Hybrid control strategy for input-to-state stability of nonlinear systems with mixed impulses

Abstract

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.

Keywords

Nonlinear systems hybrid impulses input-to-state stability average impulsive gain hybrid control strategy

Introduction

Over the past few decades, linear systems have been widely used across various fields due to their simplicity and analyzability. However, a major limitation of linear systems is their inability to fully capture nonlinear characteristics, which restricts their applicability in complex dynamic systems. In fact, most practical control systems are nonlinear (Khalil and Grizzle, 2002) and exhibit complex and unpredictable relationships between their inputs and outputs. This complexity makes nonlinear systems play a crucial role in fields such as robotics (Fang et al., 2025; Han et al., 2025), aerospace (Tan et al., 2023; Zhou et al., 2024), and power systems (Ahmadi et al., 2023; Zhang et al., 2024a). To address this issue, the concept of input-to-state stability (ISS) was introduced (Sontag, 1989), providing an effective framework to analyze the stability and performance of nonlinear systems. ISS refers to the dynamic behavior of a system state as characterized by external inputs, offering a quantitative estimate of the performance of the system. Since the introduction of the ISS concept, researchers have extended it to various types of dynamic systems, including discrete-time systems (Chen et al., 2022; Shi et al., 2024), switched systems (Cui et al., 2024; De la Sen and Ibeas, 2008; Mao et al., 2025), and stochastic systems (Hu et al., 2019; Wang et al., 2022), among others.

In practical engineering, certain nonlinear systems may exhibit instantaneous jump behaviors, a phenomenon known as “impulsive behavior.” Impulsive control offers a more flexible and efficient solution when dealing with systems marked by uncertainty and complex dynamic properties, thereby establishing itself as an increasingly significant research direction in the ISS analysis of nonlinear systems. In recent years, fundamental results have been achieved regarding the study of ISS of nonlinear systems through the application of impulsive control (Chen et al., 2024; Kumar et al., 2024; Liang and Liu, 2024; Liu et al., 2025; Wang and Zhu, 2024). In prior research, the seminal work by Hespanha et al. (2008) pioneered the extension of the ISS framework to impulsive systems through the introduction of a candidate exponential ISS-Lyapunov function and the formulation of the average dwell-time (ADT) and the reverse average dwell-time (rADT) concept. Despite these advancements, previous studies primarily focused on analyzing the stabilizing or destabilizing effects of impulsive behavior on system stability while assuming that the impulsive gain remained constant across all impulsive events, which is inconsistent with reality and limits their application in practical systems. To further investigate the ISS problem in nonlinear hybrid systems exhibiting mixed impulses, Dashkovskiy and Feketa (2017) examined the effects of multiple impulses and proposed ISS-Lyapunov functions with varying rate coefficients. Building upon this foundation, Liu et al. (2021, 2022) conducted a further investigation into the ISS of nonlinear impulsive systems characterized by time delays. Although existing research on multi-impulse systems has made significant progress by overcoming traditional fixed-gain limitations, challenges persist in complex scenarios. Conventional methods remain ineffective in achieving a dynamic balance between stabilization and destabilization effects within mixed-impulse systems.

To address the persistent challenges in stability analysis under hybrid impulsive dynamics, researchers have demonstrated that the AIG method, which integrates statistical averaging principles and dynamic equilibrium mechanisms, effectively overcoming the inherent limitations of conventional fixed-gain frameworks and exhibiting significantly enhanced adaptability in addressing ISS for hybrid impulsive systems. In Wang et al. (2018), the AIG is presented to study the problem of globally exponential synchronization of coupled neural networks with hybrid impulses. Subsequently, Geng et al. (2024), Gao et al. (2025), and Ji et al. (2022a) build on this foundation to study the synchronization of neural networks with delayed impulses. By summarizing the above studies, we observe that the AIG plays a crucial role in analyzing the stability of impulsive systems, particularly those with hybrid impulses. It provides a unified framework that accounts for both the stabilizing and destabilizing effects of impulsive jumps. This approach is particularly effective for dealing with systems that have infinitely many distinct Lyapunov gains, where a separate Lyapunov gain is allowed at each impulsive moment. Lian et al. (2021) combined sampled-data control and impulsive control to study impulsive switched systems with asynchronous switching. Their findings show that hybrid controllers offer better control than single controllers. Similar hybrid paradigms have shown efficacy in impulsive switched systems (Gao et al., 2024b) and robust finite-time control (Liu et al., 2025). This hybrid paradigm not only synergistically coordinates the temporal precision of sampled-data control with the discontinuous stabilization capability of impulsive actions but also provides enhanced adaptability to heterogeneous impulse sequences, thereby enabling a balanced trade-off between transient performance and asymptotic stability in ISS analysis for nonlinear systems with hybrid impulses.

Building upon these foundations, this paper advances the application of the AIG methodology to investigate the ISS of nonlinear impulsive systems subject to hybrid impulses, while proposing a novel hybrid control framework to concurrently enhance stability and robustness. The primary contributions of this work are threefold:

In comparison with recent studies on impulsive systems, such as those by Wei and Li (2023), De la Sen et al. (2025), and Zhang et al. (2024b), this paper introduces a novel approach by utilizing the concept of average impulsive gain (AIG) to investigate the influence of hybrid impulses on the ISS of impulsive systems. Unlike traditional methods, this approach accommodates varying gains at each impulsive moment. This refinement adds a layer of flexibility and realism to the stability analysis, making it more relevant for real-world systems where hybrid impulsive forces may occur.

The concept of ADT was introduced, which no longer has any constraints on the upper bound of impulsive interval. In contrast to Wang et al. (2022), it provides a unified analytical framework, which can effectively handle hybrid impulsive behaviors, reducing the conservativeness of analysis and enhancing system robustness. It is also convenient for practical applications and numerical solutions.

In contrast to single control methods (Shen et al., 2024, 2025; Shi et al., 2022; Zhang et al., 2024b), hybrid control integrates the rapid response characteristics of impulsive control with the precise regulation capabilities of sampled-data control. This combined approach provides a more effective solution for addressing system uncertainties and external disturbances, thereby enhancing the overall robustness and stability of the system. The synergy between impulsive and sampled-data control not only improves the adaptability of system but also ensures better performance in real-world applications where both quick adjustments and fine-tuned regulation are essential.

The remaining parts of this paper are organized as follows: Section “Preliminaries” describes the preliminaries. Section “Main results” derives the main results. Two examples are presented in section “Examples.” Section “Conclusion” is a summary of the paper and future research directions.

Notations. Let $ℤ_{+}$ represent the set of positive integers, ℝ the set of real numbers, and $ℝ_{+}$ the set of positive real numbers. $ℝ^{n}$ and $ℝ^{m \times n}$ denote the n-dimensional and $m \times n$ -dimensional real spaces equipped with the Euclidean norm |⋅|. I denotes the identity matrix with appropriate dimensions and ⋆ denotes the symmetric block in one symmetric matrix. A function $φ : [0, \infty) \to [0, \infty)$ is of class K if φ is continuous, strictly increasing, and $φ (0) = 0$ . If φ is also unbounded, it is of class $K_{\infty}$ . A function $χ : [0, \infty) \times [0, \infty) \to [0, \infty)$ is of class $KL$ , if $χ (\cdot, t)$ is of class K for each fixed $t \geq 0$ , and $χ (r, t)$ decreases to 0 as $t \to \infty$ for each fixed $r \geq 0$ .

