Stability of non-instantaneous impulsive differential equations with supremum

Asymptotic stability of non‐instantaneous impulsive systems and T‐S fuzzy non‐instantaneous impulsive control for nonlinear systems

“…Agarwal [49] studied the Lipschitz stability of neural networks with non-instantaneous impulses via the comparison system method, but the conditions are too conservative and difficult to verify. Hristova et al [50] provided a method of dealing with the stability problem of non-instantaneous impulsive differential equations with supremum via the comparison systems, but the specific conditions of stability for non-instantaneous impulsive differential equations with supremum were not given. Agarwal et al [51] first studied Lipschitz stability of solutions of nonlinear non-instantaneous Caputo delay fractional differential equations via Lyapunov-like functions, but the relationship between the impulsive time series and the stability of the system solution is not revealed.…”

“…From the above discussions, inspired by [50,51,56,57], this paper focuses on the asymptotic stability of nonlinear non-instantaneous impulsive perturbation systems and the non-instantaneous impulsive control for some nonlinear systems. Main contributions of this paper can be summarized as follows:…”

“…Hristova et al. [50] provided a method of dealing with the stability problem of non‐instantaneous impulsive differential equations with supremum via the comparison systems, but the specific conditions of stability for non‐instantaneous impulsive differential equations with supremum were not given. Agarwal et al.…”

“…From the above discussions, inspired by [50, 51, 56, 57], this paper focuses on the asymptotic stability of nonlinear non‐instantaneous impulsive perturbation systems and the non‐instantaneous impulsive control for some nonlinear systems. Main contributions of this paper can be summarized as follows: (1)Based on the existing linear non‐instantaneous impulsive models and nonlinear non‐instantaneous impulsive models, we propose a more universal nonlinear non‐instantaneous impulsive model which can explain the whole evolution process of the system more accurately. (2)In view of the difficulty in verifying the stability conditions for nonlinear non‐instantaneous impulsive systems in [50, 51], a class of concrete comparison system is constructed, which makes the obtained stability conditions depend only on the parameters of the comparison system, the impulsive control gain and the impulsive time series, thus reducing the conservatism of the conditions and making them easy to be verified. (3)The results in reference [57] are extended to the case of non‐instantaneous impulsive control for nonlinear systems, which is closer to the actual situation and can theoretically provide more alternative control schemes. …”