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Epilepsy affects approximately 50 million people worldwide. Despite its prevalence, the recurrence of seizures can be mitigated only 70% of the time through medication. Furthermore, surgery success rates range from 30% - 70% because of our limited understanding of how a seizure starts. However, one leading hypothesis suggests that a seizure starts because of a critical transition due to an instability. Unfortunately, we lack a meaningful way to quantify this notion that would allow physicians to not only better predict seizures but also to mitigate them. Hence, in this paper, we develop a method to not only characterize the instability of seizures but also to leverage these conditions to stabilize the system underlying these seizures. Remarkably, evidence suggests that such critical transitions are associated with long-term memory dynamics, which can be captured by considering linear fractional-order systems. Subsequently, we provide for the first time tractable necessary and sufficient conditions for the global asymptotic stability of discrete-time linear fractional-order systems. Next, we propose a method to obtain a stabilizing control strategy for these systems using linear matrix inequalities. Finally, we apply our methodology to a real-world epileptic patient dataset to provide insight into mitigating epilepsy and designing future cyber-neural systems.