Preliminaries

We consider the following impulsive system with external inputs

\begin{matrix} {\begin{matrix} ẋ (t) = f (x (t), μ (t)), & t \neq t_{k}, \\ x (t) = g_{k} (x^{-} (t), μ^{-} (t)), & t = t_{k}, k \in ℤ_{+}, \end{matrix} \end{matrix}

(1)

where ${t_{k}}_{k \in ℤ_{+}}$ denote a strictly increasing sequence of impulsive instants within the interval $(t_{0}, \infty)$ for some initial time $t_{0}$ ; the state $x (t) \in ℝ^{n}$ is absolutely continuous between consecutive impulses; input $μ (t) \in ℝ^{m}$ is a locally bounded and Lebesgue-measurable function; for every $k \in ℤ_{+}$ , f and $g_{k}$ are the functions from $ℝ^{n} \times ℝ^{m}$ to $ℝ^{n}$ , with f subject to $ℓ_{i}^{-} \leq \frac{f_{i} (x_{1}, μ) - f_{i} (x_{2}, μ)}{x_{1} - x_{2}} \leq ℓ_{i}^{+}, \forall x_{1}, x_{2} \in ℝ^{n}, x_{1} \neq x_{2}, μ \in ℝ^{m}$ , $g_{k}$ subject to locally Lipschitz, and $f (0, 0) = g_{k} (0, 0) = 0, \forall k \in ℤ_{+}$ . $ℓ_{i}^{-}$ and $ℓ_{i}^{+}$ are some real constants and they may be positive, zero or, negative. The $x (t)$ and $μ (t)$ are assumed to be right-continuous with left limits at all time instants, where the notation $x^{-} (t)$ denotes the left limit at time t.

Definition 1. For a given impulsive time sequences ${t_{k}}_{k \in ℤ_{+}}$ , system (1) is said to possess the property of ISS if there exist functions $β \in KL$ and $Γ_{μ} \in K_{\infty}$ , such that for any initial condition $x (t_{0}) \in ℝ^{n}$ and measurable essentially bounded input μ, the solution of system (1) exists globally and satisfies:

\begin{matrix} | x (t) | \leq β (| x (t_{0}) |, t - t_{0}) + Γ_{μ} (∥ μ ∥_{[t_{0}, t]}), \forall t \geq t_{0}, \end{matrix}

(2)

where $∥ \cdot ∥_{J}$ denotes the supremum norm on an interval J.

Definition 2. Consider a function $V : ℝ^{n} \to ℝ$ . $V$ is referred to as a candidate exponential ISS-Lyapunov function for (1) if it satisfies the following: (1) $V$ is locally Lipschitz, positive definite, and radially unbounded; (2) for all $x \in ℝ^{n}$ and $μ \in ℝ^{m}$ , whenever $V (x) \geq Γ (| μ |)$ , the inequality $D^{+} V (t, x) \leq - cV (x)$ holds for some $c > 0$ . The upper right-hand Dini derivative of $V$ along the solution of system (1) is defined as

\begin{matrix} D^{+} V (t, x) = \underset{h \to 0^{+}}{\lim \sup} \frac{1}{h} [V (t + h, x + hf (t, x, u)) - V (t, x)] . \end{matrix}

Definition 3. The ADT condition given by

\begin{matrix} \begin{matrix} N [T, t] \leq \frac{(T - t)}{τ^{*}} + N_{0}, \forall T \geq t \geq t_{0}, \end{matrix} \end{matrix}

(3)

where $N_{0} > 0$ represents the chatter bound, $τ^{*} > 0$ denotes the ADT constant, and $N (T, t)$ stands for the number of impulsive events occurring within the time interval $(t, T)$ . Notably, when $N_{0} = 1$ , this condition enforces a minimum separation of at least $τ^{*}$ time units between any two consecutive impulses.

Definition 4. The rADT condition given by

\begin{matrix} N [T, t] \geq \frac{(T - t)}{τ^{*}} - N_{0}, \forall T \geq t \geq t_{0}, \end{matrix}

where $N_{0} > 0$ represents the chatter bound, and $τ^{*} > 0$ denotes the rADT constant.

Definition 5. The average impulsive gain of system (1) with respect to d is defined as follows

\begin{matrix} \begin{matrix} K = & \lim_{t \to \infty} \frac{d_{1} + d_{2} + \dots + d_{N [t, t_{0}]}}{N [t, t_{0}]}, K > 0, \end{matrix} \end{matrix}

(4)

where $d_{k}$ represents the impulsive gain at the k-th impulsive instant $t_{k}$ , $k \in ℤ_{+}$ .

Lemma 1. (Ji et al., 2022b) For any vector $A, B \in ℝ^{n}$ , a diagonal matrix $Q > 0$ and scalar $ϵ > 0,$ we have

\begin{matrix} 2 A^{T} B \leq ϵ A^{T} Q A + ϵ^{- 1} B^{T} Q^{- 1} B . \end{matrix}

Lemma 2. (Sekine et al., 2020) For a given symmetric matrix

\begin{matrix} U = [\begin{matrix} U_{11} & U_{12} \\ U_{12}^{T} & U_{22} \end{matrix}], \end{matrix}

where $U_{11}$ is $n \times n$ dimensional, the following three conditions are equivalent:

$U < 0$ ;

$U_{11} < 0, U_{22} - U_{12}^{T} U_{11}^{- 1} U_{12} < 0$ ;

$U_{22} < 0, U_{11} - U_{12} U_{22}^{- 1} U_{12}^{T} < 0$ .

Lemma 3. (Lv and Li, 2017) Let $M = M^{T} > 0$ be an arbitrary real symmetric positive-definite matrix of compatible dimensions, and consider a vector-valued function $F (\cdot) : [t, T] \to ℝ^{n}$ such that all relevant integrals are well-defined, then

\begin{matrix} {[\int_{t}^{T} F (h) dh]}^{T} M [\int_{t}^{T} F (h) dh] \\ \leq (T - t) [\int_{t}^{T} F^{T} (h) M F (h) dh] . \end{matrix}

Main results

Theorem 1. Let $V : ℝ^{n} \to ℝ$ be a candidate exponential ISS-Lyapunov function for the system (1) with rate coefficients $c > 0, d_{1}, d_{2} \dots, d_{N} \in ℝ$ with $d_{k} \neq 0, k = 1, 2, \dots, N$

\begin{matrix} \overset{\cdot}{V} (x) & \leq cV (x) + Γ (| μ |), \forall x a . e ., \forall μ, \end{matrix}

(5)

\begin{matrix} V (g_{k} (x, μ)) & \leq e^{- d_{k}} V (x) + Γ (| μ |), \forall x, μ, k = 1, 2, \dots, N, \end{matrix}

(6)

and when the ADT constant $τ_{1}^{*}$ satisfies

\begin{matrix} \begin{matrix} c - \frac{K}{τ_{1}^{*}} \leq 0 . \end{matrix} \end{matrix}

(7)

