<p>This paper proposes generalized impulsive control systems where the time delay depends on the sequence of impulsive instants, and the interval between adjacent impulsive instants increases monotonically (either bounded or unbounded, i.e., sparse impulsive instants). For these two scenarios, sufficient Lyapunov-based conditions for the exponential stability of the systems are derived via impulsive control theory. The sufficient conditions reveal that the impulsive control strength <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\omega _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> is quantitatively dependent on the length of the adjacent impulsive interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t_{k+1}-t_k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, with its upper bound analytically determined as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\omega _k\le e^{a^{\frac{1}{\lambda }\ln \theta }\cdot (t_{k+1}-t_k)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mi>k</mi> </msub> <mo>≤</mo> <msup> <mi>e</mi> <mrow> <msup> <mi>a</mi> <mrow> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> <mo>ln</mo> <mi>θ</mi> </mrow> </msup> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> (where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$0<\lambda \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>λ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$0<\theta <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>θ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$a>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> denotes the non-average impulsive interval (NAII)). Specifically, when the impulsive instants follow a sparse distribution (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda =\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$t_k=k^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>=</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>), the designed impulsive strength <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\omega _k=0.12^{2k+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <msup> <mn>12</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> stabilizes the system with delay (<InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\tau _k=4.6-\frac{1}{k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>τ</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>4.6</mn> <mo>-</mo> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>), achieving an exponential convergence rate of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$k=0.0126$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>0.0126</mn> </mrow> </math></EquationSource> </InlineEquation> (verified by Example <InternalRef RefID="FPar23">1</InternalRef>). For systems with delay (<InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\tau _k=3(k-1)+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>τ</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>3</mn> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) and sparse impulsive instants (<InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\lambda =\frac{1}{2}, t_k=2k^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>2</mn> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>), the impulsive strength <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\omega _k=0.43^{4k+2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <msup> <mn>43</mn> <mrow> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> ensures global uniform exponential stability (GUES), with the convergence rate reaching <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$k=0.0044$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>0.0044</mn> </mrow> </math></EquationSource> </InlineEquation> (validated by Example <InternalRef RefID="FPar26">2</InternalRef>). Key parameters analysis shows that the impact factor <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$T_0=(t_1-t_0)a^{\frac{1}{\lambda }}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <msup> <mi>a</mi> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </msup> </mrow> </math></EquationSource> </InlineEquation> (related to <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$t_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> and the first impulsive instant <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$t_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>) influences the stability of large-delay systems, and <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\lambda <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> requires time-varying impulsive strength (constant <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> leads to instability). Two comparative numerical examples confirm that the proposed method outperforms the traditional average impulsive interval (AII) method for sparse impulsive instants, extending previous results on isometric impulsive sequences and providing a quantitative dynamic control strategy for generalized impulsive instant sequences.</p>

Exponential Stability of Nonlinear Impulsive Time-Delay Systems with Sparse Impulsive Instants

期刊论文
发表日期：

2026年5月24日

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Yize Chen,
Guangsheng Wei,
Juhua Liang

摘要

This paper proposes generalized impulsive control systems where the time delay depends on the sequence of impulsive instants, and the interval between adjacent impulsive instants increases monotonically (either bounded or unbounded, i.e., sparse impulsive instants). For these two scenarios, sufficient Lyapunov-based conditions for the exponential stability of the systems are derived via impulsive control theory. The sufficient conditions reveal that the impulsive control strength $\omega _k$ $ω_{k}$ is quantitatively dependent on the length of the adjacent impulsive interval $t_{k+1}-t_k$ $t_{k + 1} - t_{k}$ , with its upper bound analytically determined as $\omega _k\le e^{a^{\frac{1}{\lambda }\ln \theta }\cdot (t_{k+1}-t_k)}$ $ω_{k} \leq e^{a^{\frac{1}{λ} ln θ} \cdot (t_{k + 1} - t_{k})}$ (where $0<\lambda \le 1$ $0 < λ \leq 1$ , $0<\theta <1$ $0 < θ < 1$ , and $a>0$ $a > 0$ denotes the non-average impulsive interval (NAII)). Specifically, when the impulsive instants follow a sparse distribution ( $\lambda =\frac{1}{2}$ $λ = \frac{1}{2}$ , $t_k=k^2$ $t_{k} = k^{2}$ ), the designed impulsive strength $\omega _k=0.12^{2k+1}$ $ω_{k} = 0 . 12^{2 k + 1}$ stabilizes the system with delay ( $\tau _k=4.6-\frac{1}{k}$ $τ_{k} = 4.6 - \frac{1}{k}$ ), achieving an exponential convergence rate of $k=0.0126$ $k = 0.0126$ (verified by Example 1). For systems with delay ( $\tau _k=3(k-1)+1$ $τ_{k} = 3 (k - 1) + 1$ ) and sparse impulsive instants ( $\lambda =\frac{1}{2}, t_k=2k^2$ $λ = \frac{1}{2}, t_{k} = 2 k^{2}$ ), the impulsive strength $\omega _k=0.43^{4k+2}$ $ω_{k} = 0 . 43^{4 k + 2}$ ensures global uniform exponential stability (GUES), with the convergence rate reaching $k=0.0044$ $k = 0.0044$ (validated by Example 2). Key parameters analysis shows that the impact factor $T_0=(t_1-t_0)a^{\frac{1}{\lambda }}$ $T_{0} = (t_{1} - t_{0}) a^{\frac{1}{λ}}$ (related to $t_0$ $t_{0}$ and the first impulsive instant $t_1$ $t_{1}$ ) influences the stability of large-delay systems, and $\lambda <1$ $λ < 1$ requires time-varying impulsive strength (constant $\omega $ $ω$ leads to instability). Two comparative numerical examples confirm that the proposed method outperforms the traditional average impulsive interval (AII) method for sparse impulsive instants, extending previous results on isometric impulsive sequences and providing a quantitative dynamic control strategy for generalized impulsive instant sequences.