Optimal Control for a Nonlinear System of Integro-Differential Equations of Fractional Order
Keywords:
Fractional equation, product of a nonlinear function with a nonlinear integral, Pontryagin function, method of contracted mappings, existence, uniqueness and stability of a solutionAbstract
This paper investigates an optimal control problem for a nonlinear system of
fractional-order integro-differential equations involving the Gerasimov–Caputo operator. The
considered model contains a nonlinear integral interaction represented by the product of two
nonlinear functions, which substantially complicates both the analytical structure of the system
and the derivation of optimality conditions.
Unlike the classical approach in optimal control theory, where the control function is assumed to
be known when determining the state trajectory, the present work constructs the state function
and the control function simultaneously as a coupled solution pair of a nonlinear functionalintegral system.
Using fractional integration techniques, the original fractional integro-differential equation is
reduced to an equivalent nonlinear integral equation for the state function. By introducing an
adjoint final-value problem and constructing the Pontryagin function, a second nonlinear integral equation for the control function is derived. As a result, the optimal control problem is
transformed into a strongly coupled nonlinear system of integral equations.
A constructive analytical method based on successive approximations and the contraction mapping principle is developed. Explicit sufficient conditions ensuring existence and uniqueness of
the vector-valued control and state functions are established. Furthermore, continuous dependence of the solution on the initial vector is proved.
The obtained results provide a constructive framework for the analysis of nonlinear fractional
optimal control systems with memory effects and nonlocal interactions and extend several known
solvability results for fractional dynamical systems.
References
K. Alymkulov, D. A. Tursunov, ”A method for constructing asymptotic expansions of bisingularly perturbed problems”, Russian Mathematics 60 (12), 1–8 (2016).
https://doi.org/10.3103/S1066369X1612001X
K. Alymkulov, D. A. Tursunov, B. A. Azimov ”Generalized method of boundary layer function for
bisingularly perturbed differential cole equation”, Far East Journal of Mathematical Sciences 101
(3), 507–516 (2017). https://doi.org/10.17654/MS101030507
L. Bin, L. Xinzhi, L. Xiaoxin, ”Robust global exponential stability of uncertain impulsive systems”,
Acta Mathematika Scientia 25 (1), 161–169 (2005).
Ch. Bai, D. Yang, ”Existence of solutions for second-order nonlinear impulsive differential equations
with periodic boundary value conditions”, Boundary Value Problems (Hindawi Publishing Corporation) 2007 (41589), 1–13 (2007).
X. Li, M. Bohner, Ch.-K. Wang, ”Impulsive differential equations: Periodic solutions and applications”, Automatica 52, 173–178 (2015).
J. Chen, Ch. C. Tisdell, R. Yuan, ”On the solvability of periodic boundary value problems with
impulse”, J. of Math. Anal. and Appl. 331, 902–912 (2007).
S. G. Lobanov, ”Picard’s theorem for ordinary differential equations in locally convex
spaces”, Izvestiya Mathematics. Russian Academy of Sciences 41 (3), 465–487 (1993).
https://doi.org/10.1070/IM1993v041n03ABEH002272
S. G. Lobanov, O. G. Smolyanov, ”Ordinary differential equations in locally convex spaces”, Russian
Math. Surveys 49 (3), 97–175 (1994). https://doi.org/10.1070/RM1994v049n03ABEH002258
K. G. Kozhobekov, D. A. Tursunov, ”Asymptotics of the solution to the boundary-value problem
when the limit equation has an irregular singular point”, Vestnik Udmurtskogo Universiteta: Matematika, Mekhanika, Komp’yuternye Nauki, 29 (3), 332–340. https://doi.org/10.20537/vm190304
S. N. Man’ko, ”Vector-valued functions generated by the operator of finite order and their application to solving operator equations in locally convex spaces”, Russian Math. 62 (3), 34–44 (2018).
https://doi.org/10.3103/S1066369X18030052
S. N. Mishin, ”Generalization of the Lagrange method to the case of second-order linear differential
equations with constant operator coefficients in locally convex spaces”, Math. Notes 103 (1), 75–88
(2018). https://doi.org/10.1134/S0001434618010091
S. N. Mishin, ”On a class of operator equations in locally convex spaces”, Russian Math. 62 (11),
–44 (2018). https://doi.org/10.3103/S1066369X1811004X
M. F. Sukhinin, ”On the differentiability with respect to the initial condition of a solution to an ordinary differential equation”, Math. USSR-Sb. 35 (1), 115–129 (1979).