For some function $Γ \in K_{\infty}$ , the system (1) is ISS with respect to $β (r, t) = ϕ_{1}^{- 1} (e^{\hat{μ} + (ĉ - \frac{\hat{K}}{τ^{*}}) t} ϕ_{2} (r))$ and $Γ_{μ} (r) = ϕ_{1}^{- 1} (e^{\hat{μ}} (h e^{| d | + 1}) Γ (r))$ . Where $N_{0} = \frac{μ}{K},$ in equation (5), “ $\forall x a . e”$ is to be interpreted as “for every $x \in ℝ^{n}$ except, possibly, on a set of zero Lebesgue- measure in $ℝ^{n}$ . ”

Proof. We can rewrite equation (5) as

\begin{matrix} \overset{\cdot}{V} (x) & \leq ĉV (x) - (ĉ - c) V (x) + Γ (| μ |) . \end{matrix}

Then

\begin{matrix} (ĉ - c) V (x (t)) \geq Γ (| μ (t) |) \Rightarrow \overset{\cdot}{V} (x (t), μ (t)) \leq ĉV (x (t)), \end{matrix}

(8)

for each impulse time $t_{k} \in T$ , as similarly with equation (6), we obtain

\begin{matrix} V (g_{k} (x^{-} (t_{k}), μ^{-} (t_{k}))) \leq e^{- {\hat{d}}_{k}} V (x^{-} (t_{k})) - (e^{- {\hat{d}}_{k}} \\ - e^{- d_{k}}) V (x^{-} (t_{k})) + Γ (| μ (t_{k}) |) . \end{matrix}

Then

\begin{matrix} (e^{- {\hat{d}}_{k}} - e^{- d_{k}}) & V (x^{-} (t_{k})) \geq Γ (| μ (t_{k}) |) \Rightarrow \\ V (g_{k} (x^{-} (t_{k}), μ^{-} (t_{k}))) \leq e^{- {\hat{d}}_{k}} V (x^{-} (t_{k})) . \end{matrix}

(9)

Owing to the right continuity of x and μ, there exists a sequence of times $t_{0} : = \overset{ˇ}{t_{0}} \leq \bar{t_{1}} < \overset{ˇ}{t_{1}} < \bar{t_{2}} < \overset{ˇ}{t_{2}} < \dots$ , which divides the interval $[t_{0}, \infty)$ into a collection of disjoint subintervals and

\begin{matrix} V (x (t)) & \geq hΓ (∥ μ ∥_{[t_{0}, t]}), \forall t \in [\overset{ˇ}{t_{i}}, {\bar{t}}_{i + 1}), i = 0, 1, \dots, \end{matrix}

(10)

\begin{matrix} V (x (t)) & \leq hΓ (∥ μ ∥_{[t_{0}, t]}), \forall t \in [{\bar{t}}_{i}, \overset{ˇ}{t_{i}}), i = 1, 2, \dots, \end{matrix}

(11)

which $h : = {(\min {ĉ - c, e^{- {\hat{d}}_{k}} - e^{- d_{k}}})}^{- 1}$ , and $ĉ > c, {\hat{d}}_{k} < d_{k}$ , $\forall k \in ℤ_{+}$ . Assume that interval $[t_{0}, {\bar{t}}_{1})$ is nonempty. Otherwise, proceed directly to the line following equation (12). Between two consecutive impulses $t_{k - 1}, t_{k} \in (t_{0}, {\bar{t}}_{1}]$ with equations (8) and (10), we have $\overset{\cdot}{V} (x (t), μ (t)) \leq ĉV (x (t)), \forall t \in (t_{k - 1}, t_{k})$ a.e., and therefore

\begin{matrix} V (x^{-} (t_{k})) \leq e^{ĉ (t_{k} - t_{k - 1})} V (x (t_{k - 1})) . \end{matrix}

Consequently, it follows that using equations (9) and (10), we can deduce that $V (x (t_{k})) \leq e^{- {\hat{d}}_{k}} V (x^{-} (t_{k}))$ . It follows that

\begin{matrix} V (x (t_{k})) \leq e^{- {\hat{d}}_{k}} e^{ĉ (t_{k} - t_{k - 1})} V (x (t_{k - 1})), \end{matrix}

by iterating over $N (t, t_{0})$ impulses in the interval $(t_{0}, t]$ , we obtain the bound

\begin{matrix} V (x (t)) \leq e^{ĉ (t - t_{0}) - \sum_{i = 1}^{n} \bar{d_{k}}} V (x (t_{0})), \forall t \in (t_{0}, {\bar{t}}_{1}] . \end{matrix}

From equation (4), we can obtain

\begin{matrix} V (x (t)) \leq e^{ĉ (t - t_{0}) - \hat{K} N (t, t_{0})} V (x (t_{0})), \forall t \in (t_{0}, {\bar{t}}_{1}] . \end{matrix}

Utilizing the ADT condition (3)

\begin{matrix} V (x (t)) & \leq e^{\hat{K} N_{0} + (ĉ - \frac{\hat{K}}{τ^{*}}) (t - t_{0})} V (x (t_{0})) \\ = e^{\hat{μ} + (ĉ - \frac{\hat{K}}{τ^{*}}) (t - t_{0})} V (x (t_{0})), \forall t \in (t_{0}, {\bar{t}}_{1}] . \end{matrix}

(12)

For any subinterval of the form $[{\bar{t}}_{i}, \overset{ˇ}{t_{i}})$ , the state bounds derived from inequality (11) remain valid. If $\overset{ˇ}{t_{i}}$ is not an impulse instant, then equation (11) directly serves as the state estimate at $t = \overset{ˇ}{t_{i}}$ . However, if $ť_{i}$ coincides with an impulsive instant, then

\begin{matrix} V (x (ť_{i})) \leq h e^{- d_{k}} Γ (∥ μ ∥_{[t_{0}, ť_{i}]}) + Γ (| μ^{-} (ť) |), \end{matrix}

and in either case

\begin{matrix} V (x (t)) \leq (h e^{| d_{k} |} + 1) Γ (∥ μ ∥_{[t_{0}, t]}), \forall t \in [{\bar{t}}_{i}, \overset{ˇ}{t_{i}}], i \geq 1, \end{matrix}

(13)

when $ť_{i} = \infty$ , this bound holds for all $t \geq {\bar{t}}_{i}$ . To analyze the system behavior within any subinterval of the form $[\overset{ˇ}{t_{i}}, {\bar{t}}_{i + 1}), i \geq 1$ . By applying the same reasoning used to derive equation (12), we get

\begin{matrix} V (x (t)) & \leq e^{\hat{μ} + (ĉ - \frac{\hat{K}}{τ^{*}}) (t - ť_{i})} V (x (ť_{i})) . \end{matrix}

Conversely, by substituting $ť_{i}$ for $t_{0}$ and applying equation (13) with $t = \overset{ˇ}{t_{i}}$ , we get

\begin{matrix} V (x (t)) \leq e^{\hat{μ}} (h e^{| d_{k} |} + 1) Γ (∥ μ ∥_{[t_{0}, ť_{i}]}), \forall t \in (\overset{ˇ}{t_{i}}, {\bar{t}}_{i + 1}], i \geq 1 . \end{matrix}

By integrating the analytical results from equations (12) and (13), we derive the global bound $\forall t \geq t_{0}$

\begin{matrix} V (x (t)) \leq & \max_{k = 1, k \in Z_{+}} {e^{\hat{μ} + (ĉ - \frac{\hat{K}}{τ^{*}}) (t - t_{0})} V (x (t_{0})), \\ e^{\hat{μ}} (h e^{| d_{k} |} + 1) Γ (∥ μ ∥_{[t_{0}, t]})} . \end{matrix}

