
Search in Google Scholar Back to article
FEČKAN, M.: On a certain type of functional differential equations, Math. Slovaca 43 (1993), 39–43.
Search in Google Scholar Back to article
GHALIA, S.—AFFANE, D.: Control problem governed by an iterative differential inclusion,Rend. Circ.Mat.Palermo (2) 72 (2023), no. 4, 2621–2642.
Search in Google Scholar Back to article
GHALIA, S.—AFFANE, D.: On the attainable set of iterative differential inclusions, Math. Slovaca 73 (2023), 1479–1498.
Search in Google Scholar Back to article
KAUFMANN, E. R.: Existence and uniqueness of solutions for a second-order iterative boundary-value problem, Electron. J. Differ. Equ. 2018 (2018), 1–6.
Search in Google Scholar Back to article
KUMLIN, P.: A note on fixed point theory, Functional analysis lecture (2004).
Search in Google Scholar Back to article
KUNZE, M.—MONTEIRO MARQUES, M. D. P.: BV solutions to evolution problems with time-dependent domains, Set-Valued Anal. 5 (1997), 57–72.
Search in Google Scholar Back to article
LAURAN, M.: Existence results for some differential equations with deviating argument, Filomat 25 (2011), 21–31.
Search in Google Scholar Back to article
OPREA, N.: Numerical solutions of first order iterative functional-differential equations by spline functions of even degree, Acta Marisiensis Ser. Technol. 6 (2009), 34–37.
Search in Google Scholar Back to article
PEDREGAL, P.: On the generality of variational principles,Milan J.Math. 71 (2003), 319–356.
Search in Google Scholar Back to article
PELCZAR, A.: On some iterative-differential equations. I, Zesz. Nauk. Uniw. Jagielloń. Pr. Mat. 167 (1968), 53–56.
Search in Google Scholar Back to article
PODISUK, M.: On simple iterative ordinary differential equations, Science Asia. 28 (2002), 191–198.
Search in Google Scholar Back to article
SI, J.-G.—WANG, X.-P.—CHENG, S. S.: Nondecreasing and convex C²-solutions of an iterative functional-differential equation, Aequationes Math. 60 (2000), 38–56.
Search in Google Scholar Back to article
WANG, K.: On the equation x (t)= f (x(x(t))), Funkcial. Ekvacioj, Ser. Int. 33 (1990), no. 3, 405–425.
Search in Google Scholar Back to article
YANG, D.—ZHANG, W.: Solutions of equivariance for iterative differential equations, Appl. Math. Lett. 17 (2004), 759–765.
Search in Google Scholar Back to article
ZHANG, P.: Analytic solutions for iterative functional differential equations, Electron. J. Differ. Equ. 2012 (2012), 1–7.
Search in Google Scholar Back to article
ZHANG, P.—GONG, X.: Existence of solutions for iterative differential equations, Electron. J. Differ. Equ. 2014 (2014), 1–10.
Search in Google Scholar Back to article
ZHOU, J.—SHEN, J.: Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations, Discrete Contin. Dyn. Syst., Ser. B 27 (2022), 3605–3624.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2024-0008 | Journal eISSN: 1338-9750 (formerly 1210-3195) | Journal ISSN: 1210-3195

Journal RSS Feed

Language: English

Page range: 85 - 108

Submitted on: Sep 6, 2023

Accepted on: Mar 21, 2024

Published on: May 29, 2024

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

iterative,

differential inclusions,

maximal monotone operator,

calculus of variations,

almost-convex function

MSC:

39B12,

34K09,

49J21

Related subjects:

Mathematics,

General mathematics

© 2024 Sara Attab, Doria Affane, Mustapha Fateh Yarou, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 88 (2024): Issue 3 (December 2024)