Delay-range-dependent local adaptive and robust adaptive synchronization approaches for time-delay chaotic systems

This paper studies the local adaptive and robust adaptive control methodologies for the synchronization of the chaotic drive and the response systems with finite time lags, bounded delay rates, unknown parameters, and perturbations. A local adaptive coherence control condition under bounded initial condition is developed to synthesize a feedback controller capable of executing successful synchronization of the drive and the response systems with uncertain parameters and delay belonging to a range (with either zero or nonzero lower bound). A sufficiency condition for the adaptive control approach is inculcated through an amendment in the conventional Lyapunov–Krasovskii functional, which enables to develop the adaptation laws for the unknown parameters associated with the nonlinearity in the dynamical behavior of the drive and the response systems. Further, a robust adaptive synchronization control condition for asymptotic convergent behavior of the synchronization error to a bounded region under perturbations as well as disturbances is provided. Indifferent to the existing works, delay-range-dependent adaptive synchronization methodologies are derived in this paper, and the rendered controller design conditions are valid for both the slow and the fast variations in time delays. Numerical simulations for the chaotic delayed Hopfield neural networks are presented to elaborate the efficacy of the proposed delay-range-dependent adaptive and robust adaptive synchronization control methodologies.