(14)

Given that $V$ is positive definite and radially unbounded, there exist functions $ϕ_{1}, ϕ_{2} \in K_{\infty}$ such that the inequality chain $ϕ_{1} (| x (t) |) \leq V (t, x) \leq ϕ_{2} (| x (t) |)$ holds for all $x \in ℝ^{n}$ . Consequently, inequality (14) implies the stability criterion in equation (2) with the explicit forms: $β (r, t) = ϕ_{1}^{- 1} (e^{\hat{μ} + (ĉ - \frac{\hat{K}}{τ^{*}}) t} ϕ_{2} (r))$ and $Γ_{μ} (r) = ϕ_{1}^{- 1} (e^{\hat{μ}} (h e^{| d | + 1}) Γ (r))$ . The global existence of solutions is guaranteed by the uniform boundedness of the state. Specifically, the state of the system is guaranteed to remain bounded uniformly over time, which ensures that the system does not experience unbounded growth or instability. Moreover, the construction of β and $Γ_{μ}$ is independent of the specific impulsive sequence, thereby ensuring the uniformity of the ISS property. This completes the proof. □

Remark 1. Theorem 1 establishes a bidirectional ISS criterion: for systems where either discrete dynamics satisfy ISS or continuous dynamics do not, the proposed framework integrates the AIG and dynamic Lyapunov gain mechanisms to balance the effects of hybrid impulses (coexisting stabilizing/destabilizing actions). Specifically, the parameter K, acting as a global balancing factor, quantifies the ratio between the cumulative intensity of destabilizing impulses and their occurrence frequency, thereby providing a unified metric for stability analysis under heterogeneous impulse sequences. Compared to fixed-gain or uniform-interval approaches in Wang et al. (2023), Chen et al. (2023), and Wang and Zhu (2024), our method achieves reduced conservatism through two innovations: (1) independent configuration of Lyapunov gains $d_{k}$ at each impulse instant $t_{k}$ , enabling adaptive responses to different impulse types (e.g. positive gains enhance stability, while negative gains model perturbations) and (2) integration of a relaxed ADT condition that replaces the traditional minimum interval constraint with a statistical average interval requirement, thereby accommodating high-frequency impulse scenarios. These advancements not only improve the universality of the theoretical framework but also offer a foundation for dynamic scheduling of impulse controllers in practical engineering applications.

Theorem 2. Let $V : R^{n} \to R$ be a candidate exponential ISS-Lyapunov function for the system (1) with rate coefficients $c > 0, d_{1}, d_{2} \dots, d_{N} \in R$ with $d_{k} \neq 0, k = 1, 2, \dots, N$

\begin{matrix} \overset{\cdot}{V} (x) & \leq - cV (x) + Γ (| μ |), \forall x a . e ., \forall μ, \end{matrix}

(15)

\begin{matrix} V (g_{k} (x, μ)) & \leq e^{d_{k}} V (x) + Γ (| μ |), \forall x, μ, k = 1, 2, \dots, N, \end{matrix}

(16)

and when the rADT constant $τ_{2}^{*}$ satisfies

\begin{matrix} \begin{matrix} c - \frac{K}{τ_{2}^{*}} \geq 0 . \end{matrix} \end{matrix}

(17)

For some function $Γ \in K_{\infty}$ , the system (1) is ISS with respect to $β (r, t) = ϕ_{1}^{- 1} (e^{\hat{μ} - (c - \frac{K}{τ_{2}^{*}}) t} ϕ_{2} (r))$ and $Γ_{μ} (r) = ϕ_{1}^{- 1} (e^{\hat{μ}} (p e^{| d | + 1}) Γ (r))$ .

Proof. We can rewrite equation (17) as

\begin{matrix} \overset{\cdot}{V} (x) & \leq - ĉV (x) - (c - ĉ) V (x) + Γ (| μ |) . \end{matrix}

Then

\begin{matrix} (c - ĉ) V (x (t)) \geq Γ (| μ (t) |) \Rightarrow \overset{\cdot}{V} (x (t), μ (t)) \leq - ĉV (x (t)), \end{matrix}

(18)

for each impulse time $t_{k} \in T$ , as similarly with equation (16),

\begin{matrix} V (g_{k} (x^{-} (t_{k}), μ^{-} (t_{k}))) \leq e^{{\hat{d}}_{k}} V (x^{-} (t_{k})) - (e^{{\hat{d}}_{k}} - e^{d_{k}}) \\ V (x^{-} (t_{k})) + Γ (| μ (t_{k}) |) . \end{matrix}

Then

\begin{matrix} (e^{{\hat{d}}_{k}} - e^{d_{k}}) & V (x^{-} (t_{k})) \geq Γ (| μ (t_{k}) |) \Rightarrow \\ V (g_{k} (x^{-} (t_{k}), μ^{-} (t_{k}))) \leq e^{{\hat{d}}_{k}} V (x^{-} (t_{k})) . \end{matrix}

(19)

\begin{matrix} V (x (t)) & \geq pΓ (∥ μ ∥_{[t_{0}, t]}), \forall t \in [\overset{ˇ}{t_{i}}, {\bar{t}}_{i + 1}), i = 0, 1, \dots, \end{matrix}

(20)

\begin{matrix} V (x (t)) & \leq pΓ (∥ μ ∥_{[t_{0}, t]}), \forall t \in [{\bar{t}}_{i}, \overset{ˇ}{t_{i}}), i = 1, 2, \dots, \end{matrix}

(21)

which $p : = {(\min {c - ĉ, e^{{\hat{d}}_{k}} - e^{d_{k}}})}^{- 1}$ and $ĉ < c, {\hat{d}}_{k} > d_{k}, \forall k \in ℤ_{+}$ . The proof method is the same as in Theorem 1. We get $β (r, t) = ϕ_{1}^{- 1} (e^{\hat{μ} - (c - \frac{K}{τ_{2}^{*}}) t} ϕ_{2} (r))$ and $Γ_{μ} (r) = ϕ_{1}^{- 1} (e^{\hat{μ}} (p e^{| d | + 1}) Γ (r))$ . This completes the proof. □

Consider the following nonlinear systems

\begin{matrix} {\begin{matrix} ẋ (t) = Ax (t) + Bf (x (t)) + K_{1} x (t_{k}) + Dμ (t), t \geq 0, t \neq t_{k}, \\ △ x (t_{k}) = {K_{2}}_{k} x (t_{k}^{-}), k \in ℤ_{+}, \end{matrix} \end{matrix}

(22)

where $△ x = x (t_{k}) - x (t_{k}^{-}), x (t_{k}) = x (t_{k}^{+})$ and $x (t_{k}^{-}) = \lim_{t \to t_{k}^{-}} x (t)$ , $x \in ℝ^{n}$ denote the state, we denote

\begin{matrix} F_{1} = diag (ℓ_{1}^{-} ℓ_{1}^{+}, ℓ_{2}^{-} ℓ_{2}^{+}, \dots, ℓ_{n}^{-} ℓ_{n}^{+}), \end{matrix}

and

\begin{matrix} F_{2} = diag (\frac{ℓ_{1}^{-} + ℓ_{1}^{+}}{2}, \frac{ℓ_{2}^{-} + ℓ_{2}^{+}}{2}, \dots, \frac{ℓ_{n}^{-} + ℓ_{n}^{+}}{2}), \end{matrix}

${t_{k}}_{k \in Z_{+}}$ is the impulse time sequence. Matrices $A, B, D \in ℝ^{n \times n}$ are known.

