This paper investigates the input-to-state stability (ISS) of nonlinear impulsive systems with hybrid impulses and proposes a unified analytical framework based on the average impulsive gain (AIG) method. Addressing the limitations of existing methods in handling hybrid impulses (the coexistence of stabilizing and destabilizing impulses) and time-varying impulsive gains, this paper incorporates the AIG method into ISS analysis. It allows for different Lyapunov gains at each impulsive moment and combines impulsive control with sampled-data control in a hybrid strategy, significantly enhancing the robustness of the system. By introducing time-varying Lyapunov gains and AIG conditions, sufficient conditions for achieving ISS are derived, along with linear matrix inequality (LMI)-based conditions. The results show that ISS of nonlinear impulsive systems can be ensured in the presence of hybrid impulses, using appropriate impulsive intervals and hybrid control strategies. Finally, two numerical examples are provided to validate the effectiveness of the results.
AhmadiFBatmaniYBevraniH, et al. (2023) Finite-time synergetic controller design for DC microgrids with constant power loads. IEEE Transactions on Smart Grid14(5): 3352–3361.
2.
ChenDLiuXSongY (2022) Input-to-state stability for discrete-time monotone systems via max-separable Lyapunov functions. Procedia Computer Science199: 1024–1030.
3.
ChenHShiPLimCC (2024) Stability analysis for nonautonomous impulsive hybrid stochastic delay systems. Systems & Control Letters187: 105785.
4.
ChenWHLiXNiuS, et al. (2023) Input-to-state stability of positive delayed neural networks via impulsive control. Neural Networks164: 576–587.
5.
CuiQCaoJAbdel-AtyM (2024) Stability of state-dependent switched systems under hybrid delayed impulsive control. Proceedings of the Royal Society A480(2304): 20240286.
6.
DashkovskiySFeketaP (2017) Input-to-state stability of impulsive systems and their networks. Nonlinear Analysis: Hybrid Systems26: 190–200.
7.
De la SenMIbeasA (2008) On the global asymptotic stability of switched linear time-varying systems with constant point delays. Discrete Dynamics in Nature and Society2008(1): 231710.
8.
De la SenMIbeasAGarridoAJ, et al. (2025) Fixed points of self-mappings with jumping effects: Application to stability of a class of impulsive dynamic systems. Mathematics13(7): 1157.
9.
FangYZhangJLiY (2025) Neural networks adaptive predefined-time control for pure-feedback nonlinear systems: A case study on robotic exoskeleton systems. Scientific Reports15(1): 6041.
10.
GaoKLuJZhengWX, et al. (2025) Synchronization in coupled neural networks with hybrid delayed impulses: Average impulsive delay-gain method. IEEE Transactions on Neural Networks and Learning Systems36(2): 3608–3617.
11.
GaoLLiuZYangS (2024a) Input-to-state stability in probability of constrained impulsive switched delayed systems with unstable subsystems. Applied Mathematics and Computation477: 128820.
12.
GaoLXieYYangS (2024b) Period-sampled switching event-triggered control of discrete-time impulsive switched systems under two asynchronous behaviors. Chaos, Solitons & Fractals185: 115096.
13.
GengZTangZDingD, et al. (2024) Delay-compensatory synchronization on uncertain chaotic neural networks via average delayed impulsive gains. Journal of the Franklin Institute361(17): 107235.
14.
HanQWangLChenZ (2025) An ISS-based design of global robust adaptive output regulators for nonlinear systems. IEEE Transactions on Automatic Control70(5): 3354–3361.
15.
HespanhaJPLiberzonDTeelAR (2008) Lyapunov conditions for input-to-state stability of impulsive systems. Automatica44(11): 2735–2744.
16.
HuZYangZMuX (2019) Stochastic input-to-state stability of random impulsive nonlinear systems. Journal of the Franklin Institute356(5): 3030–3044.
17.
JiXLuJJiangB, et al. (2022a) Distributed synchronization of delayed neural networks: Delay-dependent hybrid impulsive control. IEEE Transactions on Network Science and Engineering9(2): 634–647.
18.
JiXLuJJiangB, et al. (2022b) Network synchronization under distributed delayed impulsive control: Average delayed impulsive weight approach. Nonlinear Analysis: Hybrid Systems44: 101148.
KumarRFeketaPMeurerT (2024) Input-to-state stability of nonlinear impulsive systems with hybrid impulses. IFAC-PapersOnLine58(17): 184–189.
