Abstract
In this paper, the global Mittag-Leffler synchronization (GMLS) for fractional-order neural networks with time-varying delay (FNNTVD) is studied based on hybrid sliding mode control (HSMC). First, a class of sliding surface is designed to study the dynamic behavior of FNNTVD by using the integer order Lyapunov direct method, which resolves the problem that the activation function of FNNTVD cannot be applied to the construction of Lyapunov function in the previous literature. Then, a new HSMC with the fixed signal transmission time delay is developed to ensure GMLS of FNNTVD. Next, based on Lyapunov direct method and the designed HSMC, some criteria are put forward to achieve GMLS of FNNTVD. Finally, two examples are given to verify the effectiveness of the proposed synchronization criteria for FNNTVD.
Highlights
Fractional order system, as a more general dynamic system, has attracted increasing concerns [1]–[5] owing to their special properties: (1) the fractional order parameters can help analyze the dynamic behavior of systems by adding a degree of freedom [1]–[3]; (2) fractional-order systems have merits of memory and hereditary [4], [5]
In the proof process of Theorem 1, we find that the system (3) is still global Mittag-Leffler synchronization (GMLS) under Assumption 1 and Assumption 3
According to Theorem 1, we can obtain that the drive-response systems (42) and (43) are GMLS based on hybrid sliding mode control (HSMC)
Summary
Fractional order system, as a more general dynamic system, has attracted increasing concerns [1]–[5] owing to their special properties: (1) the fractional order parameters can help analyze the dynamic behavior of systems by adding a degree of freedom [1]–[3]; (2) fractional-order systems have merits of memory and hereditary [4], [5]. In [13], finite-time stability problem for fractional-order complex-valued neural networks with time delay is discussed. Only few studies explore the fractional-order neural networks with time-varying delay (FNNTVD) [21]–[29]. In [40], a non-fragile observer-based adaptive SMC is designed to achieve stability for fractional-order Markovian jump systems with time delay and input nonlinearity. Fixed-time synchronization problem of fractional-order memristive MAM neural networks is discussed by using SMC in [41]. In [42], adaptive SMC for a class of fractional-order chaotic systems is proposed to ensure the fuzzy neural network-based chaos synchronization. Mittag-Leffler synchronization (GMLS) for FNNTVD is studied by using the integer order Lyapunov direct method (IOLDM) and hybrid sliding mode control (HSMC).
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