Theorem 3. Assume that there exist positive-definite matrices $P \in ℝ^{n \times n}, Q \in ℝ^{n \times n}, M \in ℝ^{n \times n}$ , positive-definite diagonal matrix $R \in ℝ^{n \times n}$ , symmetric matrix S, real matrix $G_{k} \in ℝ^{n \times n}$ , arbitrary matrices G with compatible dimensions and

\begin{matrix} [\begin{matrix} PA + A^{T} P - cP + S & U & P \\ ⋆ & - cQ & 0 \\ ⋆ & ⋆ & - \frac{1}{ξ} I_{n} \end{matrix}] & < 0, \end{matrix}

(23)

\begin{matrix} [\begin{matrix} - F_{1} R - S & F_{2} R + PB \\ ⋆ & - R \end{matrix}] & < 0, \end{matrix}

(24)

\begin{matrix} [\begin{matrix} - e^{- d_{k}} P & G_{k} \\ ⋆ & - (P + Q) \end{matrix}] & < 0, \end{matrix}

(25)

when (7) holds, the system (22) is ISS, where $K_{1} = P^{- 1} U, {K_{2}}_{k} = {(P + Q)}^{- 1} G_{k}^{T} - I_{n} .$

Proof. Choose $V (t, x (t)) = x^{T} (t) Px (t) + x^{T} (t_{k}) Qx (t_{k})$ . Then, based on equations (23), (24), and the Schur complement, it also follows from equation (25) that

\begin{matrix} [\begin{matrix} - e^{- d_{k}} P & G_{k} \\ ⋆ & - (P + Q) \end{matrix}] < 0 \\ \Leftrightarrow & [\begin{matrix} I_{n} & {({K_{2}}_{k} + I_{n})}^{T} \\ 0 & I \end{matrix}] [\begin{matrix} - e^{- d_{k}} P & G_{k} \\ ⋆ & - (P + Q) \end{matrix}] \\ [\begin{matrix} I_{n} & 0 \\ {({K_{2}}_{k} + I_{n})}^{T} & I \end{matrix}] < 0 \\ \Leftrightarrow & [\begin{matrix} - e^{- d_{k}} P + {({K_{2}}_{k} + I_{n})}^{T} P ({K_{2}}_{k} + I_{n}) & 0 \\ ⋆ & - (P + Q) \end{matrix}] < 0 \\ \Leftrightarrow & - e^{- d_{k}} P + {({K_{2}}_{k} + I_{n})}^{T} (P + Q) ({K_{2}}_{k} + I_{n})) < 0, \end{matrix}

thus,

\begin{matrix} V (t_{k}) & = x^{T} (t_{k}) (P + Q) x (t_{k}) \\ = x^{T} (t_{k}^{-}) {({K_{2}}_{k} + I_{n})}^{T} (P + Q) ({K_{2}}_{k} + I_{n}) x (t_{k}^{-}) \\ \leq e^{- d_{k}} V (t_{k}^{-}), \end{matrix}

and

\begin{matrix} \overset{\cdot}{V} (t) & = 2 x^{T} (t) Pẋ (t) \\ = x^{T} (t) (A^{T} P + PA) x (t) + 2 x^{T} (t) PDμ (t) \\ + 2 x^{T} (t) PBf (x (t)) + 2 x^{T} (t) P K_{1} x (t_{k}^{-}) \\ = 2 x^{T} (t) PBf (x (t) + 2 x^{T} (t) PDμ (t) \\ + {[\begin{matrix} x (t) \\ x (t_{k}) \end{matrix}]}^{T} [\begin{matrix} PA + A^{T} P & P K_{1} \\ ⋆ & 0 \end{matrix}] [\begin{matrix} x (t) \\ x (t_{k}) \end{matrix}] . \end{matrix}

(26)

Note that

\begin{matrix} (f_{i} (x_{i} (t)) - ℓ_{i}^{-} (x_{i} (t)) (f_{i} (x_{i} (t)) - ℓ_{i}^{+} (x_{i} (t)) \leq 0, \end{matrix}

for every $i \in N$ , which is equivalent to

\begin{matrix} {[\begin{matrix} x (t) \\ f (x (t)) \end{matrix}]}^{T} & [\begin{matrix} ℓ_{i}^{-} ℓ_{i}^{+} e_{i} e_{i}^{T} & - \frac{ℓ_{i}^{-} + ℓ_{i}^{+}}{2} e_{i} e_{i}^{T} \\ - \frac{ℓ_{i}^{-} + ℓ_{i}^{+}}{2} e_{i} e_{i}^{T} & e_{i} e_{i}^{T} \end{matrix}] [\begin{matrix} x (t) \\ f (x (t)) \end{matrix}] \leq 0, \end{matrix}

for every $i \in N$ , where $e_{i}$ denotes the unit-column vector with a single element on its ith row and zero elsewhere. Let $R = diag (r_{1}, r_{2}, \dots, r_{n}) > 0$ . Then

\begin{matrix} \sum_{i = 1}^{n} r_{i} {[\begin{matrix} x (t) \\ f (x (t)) \end{matrix}]}^{T} [\begin{matrix} ℓ_{i}^{-} ℓ_{i}^{+} e_{i} e_{i}^{T} & - \frac{ℓ_{i}^{-} + ℓ_{i}^{+}}{2} e_{i} e_{i}^{T} \\ - \frac{ℓ_{i}^{-} + ℓ_{i}^{+}}{2} e_{i} e_{i}^{T} & e_{i} e_{i}^{T} \end{matrix}] \\ [\begin{matrix} x (t) \\ f (x (t)) \end{matrix}] \leq 0, \end{matrix}

which is equivalent to

\begin{matrix} {[\begin{matrix} x (t) \\ f (x (t)) \end{matrix}]}^{T} [\begin{matrix} - F_{1} R & F_{2} R \\ ⋆ & - R \end{matrix}] [\begin{matrix} x (t) \\ f (x (t)) \end{matrix}] \geq 0 . \end{matrix}

(27)

By equations (26) and (27), we can obtain

\begin{matrix} \overset{\cdot}{V} (t) & \leq 2 x^{T} (t) PBf (x (t) + 2 x^{T} (t) PDμ (t) \\ + {[\begin{matrix} x (t) \\ x (t_{k}) \end{matrix}]}^{T} [\begin{matrix} PA + A^{T} P & P K_{1} \\ ⋆ & 0 \end{matrix}] [\begin{matrix} x (t) \\ x (t_{k}) \end{matrix}] \\ + {[\begin{matrix} x (t) \\ f (x (t)) \end{matrix}]}^{T} [\begin{matrix} - F_{1} R & F_{2} R \\ ⋆ & - R \end{matrix}] [\begin{matrix} x (t) \\ f (x (t)) \end{matrix}] \\ \leq {[\begin{matrix} x (t) \\ x (t_{k}) \end{matrix}]}^{T} [\begin{matrix} cP & 0 \\ ⋆ & cQ \end{matrix}] \\ + [\begin{matrix} PA + A^{T} P - cP + S + P P^{T} & P K_{1} \\ ⋆ & - cQ \end{matrix}] \\ [\begin{matrix} x (t) \\ x (t_{k}) \end{matrix}] + {[\begin{matrix} x (t) \\ f (x (t)) \end{matrix}]}^{T} [\begin{matrix} - F_{1} R - S & F_{2} R + PB \\ ⋆ & - R \end{matrix}] \\ [\begin{matrix} x (t) \\ f (x (t)) \end{matrix}] + μ^{T} (t) D^{T} Dμ (t) \\ \leq {[\begin{matrix} x (t) \\ x (t_{k}) \end{matrix}]}^{T} [\begin{matrix} cP & 0 \\ ⋆ & cQ \end{matrix}] [\begin{matrix} @x (t) \\ x (t_{k}) \end{matrix}] + λ_{\max} (D^{T} D) | μ (t) |^{2} \\ = cV (x (t)) + λ_{\max} (D^{T} D) | μ (t) |^{2} . \end{matrix}

