Asymptotic stabilization for the full-discrete temporal–spatial-fractional reaction–diffusion system with time-varying delay

Under the designed state-feedback controller, the asymptotic stabilization problem is studied for the continuous, spatial semi-discrete and temporal–spatial full-discrete temporal–spatial-fractional reaction–diffusion systems with time-varying delay (DTSFRDSs). For the continuous DTSFRDS, the sufficient condition of asymptotic stability is developed through the fractional Halanay inequality based on the Lyapunov functional method. Using the finite element scheme, we construct the spatial semi-discrete DTSFRDS and derive a sufficient criterion to ensure asymptotic stability. Combined with the L1 scheme, the temporal–spatial full-discrete DTSFRDS is built. By deriving the discrete fractional lemma and the discrete fractional Halanay inequality, we provide a sufficient condition of asymptotic stability for the temporal–spatial full-discrete DTSFRDS. According to the obtained results, when the spatial step size h is sufficiently small, the stability of the continuous DTSFRDS can be inherited by the discrete DTSFRDSs under the state-feedback controller. Numerical example are provided to confirm the validity of our results.