21.
LiPLiXLuJ (2021) Input-to-state stability of impulsive delay systems with multiple impulses. IEEE Transactions on Automatic Control66(1): 362–368.
22.
LianJLiCZhaiG (2021) Stability of impulsive switched systems with sampled-data control. IET Control Theory & Applications15(4): 523–533.
23.
LiangZLiuX (2024) Input-to-state hybrid impulsive formation stabilization for multi-agent systems with impulse delays. Communications in Nonlinear Science and Numerical Simulation139: 108323.
24.
LiuBLiMGShiYD, et al. (2025) Robust finite-time input-to-state stability via impulsive hybrid control for uncertain dynamical systems with disturbances. ISA Transactions160: 97–110.
25.
LiuWLiPLiX (2022) Impulsive systems with hybrid delayed impulses: Input-to-state stability. Nonlinear Analysis: Hybrid Systems46: 101248.
26.
LvXLiX (2017) Delay-dependent dissipativity of neural networks with mixed non-differentiable interval delays. Neurocomputing267: 85–94.
27.
MaoXZhengQXiaoZ (2025) Hybrid resilient observer-based guaranteed cost control for nonlinear impulsive switched systems with multipath quantizations and packet dropouts. International Journal of Robust and Nonlinear Control35: 4754–4771.
28.
SekineKNakaoMTOishiS (2020) A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: Without direct estimation for an inverse of the linearized operator. Numerische Mathematik146: 907–926.
29.
ShenHYuXYanH, et al. (2025) Robust fixed-time sliding mode attitude control for a 2-DOF helicopter subject to input saturation and prescribed performance. IEEE Transactions on Transportation Electrification11(1): 1223–1233.
30.
ShenHZhaoWCaoJ, et al. (2024) Predefined-time event-triggered tracking control for nonlinear servo systems: A fuzzy weight-based reinforcement learning scheme. IEEE Transactions on Fuzzy Systems32(8): 4557–4569.
31.
ShiFLiuYLiY, et al. (2022) Input-to-state stability of nonlinear systems with hybrid inputs and delayed impulses. Nonlinear Analysis: Hybrid Systems44: 101145.
32.
ShiJPengCGuoY, et al. (2024) Input-to-state stabilization of discrete-time delayed fuzzy systems via aperiodically intermittent control. ISA Transactions155: 205–216.
SunWWangMXiaJ, et al. (2025) Input-to-state stabilization of nonlinear systems with impulsive disturbance via event-triggered impulsive control. Applied Mathematics and Computation497: 129365.
35.
TanJZhangKLiB, et al. (2023) Event-triggered sliding mode control for spacecraft reorientation with multiple attitude constraints. IEEE Transactions on Aerospace and Electronic Systems59(5): 6031–6043.
36.
WangMLiPLiX (2023) Event-triggered delayed impulsive control for input-to-state stability of nonlinear impulsive systems. Nonlinear Analysis: Hybrid Systems47: 101277.
37.
WangNLiXLuJ, et al. (2018) Unified synchronization criteria in an array of coupled neural networks with hybrid impulses. Neural Networks101: 25–32.
38.
WangPGuoWSuH (2022) Improved input-to-state stability analysis of impulsive stochastic systems. IEEE Transactions on Automatic Control67(5): 2161–2174.
39.
WangPWangXSuH (2021) Input-to-state stability of impulsive stochastic infinite dimensional systems with Poisson jumps. Automatica128: 109553.
40.
WangZZhuQ (2024) Stability of impulsive and switched stochastic delay systems via the event-triggered impulsive mechanism. Systems & Control Letters188: 105806.
ZhangHYuCZengM, et al. (2024a) Homomorphic encryption-based resilient distributed energy management under cyber-attack of micro-grid with event-triggered mechanism. IEEE Transactions on Smart Grid15(5): 5115–5126.
43.
ZhangTCaoJLiX, et al. (2024b) Finite-time input-to-state stability and settling-time estimation of impulsive switched systems with multiple impulses. Journal of the Franklin Institute361(17): 107205.
44.
ZhouJDongJXuS, et al. (2024) Input-to-state stabilization for Markov jump systems with dynamic quantization and multimode injection attacks. IEEE Transactions on Systems, Man, and Cybernetics: Systems54(4): 2517–2529.
