On General Fractional-Order Discrete-Time Reaction-Diffusion Systems: Finite Time Stability and Simulations

Finite-time stability (FTS) in fractional-order reaction-diffusion systems (FO-RDs) is critically important for biochemical applications where rapid stabilization prevents metabolic imbalances, such as in glucose-insulin regulatory systems where delayed convergence can lead to diabetic complications. This work advances the field by providing the first comprehensive stability framework for discrete-time FO-RDs that guarantees convergence within a predetermined finite time, unlike previous approaches that only established asymptotic stability. We introduce a novel Lyapunov functional (LF) and, through careful eigenvalue analysis of the discrete Laplacian and boundedness of nonlinear terms, we establish that its fractional derivative remains strictly negative for all states within a neighborhood of the equilibrium point, with explicit bounds derived from system parameters. This approach, which integrates discrete Green’s formula and eigenvalue estimates, rigorously addresses the challenges of non-local fractional operators in spatially discretized systems. The theoretical framework is validated through numerical simulations of a fractional glycolysis model, which demonstrate rapid convergence and robustness. The results confirm the method’s potential for designing robust control strategies for complex biochemical and chemical processes that require swift stabilization.