Keywords:-
Article Content:-
Abstract
This paper investigates the Hyers–Ulam and generalized Hyers–Ulam–Rassias stability of a class of feedback-controlled fractional differential equations involving the Caputo derivative. By employing Laplace transform techniques and resolvent kernel representations, we derive explicit stability bounds that quantify the de viation between approximate and exact solutions. Sufficient conditions ensuring stability are obtained in terms of the fractional order, system parameters, time de lay, and feedback gain. Furthermore, the influence of the controller gain on stability performance is rigorously analyzed, and it is shown that increasing the feedback gain leads to a strict reduction in the Hyers–Ulam stability constant, thereby en hancing system robustness. The generalized Hyers–Ulam–Rassias stability is also established under time-varying perturbations. To validate the theoretical findings, a numerical simulation based on a predictor–corrector scheme is presented, demon strating the effectiveness of the proposed control strategy in improving stability and convergence. The obtained results significantly extend existing Hyers–Ulam stability theory to controlled fractional-order dynamical systems and provide new insights into robust control design for systems with memory effects.
References:-
References
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Frac
tional Differential Equations, Elsevier, Boston, USA (2006).
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999).
Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singa
pore (2014).
C. Alsina, R. Ger, On some inequalities and stability results related to the exponen
tial function, J. Inequal. Appl. (1998) 373–380.
S. M. Jung, Hyers–Ulam stability of linear differential equations of first order, Appl.
Math. Letters 17(10) (2004) 1135-1140.
H. Choda, T. Miura, S.-E. Takahasi, On the Hyers–Ulam stability of real continuous
function valued differentiable maps, Tokyo J. Math. 24(2) (2001) 467–476.
S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960.
D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad.
Sci. USA27 (1941) 222–224.
Th. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer.
Math. Soc. 72 (1978) 297–300.
A. M. Elshenhab and X. Wang, Controllability and Hyers–Ulam Stability of Differ
ential Systems with Pure Delay, Mathematics 10 (2022) 1248.
A. M. A. El-Sayed, E. M. Al-Barg and H. R. Ebead, A comprehensive study of
a feedback control problem with a state-dependent implicit pantograph equation,
AIMS Mathematics 10 (2025) 951–971.
J. Wang and Y. Zhou, Mittag-Leffler-Ulam stabilities of fractional evolution equa
tions, Applied Mathematics Letters 25 (2012) 723-728.
S. Moualkia, Y. Liu, J. Qiu and J. Cao, Mittag-Leffler-Ulam stabilities for variable
fractional-order differential equations driven by Levy noise, Nonlinear Dynamics 113
(2025) 13043–13059.
R. Shah and N. Irshad, Ulam–Hyers–Mittag–Leffler Stability for a Class of Non
linear Fractional Reaction-Diffusion Equations with Delay, International Journal of
Theoretical Physics 64 (2025) 20.
S. Kundu and S. N. Bora, On the Ulam-Hyers-Rassias-Mittag-Leffler Stability of
the Solution to a Ψ-Hilfer Abstract Fractional Differential Equation, International
Journal of Theoretical Physics 64 (2025) 190.
K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists
and Engineers, Princeton University Press (2008).
C. Xiao, B. Zeng and Z. Liu, Feedback control for fractional impulsive evolution
systems, Applied Mathematics and Computation 268 (2015) 924- 936.
A. I. Mees, Dynamics of Feedback Systems, Wiley, New York (1981).
G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic
Systems, Pearson (2015).
S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analy
sis, Springer, New York (2011).