https://doi.org/10.1070/SM1979v035n01ABEH001456
D. A. Tursunov, ”The asymptotic solution of the three-band bisingularly problem” Lobachevskii J.
Mathematics 38 (3), 542–546 (2017). https://doi.org/10.1134/S1995080217030258
D. A. Tursunov, M. O. Orozov, ”Asymptotics of the solution to the Roben problem for a
Ring with regularly singular boundary”, Lobachevskii J. Mathematics 41 (1), 89–95 (2020).
https://doi.org/10.1134/S1995080220010126
D. A. Tursunov, M. O. Orozov ”Asymptotic solution of the dirichlet problem for a ring, when the corresponding unperturbed equation has a regular special circle”, Vestnik Tomskogo Gosudarstvennogo
Universiteta, Matematika i Mekhanika 63, 37–46 (2019). https://doi.org/10.17223/19988621/63/4
D. A. Tursunov, U. Z. Erkebaev, ”Asymptotic expansions of solutions to Dirichlet problem for elliptic equation with singularities”, Ufa Mathematical J. 8 (1), 97–107 (2016).
https://doi.org/10.13108/2016-8-1-97
T. K. Yuldashev, A. K. Fayziev, ”On a nonlinear impulsive differential equations with maxima”,
Bull. Inst. Math. 4 (6), 42–49 (2021).
T. K. Yuldashev, A. K. Fayziev, ”On a nonlinear impulsive system of integro-differential equations
with degenerate kernel and maxima”, Nanosystems: Phys. Chem. Math. 13 (1), 36–44 (2022).
V. M. Abdullayev, ”Numerical solution to optimal control problems with multipoint and integral
conditions”, Proceedings of the Institute of Mathematics and Mechanics 44 (2), 171–186 (2018).
E. F. Abdyldaeva, A. Kerimbekov, T. K. Yuldashev, M. Kamali, ”On the convergence of the approximate solution to the optimization problem for oscillatory processes”, Bulletin of the Karaganda
University. Mathematics Series 119 (3), 22–33 (2025). https://doi.org/10.31489/2025M3/22-33
S. Albeverio, Sh. A. Alimov, ”On a time-optimal control problem associated with the
heat exchange process”, Applied Mathematics and Optimization 57 (1), 58–68 (2008).
https://doi.org/10.1007/s00245- 007-9008-7
Sh. A. Alimov, G. I. Ibragimov, ”Time optimal control problem with integral constraint for the
heat transfer process”, Eurasian Math. J. 15 (1), 8–22 (2024). https://doi.org/10.32523/2077-9879-
-15-1-08-22
B. Y. Ashirbaev, T. K. Yuldashev, ”Derivation of a controllability criteria for a linear singularly
perturbed discrete system with small step”, Lobachevskii J. Mathematics 45 (3), 938–948 (2024).
https://doi.org/10.1134/S1995080224600547
A. A. Azamov, G. I. Ibragimov, K. Mamayusupov, M. B. Ruzibaev, ”On the stability and null
controllability of an infinite system of linear differential equations”, J. Dynamical and Control
Systems (2021). https://doi.org/10.1007/s10883-021-09587-6
A. A. Azamov, M. B. Ruziboev, ”The time-optimal problem for evolutionary partial
differential equations”, J. Applied Mathematics and Mechanics 77 (2), 220–224 (2013).
https://doi.org/10.1016/j.jappmathmech.2013.07.013
A. G. Butkovsky, L. M. Pustylnikov, Theory of mobile control of systems with distributed parameters,
(Nauka, Moscow, 1980). [in Russian]
V. S. Deineka, ”Optimal control of the dynamic state of a thin compound plate”, Cybernetics and
Systems Analysis 42 (4), 151–157 (2006). https://doi.org/10.1007/s10559-006-0096-1
R. Deumlich, K. H. Elster, ”Duality theorems and optimality, conditions for nonconvex optimization
problems”, Mathematische Operationsforschung und Statistik. Series Optimization 11 (2), 181–219
(1980).
