## 1 Introduction
### Motivation and Background  
The stability and control of FOSs are critical for their effective deployment in practical scenarios, particularly in systems where nonlinearity, input saturation, and time delays play a significant role. Fractional derivatives provide a compelling physical interpretation, wherein the system's present state depends not merely on the instantaneous rate of change but also on the cumulative historical dynamics. This attribute underscores the indispensability of FC in modeling and analyzing systems with long-term dependencies. A significant challenge in the control of FOSs arises from actuator saturation, a phenomenon that stems from the physical limitations of actuators [17]. If left unaddressed, actuator saturation can lead to degraded performance and, in extreme cases, instability of the closed-loop system. Extensive research has focused on mitigating these effects through advanced control strategies. For instance, stability analysis under saturation has been well studied in linear FOSs, leveraging analytical methods to estimate the domain of attraction while incorporating actuator constraints [18; 19; 20]. Nonlinear FOSs introduce additional complexities due to their sensitivity to initial conditions and external perturbations. The authors of [21] addressed this issue by deriving sufficient stabilization conditions using the Gronwall-Bellman lemma and sector-bounded conditions. Other studies have explored uncertain FOSs with saturation constraints, employing the Lipschitz condition to design state feedback controllers that ensure stability [22]. Recent advancements include the development of adaptive backstepping control strategies tailored to incommensurate FOSs under input saturation [23]. These strategies utilize frequency-distributed models and indirect Lyapunov methods to account for control constraints. Moreover, the stochastic stability of FOSs with actuator saturation has been rigorously analyzed, with stability criteria derived using stochastic system theory and FO calculus [24]. Additional research has explored delay-dependent stability in uncertain FOS, particularly those with distributed delays and input saturation. These studies employ Lyapunov-Krasovskii functionals to derive robust stability conditions, expressed as linear matrix inequalities (LMIs), incorporating time delays into their analysis [25]. Despite these advancements, the combined effects of time delays and saturation control in FOSs remain underexplored, underscoring the need for comprehensive stability analyses that integrate these critical factors. Time delays present another major challenge in the stability analysis of dynamical systems. Considerable research has addressed this issue, often focusing on FOSs with linear components and pure delays. For instance, characteristic polynomial methods have been employed to derive stability conditions for such systems [26]. Additionally, the Mittag-Leffler function has been utilized to develop decentralized FO controllers for nonlinear FOSs, accounting for its effects in stabilizing time-delayed systems [27]. The problem of asymptotic stabilization in nonlinear FOSs with time delays has also been addressed using innovative control strategies. A notable contribution in this area was the development of two novel stabilization methods for FOSs with multiple time delays [28]. This work introduced a stabilization control criterion that integrates a Lyapunov-like function with the M-matrix method, offering a more robust solution for managing the complexities of time delays in nonlinear systems. Another key study extended the application of vector Lyapunov functions to FOSs with time delays, proposing a new stability condition that specifically addressed the challenges posed by time delays in FOSs [29]. A time-delay feedback controller was designed in [30] to suppress the chaos of the FO chaotic jerk system, where delay-independent stability and bifurcation conditions were established.