Therefore, based on the results presented in Theorem 1, the ISS of system (22) is guaranteed. This completes the proof. □

Remark 2. Theorem 3 provides ISS conditions for nonlinear systems with hybrid impulses using the linear matrix inequality (LMI) method, further demonstrating the practicality and superiority of the approach presented in this paper. Compared to existing studies (Gao et al., 2024a; Sun et al., 2025; Wang et al., 2021), this paper not only considers the hybrid effects in impulsive control (stabilizing and destabilizing impulses) but also significantly reduces the conservatism of the analysis by introducing time-dependent Lyapunov functions and AIG conditions. Moreover, the hybrid control strategy proposed in this paper effectively handles systems with time-varying characteristics and uncertainties, offering more reliable stability guarantees for complex dynamic systems in practical applications. The proposed hybrid control strategy offers a novel framework and methodology for stability analysis and control design of complex dynamic systems in practical engineering applications. Consequently, the research outcomes presented in this paper not only demonstrate significant theoretical innovation and depth but also exhibit broad applicability and critical engineering significance in real-world scenarios.

Consider the following nonlinear systems

\begin{matrix} {\begin{matrix} ẋ (t) = Ax (t) + Bf (x (t)) + K_{1} x (t - ϵ (t)) + Dμ (t), t \geq 0, t \neq t_{k}, \\ △ x (t_{k}) = {K_{2}}_{k} x (t_{k}^{-}) + Eμ (t_{k}), k \in ℤ_{+}, \end{matrix} \end{matrix}

(28)

where $ϵ (t) = t - t_{k}$ , $t \in [t_{k}, t_{k + 1}], t - t_{k} > η$ .

Corollary 1. Assume that there exist positive-definite matrices $P \in ℝ^{n \times n}, Q \in ℝ^{n \times n}$ , arbitrary matrices $S_{1}, S_{2}$ with compatible dimensions and

\begin{matrix} M = [\begin{matrix} M_{11} & M_{12} & M_{13} & M_{14} & P & S_{1} & S_{2} \\ ⋆ & M_{22} & W & 0 & 0 & 0 & 0 \\ ⋆ & ⋆ & M_{33} & M_{34} & 0 & 0 & 0 \\ ⋆ & ⋆ & ⋆ & - R & 0 & 0 & 0 \\ ⋆ & ⋆ & ⋆ & ⋆ & - ξ_{1} I_{n} & 0 & 0 \\ ⋆ & ⋆ & ⋆ & ⋆ & ⋆ & - ξ_{2} I_{n} & 0 \\ ⋆ & ⋆ & ⋆ & ⋆ & ⋆ & ⋆ & - ξ_{3} I_{n} \end{matrix}] < 0, \end{matrix}

\begin{matrix} [\begin{matrix} - e^{d_{k}} P & \sqrt{2} G_{k} \\ ⋆ & - P \end{matrix}] < 0, \end{matrix}

(29)

when ( 17 ) holds, the system ( 28 ) is ISS, where $K_{1} = P^{- 1} U = S_{1}^{- 1} V = S_{2}^{- 1} W, {K_{2}}_{k} = P^{- 1} G_{k}^{T} - I .$

Where

$M_{11} = PA + A^{T} P + cP - \frac{e^{- cη}}{η} Q - F_{1} R,$

$M_{12} = U + \frac{e^{- cη}}{η} Q - V$ ,

$M_{13} = S_{1} - S_{2} A,$

$M_{14} = PB + F_{2} R - S_{1} B,$

$M_{22} = - \frac{e^{- cη}}{η} Q,$

$M_{33} = 2 S_{2} + (t_{k + 1} - t) Q,$

$M_{34} = - S_{2} B .$

Proof. We propose the following Lyapunov–Krasovskii functional as a candidate for analyzing the system

\begin{matrix} V (t, x (t)) = V_{1} + V_{2}, \end{matrix}

where

\begin{matrix} V_{1} = x^{T} Px, \\ V_{2} = (t_{k + 1} - t) \int_{t - ϵ (t)}^{t} e^{c (σ - t)} ẋ^{T} (σ) Qẋ (σ) dσ, \end{matrix}

So, we can get

\begin{matrix} \begin{matrix} {\overset{\cdot}{V}}_{1} (t) = & x^{T} (t) (A^{T} P + PA) x (t) + 2 x^{T} (t) P K_{1} x (t - ϵ (t)) \\ + 2 x^{T} (t) PBf (x (t)) + 2 x^{T} (t) PDμ (t), \end{matrix} \\ \begin{matrix} {\overset{\cdot}{V}}_{2} (t) = & (t_{k + 1} - t) [ẋ^{T} (t) Qẋ (t) - c \int_{t - ϵ (t)}^{t} ẋ^{T} (σ) Qẋ (σ) dσ] \\ - \int_{t - ϵ (t)}^{t} e^{c (σ - t)} ẋ^{T} (σ) Qẋ (σ) dσ, \end{matrix} \end{matrix}

\begin{matrix} \overset{\cdot}{V} + cV \leq & x^{T} (t) (A^{T} P + PA + cP) x (t) + 2 x^{T} (t) PDμ (t) \\ + 2 x^{T} (t) P K_{1} x (t - ϵ (t)) + 2 x^{T} (t) PBf (x (t)) \\ - \int_{t - ϵ (t)}^{t} e^{c (σ - t)} ẋ^{T} (σ) Qẋ (σ) dσ \\ + (t_{k + 1} - t) ẋ^{T} (t) Qẋ (t) . \end{matrix}

Applying Lemma 3 and the Newton–Leibniz formula

\begin{matrix} \int_{t_{k}}^{t} ẋdσ = x (t) - x (t_{k}) = x (t) - x (t - ϵ (t)) . \end{matrix}

We have

\begin{matrix} - \int_{t - ϵ (t)}^{t} e^{c (σ - t)} ẋ^{T} (σ) Qẋ (σ) dσ \leq - e^{- cϵ (t)} \\ \int_{t - ϵ (t)}^{t} ẋ^{T} (σ) Qẋ (σ) dσ \\ \leq - \frac{e^{- cϵ (t)}}{ϵ (t)} [\int_{t - ϵ (t)}^{t} ẋ^{T} dσ] Q [\int_{t - ϵ (t)}^{t} ẋdσ] \\ \leq - \frac{e^{- cϵ (t)}}{ϵ (t)} [x^{T} (t) - x^{T} (t - ϵ (t))] Q [x (t) \\ - x (t - ϵ (t))] . \end{matrix}

