A necessary optimality condition for Lagrange problem with fractional partial system

Marek Majewski

doi:10.3934/dcdsb.2025017

Article Contents

2025, Volume 30, Issue 11: 4394-4404. Doi: 10.3934/dcdsb.2025017

This issue Previous Article Existence and non-existence results for higher order fractional boundary value problem Next Article Existence of nonoscillatory solution on $ k $-dimensional system of delayed nonlinear discrete equations with $ p $-Laplacian

A necessary optimality condition for Lagrange problem with fractional partial system

Marek Majewski^1, ,

1.
Faculty of Mathematics and Computer Science University of, Lodz Banacha 22 90-238 Lodz, Poland

^*Corresponding author: Marek Majewski
^*Corresponding author: Marek Majewski

Received: October 2024

Revised: January 2025

Early access: February 2025

Published: November 2025

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In the paper, a necessary condition for the existence of minimum for a Lagrange problem governed by a partial fractional differential equation with a mixed Riemann-Liouville derivative is derived. The proof of the main result is based on an application of the Smooth-Convex Pontryagin Principle.

Keywords:

Mathematics Subject Classification: Primary: 26A33, 35R11, 49K20; Secondary: 49J20, 35G31.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Mathematics, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[2]	E. Fornasini and G. Marchesini, Doubly-indexed dynamical systems: State-space models and structural properties, Math. Systems Theory, 12 (1978), 59-72. doi: 10.1007/BF01776566.
[3]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.
[4]	D. Idczak, Optimal control of a coercive Dirichlet problem, SIAM Journal on Control and Optimization, 36 (1998), 1250-1267. doi: 10.1137/S0363012997296341.
[5]	D. Idczak, R. Kamocki and M. Majewski, Nonlinear continuous Fornasini–Marchesini model of fractional order with nonzero initial conditions, Journal of Integral Equations and Applications, 32 (2020), 19-34. doi: 10.1216/JIE.2020.32.19.
[6]	A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holland Publishing Company, 1979.
[7]	Z. D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems, Structural and Multidisciplinary Optimization, 38 (2009), 571-581. doi: 10.1007/s00158-008-0307-7.
[8]	R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Mathematical Methods in the Applied Sciences, 37 (2014), 1668-1686. doi: 10.1002/mma.2928.
[9]	R. Kamocki, Necessary and sufficient optimality conditions for fractional nonhomogeneous Roesser model, Optimal Control Applications and Methods, 37 (2016), 574-589. doi: 10.1002/oca.2180.
[10]	R. Kamocki, Necessary optimality conditions for a lagrange problem governed by a continuous Roesser model with Caputo derivatives, Archives of Control Sciences, 34 (2024), 513-535.
[11]	T. Kosztołowicz, Subdiffusion with particle immobilization process described by a differential equation with Riemann-Liouville-type fractional time derivative, Phys. Rev. E, 108 (2023), Paper No. 014132, 6 pp. doi: 10.1103/PhysRevE.108.014132.
[12]	T. Kosztołowicz and A. Dutkiewicz, Stochastic interpretation of $g$-subdiffusion process, Phys. Rev. E, 104 (2021), Paper No. L042101, 5 pp. doi: 10.1103/PhysRevE.104.L042101.
[13]	M. Majewski, On the existence of optimal solutions to an optimal control problem, Journal of Optimization Theory and Applications, 128 (2006), 635-651. doi: 10.1007/s10957-006-9036-5.
[14]	M. Majewski, On the existence of optimal solutions to the Lagrange problem governed by a nonlinear Goursat-Darboux problem of fractional order, Opuscula Mathematica, 43 (2023), 547-558. doi: 10.7494/OpMath.2023.43.4.547.
[15]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[16]	A. Nemati and K. Mamehrashi, The use of the Ritz method and Laplace transform for solving 2D fractional-order optimal control problems described by the Roesser model, Asian Journal of Control, 21 (2019), 1189-1201. doi: 10.1002/asjc.1791.
[17]	Ł. Płociniczak and K. Taźbierski, Fully discrete galerkin scheme for a semilinear subdiffusion equation with nonsmooth data and time-dependent coefficient, Computers and Mathematics with Applications, 165 (2024), 217-223. doi: 10.1016/j.camwa.2024.04.032.
[18]	Ł. Płociniczak and M. Świtała, Existence and uniqueness results for a time-fractional nonlinear diffusion equation, Journal of Mathematical Analysis and Applications, 462 (2018), 1425-1434. doi: 10.1016/j.jmaa.2018.02.050.
[19]	S. Pooseh, R. Almeida and D. F. M. Torres, Fractional order optimal control problems with free terminal time, Journal of Industrial and Management Optimization, 10 (2014), 363-381. doi: 10.3934/jimo.2014.10.363.
[20]	R. T. Rockafellar, Integral functionals, normal integrands and measurable selections, in Nonlinear Operators and the Calculus of Variations, Springer Berlin Heidelberg, 543 (1976), 157-207. doi: 10.1007/BFb0079944.
[21]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, CRC Press, Gordon and Breach Science Publishers, 1993.
[22]	J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst., Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.
[23]	S. Walczak, Absolutely continuous functions of several variables and their applications to differential equations, Bulletin Polish Academy of Science, Math., 35 (1987), 733-744.