A. K. Kerimbekov, R. J. Nametkulova, A. K. Kadirimbetova, ”Optimality conditions in the problem
of thermal control with integral-differential equation”, Vestnik Irkutsk Gosudarstvennogo Universiteta. Matematika 15, 50–61 (2016). [in Russian]. https://www.mathnet.ru/rus/iigum252
A. K. Kerimbekov, E. F. Abdyldaeva, ”On the solvability of the tracking problem in nonlinear vector
optimization of oscillatory processes”, Trudy Instituta Matematiki i Mekhaniki Ural. Otd. 30 (2),
–115 (2024). [in Russian] https://doi.org/10.21538/0134-4889-2024-30-2-103-115
E. N. Mahmudov, ”Infimal convolution and duality in convex optimal control problems with second
order evolution differential inclusions”, Evolution Equations and Control Theory 10 (1), 37–59
(2021). https://doi.org/10.3934/eect.2020051
A. T. Ramazanova, A. N. Abdullozhonova, T. K. Yuldashev, ”Optimal mobile control in inverse
problem for Barenblatt–Zheltov–Kochina type fractional order equation”, Lobachevskii Journal of
Mathematics 46 (1), 488–501 (2025). https://doi.org/10.1134/S1995080224608087
T. K. Yuldashev, B. Y. Ashirbaev, ”Optimal feedback control problem for a singularly perturbed discrete system”, Lobachevskii J. Mathematics 44 (2), 661–668 (2023).
https://doi.org/10.1134/S1995080223020373
T. K. Yuldashev, B. J. Kadirkulov, A. T. Ramazanova, Zh. Zh. Shermamatov, ”Nonlinear optimal
control problem in a two-point boundary regime for a pseudoparabolic equation with Samarskii–
Ionkin type conditions”, Nanosystems: Physics, Chemics, Mathematics 16 (5), 563–576 (2025).
https://doi.org/10.17586/2220-8054-2025-16-5-563-576
T. K. Yuldashev, A. T. Ramazanova, Zh. Zh. Shermamatov, ”Optimal control problem for a linear pseudoparabolic equation with final condition, degeneration and Gerasimov–Caputo operator”,
Azerbaijan J. Mathematics 15 (1), 257–283 (2025). https://doi.org/10.59849/2218-6816.2025.1.259
M. K. Mamanov, G. A. Khasanov, Z. A. Madatova, ”Optimal control problem for an impulsive
systems of functional-differential equations with a nonlinear function under the sign of a first-order
differential”, J. Osh University ”Differential Equations 1 (1), 67–77 (2026).
O. Kh. Abdullaev, O. Sh. Salmanov, T. K. Yuldashev, ”Direct and inverse problems for a parabolichyperbolic equation involving Riemann–Liouville derivatives”, Trans. Natl. Acad. Sci. Azerb. Ser.
Phys.-Tech. Math. Sci. – Mathematics 43 (1), 21–33 (2023).
I. Area, H. Batarfi, J. Losada, J. J. Nieto, W. M. Shammakh, A. Torres, ”On a fractional order
Ebola epidemic model”, Advances in Differerence Equations 278 2015.
Zh. A. Artykova, T. A. Abduvakhobov, M. D. Shukurova, ”Nonlinear fractional-differential-integral
equation with product of two nonlinear functions, degeneration and maxima”, J. Osh University
”Differential Equations” 1 (1), 53–66 (2026).
L. Boudabsa, T. Simon, P. Vallois, ”Fractional extreme distributions”, Mathematics arxiv: Probability, No 46 (2021).
K. Diethelm, A. D. Freed, ”On the solution of nonlinear fractional order differential equations used in
the modeling of viscoplasticity”, Scientifice Computing in Chemical Engineering II-Computational
Fluid Dynamics, Reaction Engineering and Molecular Properties (F. Keil, W. Mackens, H. Voss,
and J. Werther, Eds), Springer–Verlag, Heidelberg, 1999. 217–224.
K. Diethelm, N. J. Ford, ”Analysis of fractional differential equations”, J. Math. Anal. Appl. 265,
–248 (2002).
D. Delbosco, L. Rodino, ”Existence and uniqueness for a nonlinear fractional differential equation”,
J. Math. Anal. Appl. 204, 609–625 (1996).
M. M. Dzhrbashian, Integral transformation and representation of functions in complex domain,
Nauka, Moscow, 1966. [in Russian]
L. Gaul, P. Klein, S. Kempfle, ”Damping description involving fractional operators”, Mech. Systems
Signal Processing No. 5, 81–88 (1991).