From system (28), for any matrices $S_{1}, S_{2}$ we introduce some auxiliary equalities as follows

\begin{matrix} \begin{matrix} 0 = & [2 x^{T} (t) S_{1} + 2 ẋ^{T} (t) S_{2}] [ẋ (t) - Ax (t) - Bf (x (t)) \\ - K_{1} x (t - ϵ (t)) - Dμ (t)] . \end{matrix} \end{matrix}

In addition, it follows from Lemma 1 that for every $ξ > 0$

\begin{matrix} \begin{matrix} 2 x^{T} (t) PDμ (t) \leq x^{T} (t) P ξ_{1}^{- 1} P^{T} x (t) + μ^{T} (t) D^{T} ξ_{1} Dμ (t) . \end{matrix} \end{matrix}

Similarly, we have

\begin{matrix} \begin{matrix} 2 x^{T} (t) S_{1} Dμ (t) \leq x^{T} (t) S_{1} ξ_{2}^{- 1} S_{1}^{T} x (t) + μ^{T} (t) D^{T} ξ_{2} Dμ (t), \end{matrix} \\ \begin{matrix} 2 ẋ^{T} (t) S_{2} Dμ (t) \leq ẋ^{T} (t) S_{2} ξ_{3}^{- 1} S_{2}^{T} ẋ (t) + μ^{T} (t) D^{T} ξ_{3} Dμ (t) . \end{matrix} \end{matrix}

From the above and equation (24), we get

\begin{matrix} \overset{\cdot}{V} + cV \leq & x^{T} (t) (A^{T} P + PA + cP + P ξ_{1}^{- 1} P^{T} - \frac{e^{- cϵ (t)}}{ϵ (t)} Q \\ - S_{1} ξ_{2}^{- 1} S_{1}^{T}) x (t) + 2 x^{T} (t) (PB - S_{1} B) f (x (t)) \\ + 2 x^{T} (t) (P K_{1} + \frac{e^{- cϵ (t)}}{ϵ (t)} Q - S_{1} K_{1}) x (t - ϵ (t)) \\ - x^{T} (t - ϵ (t)) Qx (t - ϵ (t)) + 2 x^{T} (t) (S_{1 -} S_{2} A) \\ \times ẋ (t) + ẋ^{T} (t) (2 S_{2} + (t_{k + 1} - t) Q - S_{2} ξ_{3} S_{2}^{T}) ẋ (t) \\ - 2 ẋ^{T} (t) S_{2} Bf (x) - 2 ẋ^{T} (t) S_{2} K_{1} x (t - ϵ (t)) \\ + μ^{T} (t) D^{T} (ξ_{1} + ξ_{2} + ξ_{3}) Dμ (t) \\ - {[\begin{matrix} x (t) \\ f (x (t)) \end{matrix}]}^{T} [\begin{matrix} F_{1} R & - F_{2} R \\ ⋆ & R \end{matrix}] [\begin{matrix} x (t) \\ f (x (t)) \end{matrix}] . \end{matrix}

In other words, we have

\begin{matrix} \overset{\cdot}{V} + cV \leq Γ^{T} (t) M_{1} Γ (t) + α | μ (t) |^{2}, \end{matrix}

where

\begin{matrix} Γ (t) & = {(x^{T} (t), x^{T} (t - ϵ (t)), ẋ^{T} (t), f^{T} (x))}^{T}, \\ α & = \sum_{i = 1}^{3} ξ_{i} λ_{\max} (D^{T} D) . \end{matrix}

Since $M < 0$ , by Lemma 3 we have that $M$ and $M_{1}$ are equivalent, we have $M_{1} < 0$ . So, we obtain

\begin{matrix} \overset{\cdot}{V} \leq - cV + α | μ (t) |^{2} . \end{matrix}

At the instant of the impulse, it follows from equation (29) that

\begin{matrix} V (t_{k}) & = x^{T} (t_{k}) Px (t_{k}) \\ = {[({K_{2}}_{k} + I) x (t_{k}^{-}) + Eμ (t_{k})]}^{T} P [({K_{2}}_{k} + I) x (t_{k}^{-}) + Eμ (t_{k})] \\ = x^{T} (t_{k}^{-}) {({K_{2}}_{k} + I)}^{T} P ({K_{2}}_{k} + I) x (t_{k}^{-}) + x^{T} (t_{k}^{-}) {({K_{2}}_{k} + I)}^{T} \\ PEμ (t_{k}) + μ^{T} (t) E^{T} P ({K_{2}}_{k} + I) x (t_{k}^{-}) + μ^{T} (t) E^{T} PEμ (t_{k}) \\ \leq 2 x^{T} (t_{k}^{-}) {({K_{2}}_{k} + I)}^{T} P ({K_{2}}_{k} + I) x (t_{k}^{-}) + 2 μ^{T} (t) E^{T} PEμ (t_{k}) \\ \leq e^{d_{k}} V (t_{k}^{-}) + χ (μ (t_{k})), \end{matrix}

where $χ (μ (t_{k})) = 2 λ_{\max} (E^{T} PE) μ^{T} (t_{k}) μ (t_{k}) .$ Therefore, based on the results presented in Theorem 2, the ISS of system (28) is guaranteed. This completes the proof. □

Examples

This section presents two illustrative examples that serve to demonstrate the key findings of this paper.

Example 1. Consider system (22) with $f (x) = 0.5 (| x + 1 | - | x - 1 |)$ , $μ (t) = {[\sin (t), \sin (t), \cos (t)]}^{T}$ and

\begin{matrix} A = [\begin{matrix} - p - p m_{1} & p & 0 \\ 1 & - 1 & 1 \\ 0 & - q & 0 \end{matrix}], B = p (m_{1} - m_{0}) I, \\ D = [\begin{matrix} 1.5 & 0 & 2.2 \\ 1.1 & 1.2 & 0 \\ 0 & 1.1 & 1.2 \end{matrix}] . \end{matrix}

with $p = 10, q = 14.87, m_{0} = - 0.2$ , and $m_{1} = - 0.4$ . Clearly, system (22) with above parameters is a Chua’s circuit. By calculation, $F_{1}^{-} = F_{2}^{-} = 0, F_{1}^{+} = F_{2}^{+} = 1$ , we get

\begin{matrix} F_{1} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}], F_{2} = [\begin{matrix} 0.5 & 0 \\ 0 & 0.5 \end{matrix}] . \end{matrix}

The Lyapunov function is chosen to be $V (x (t)) = x^{T} (t) Px (t) + x^{T} (t_{k}) Qx (t_{k})$ . The feasible solutions can be obtained

\begin{matrix} P = [\begin{matrix} 0.7764 & - 0.1754 & 0.1639 \\ - 0.1754 & 8.1595 & - 0.4970 \\ 0.1639 & - 0.4970 & 0.6395 \end{matrix}], \\ Q = [\begin{matrix} 7.0730 & 0.0705 & - 0.2205 \\ 0.0705 & 3.1338 & 0.2744 \\ - 0.2205 & 0.2744 & 7.3747 \end{matrix}], \end{matrix}

with $K_{1}$ and $\bar{K_{2}}$ is derived by

\begin{matrix} K_{1} & = [\begin{matrix} 1.9809 & 1.0183 & - 0.2793 \\ 0.0663 & - 0.0566 & 0.0082 \\ 0.2211 & - 0.1330 & 0.0124 \end{matrix}], \\ \bar{K_{2}} & = [\begin{matrix} 0.0194 & 0.0096 & 0.0075 \\ 0.0096 & - 0.2911 & 0.0198 \\ 0.0075 & 0.0198 & - 0.0012 \end{matrix}] . \end{matrix}

