In this article, novel results on the maximality of discrete fractional Green's functions are established and corresponding explicit Lyapunov inequalities for delta fractional systems, with applications to chaos analysis and robust control design, are derived. For the proposed Riemann-Liouville fractional difference system with the delta boundary conditions, explicit expressions for the maximum values of the associated Green's function over its domain are obtained. These results lead to a refined Lyapunov delta-type inequality establishing a necessary condition for the existence of nontrivial solutions, where the lower bound explicitly depends on the maximum values of the fractional order and the Green's function. Furthermore, it is demonstrated that violation of this inequality implies the existence of nontrivial solutions and can induce chaotic behavior in fractional difference systems. For control applications, robust stability conditions for uncertain fractional systems are established and stabilizing state feedback controllers is designed. Finally, the numerical examples validate the emergence of chaos under inequality violation and confirm the control design's efficacy for robust stability.
Arab et al. (Tue,) studied this question.