W. G. Glockle, T. F. Nonnenmacher, ”A fractional calculus approach of selfsimilar protein dynamics”, Biophys. J. 68, 46–53 (1995).
Handbook of Fractional Calculus with Applications. Vols. 1-8. Tenreiro Machado J.A. (ed.). Berlin,
Boston, Walter de Gruyter GmbH, 2019.
R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
A. Hussain, D. Baleanu, M. Adeel, ”Existence of solution and stability for the fractional order novel
coronavirus (nCoV-2019) model”, Advances in Differerence Equations 384 (2020).
E. R. Kaufmann, E. Mboumi, ”Positive solutions of a boundary value problem for a nonlinear
fractional differential equation”, Electron. J. Qual. Theory Differ. Equ. 2007 (3), 1–11 (2007).
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204. Elsevier Science B. V., Amsterdam, 2006.
M. T. Kosmakova, T. K. Yuldashev, M. U. Sagdullaeva, ”Inverse problem for a nonlinear pseudiparabolic differential equation with final condition and Gerasimov–Caputo operator”, Journal of
Osh University ”Differential Equations” 1 (1), 1–14 (2026).
V. Lakshmikantham, A. S. Vatsala, ”Theory of fractional differential inequalities and applications”,
Commun. Appl. Anal. 11 (3-4), 395–402 (2007).
F. Mainardi, ”Some basic problems in continuum and statistical mechanics”, Fractals and Fractional
Calculus in Continuum Mechanics. (A. Carpinteri and F. Mainardi, Eds), 291–348, Springer-Verlag,
Wien, 1997.
N. K. Ochilova, T. K. Yuldashev, ”On a nonlocal boundary value problem for a degenerate parabolichyperbolic equation with fractional derivative”, Lobachevskii J. Math. 43 (1), 229–236 (2022).
https://doi.org/10.1134/S1995080222040175
I. Podlubny, ”Fractional differential equations”, Mathematics in Sciences and Engineering 198,
Academic Press, San Diego, 1999.
A. V. Pskhu, Partial differential equation of fractional order, Nauka, Moscow, 2000 (in Russian).
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives. Theory and applications, Gordon and Breach, Yverdon, 1993.
H. Sun, A. Chang, Y. Zhang, W. Chen, ”A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications”, Fractional
Calculus and Applied Analysis 22 (1), 27–59 (2019).
T. K. Yuldashev, ”Determining of coefficients and the classical solvability of a nonlocal boundaryvalue problem for the Benney—Luke integro-differential equation with degenerate kernel”, J. Mathematical Sciences 254 (6), 793–807 (2021). https://doi.org/10.1007/s10958-021-05341-2
T. K. Yuldashev, O. Kh. Abdullaev, ”Inverse problems for the loaded parabolic-hyperbolic equation involves Riemann–Liouville operator”, Lobachevskii J. Math. 44 (3), 1080–1090 (2023).
https://doi.org/10.1134/S1995080223030034.
T. K. Yuldashev, B. J. Kadirkulov, ”On a boundary value problem for a mixed type fractional
differential equations with parameters”, Proceedings of the Institute of Mathematics and Mechanics,
National Academy of Sciences of Azerbaijan 47 (1), 112–123 (2021).
T. K. Yuldashev, B. J. Kadirkulov, R. A. Bandaliyev, ”On a mixed problem for Hilfer type fractional differential equation with degeneration”, Lobachevskii J. Mathematics 43 (1), 263–274 (2022).
https://doi.org/10.1134/S1995080222040229
T. K. Yuldashev, B. J. Kadirkulov, Kh. R. Mamedov, ”An inverse problem for Hilfer type differential
equation of higher order”, Bulletin of the Karaganda University. Mathematics series 105 (1), 136–
(2022). https://doi.org/10.31489/2022m1/136-149
S. Ullah, M. A. Khan, M. Farooq, Z. Hammouch, D. Baleanu, ”A fractional model for the dynamics
of tuberculosis infection using Caputo–Fabrizio derivative”, Discrete and Continuous Dynamical
Systems. Series S. 13 (3), 975–993 (2020). https://doi.org/10.3934/dcdss.2020057