Where

\begin{matrix} K_{21} = [\begin{matrix} - 0.2355 & 0.0072 & 0.0056 \\ 0.0072 & - 0.4684 & 0.0148 \\ 0.0056 & 0.0148 & - 0.2509 \end{matrix}], \\ K_{22} = [\begin{matrix} - 0.1081 & 0.0084 & 0.0065 \\ 0.0084 & - 0.3797 & 0.0173 \\ 0.0065 & 0.0173 & - 0.1260 \end{matrix}], \\ K_{23} = [\begin{matrix} 0.0194 & 0.0096 & 0.0075 \\ 0.0096 & - 0.2911 & 0.0198 \\ 0.0075 & 0.0198 & - 0.0012 \end{matrix}], \\ K_{24} = [\begin{matrix} 0.1468 & 0.0108 & 0.0084 \\ 0.0108 & - 0.2025 & 0.0222 \\ 0.0084 & 0.0222 & 0.1237 \end{matrix}], \\ K_{25} = [\begin{matrix} 0.2742 & 0.0120 & 0.0093 \\ 0.0120 & - 0.1139 & 0.0247 \\ 0.0093 & 0.0247 & 0.2485 \end{matrix}] . \end{matrix}

By Theorem 1, we select the parameters $c = 0.48$ and $d_{k} = {3, 1.85, - 0.09, 1.1, - 1.2}$ . Consequently, we have $K = 0.932$ . From the above, we can obtain

\begin{matrix} D^{+} V (x) & \leq 0.48 V (x) + | μ (t) |, t \neq t_{k} . \end{matrix}

Setting $N_{0} = 5$ , it follows from Theorem 1 that system (22) is ISS if $τ_{1}^{*} = 1.94$ . Numerical simulations demonstrate that under the ADT constraint of $τ_{1}^{*} = 1.94$ . the system exhibits stable convergence behavior under both bounded input $μ (t) = {[\sin (t), \sin (t), \cos (t)]}^{T}$ and zero-input conditions (Figure 1(c)–(d)), while chaotic divergence occurs in the absence of impulsive control and sampled-data (Figure 1(e)). These results validate the efficacy of the proposed hybrid control strategy in balancing stabilizing and destabilizing impulse effects.

Example 2. Consider system (28) with $f_{1} (x) = \arctan (0.5 x) - 0.3 \sin (x)$ , $f_{2} (x) = \arctan (0.3 x) - 0.5 \sin (x)$ , $μ (t) = [\sin (t)$ , $\cos (t)]^{T}$ , and

\begin{matrix} A = [\begin{matrix} - 0.5 & 0.3 \\ 0 & - 1 \end{matrix}], B = [\begin{matrix} 0.11 & - 1 \\ 0 & 0.8 \end{matrix}], D = [\begin{matrix} 1.5 & 0 \\ 0 & 1 \end{matrix}] . \end{matrix}

By calculation, $F_{1}^{-} = - 0.3, F_{1}^{+} = 0.8, F_{2}^{-} = - 0.5,$ and $F_{2}^{+} = 0.8,$ then

\begin{matrix} F_{1} = [\begin{matrix} - 0.24 & 0 \\ 0 & - 0.4 \end{matrix}], F_{2} = [\begin{matrix} 0.25 & 0 \\ 0 & 0.15 \end{matrix}] . \end{matrix}

We choose Lyapunov function $V (x (t)) = x^{T} (t) Px (t) + (t_{k + 1} - t) \int_{t - ϵ (t)}^{t} e^{c (σ - t)} ẋ^{T} (σ) Qẋ (σ) dσ$ . By the LMIs in Corollary 1, feasible solutions can be obtained

\begin{matrix} P = [\begin{matrix} 10.4880 & 0.0792 \\ 0.0792 & 10.3499 \end{matrix}], Q = [\begin{matrix} 49.8223 & 4.2838 \\ 4.2838 & 46.8030 \end{matrix}], \end{matrix}

with $K_{1}$ and $\bar{K_{2}}$ being derived by

\begin{matrix} K_{1} = [\begin{matrix} 0.5838 & 0.0171 \\ 0.0134 & 0.6247 \end{matrix}], \bar{K_{2}} = [\begin{matrix} 0.0489 & - 0.0080 \\ - 0.0080 & 0.0629 \end{matrix}], \end{matrix}

where $\bar{K_{2}} = \frac{\sum_{i = 1}^{5} {K_{2}}_{i}}{5}$ , where

\begin{matrix} K_{21} = [\begin{matrix} - 0.1418 & - 0.0066 \\ - 0.0066 & - 0.1304 \end{matrix}], K_{22} = [\begin{matrix} - 0.0465 & - 0.0073 \\ - 0.0073 & - 0.0338 \end{matrix}], \end{matrix}

\begin{matrix} K_{23} = [\begin{matrix} 0.0489 & - 0.0080 \\ - 0.0080 & 0.0629 \end{matrix}], K_{24} = [\begin{matrix} 0.1442 & - 0.0088 \\ - 0.0088 & 0.1595 \end{matrix}], \\ K_{25} = [\begin{matrix} 0.2396 & - 0.0095 \\ - 0.0095 & 0.2561 \end{matrix}] . \end{matrix}

By Theorem 2, we choose $d_{k} = {3.65, 4.1, - 4.5, 2.82, - 2.5}$ and $c = - 0.45 .$ That is, $K = 0.714$ . From the above, we can obtain

\begin{matrix} D^{+} V (x) & \leq - 0.45 V (x) + | μ (t) |, t \neq t_{k} . \end{matrix}

That is, $c = - 0.45 < 0$ implies that continuous dynamics of the system is ISS. Setting $N_{0} = 5$ , it follows from Theorem 2 that system (28) is ISS if the rADT constant satisfies $τ_{2}^{*} = 1.59$ . The simulation results demonstrate that, in the presence of the external disturbance $μ (t) = {[\sin t, \cos t]}^{T}$ , the system states can still rapidly converge to the equilibrium state (Figure 2(b)), thereby achieving ISS of the system under the condition that the discrete dynamics are not ISS.

Conclusion

This paper explores the ISS of nonlinear systems with hybrid impulses by introducing the AIG method. By defining new stability conditions and control strategies, this paper offers novel theoretical tools for the stability analysis of nonlinear impulsive systems. Numerical experiments validate the effectiveness of the theoretical results and demonstrate the potential of this method in practical applications. Future research can further investigate other characteristics of hybrid impulsive systems, such as stochasticity, time-varying properties, and distributed control.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China under grant 62403265, in part by the Natural Science Foundation of Shandong Province under grant ZR2023QF089, in part by the China Postdoctoral Science Foundation under grant 2024M751566, in part by the Qingdao Postdoctoral Science Foundation under grant QDBSH20230102047, and in part by the Young Talent of Lifting engineering for Science and Technology in Shandong, China under grant SDAST2024QTA018.

ORCID iD

Xiaoxiao Lv

Data availability statement

Data sharing is not applicable to this article, as no data sets were generated or analyzed during the current study.

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