Pth moment input-to-state stability of impulsive stochastic functional differential systems with Markovian switching

Abstract

The paper introduces a unified criterion for pth moment input-to-state stability (p-ISS), pth moment integral input-to-state stability (p-iISS), and e^σt-weighted pth moment integral input-to-state stability (e^σt-p-ISS) of impulsive stochastic function differential system with Markovian switching under the perturbation of the stabilizing impulse and destabilizing impulse. The Lyapunov function approach, comparison principle, and impulsive average dwell-time method are applied in this paper. The linear coefficients of the upper bound of Lyapunov functional differential operators are time-varying functions, including the case of constants, which advances and improves the existing results. In addition, the same results were obtained by applying impulse differential inequality. At the end of this paper, we use a numerical example to verify the validity of the results.

Keywords

Input-to-state stability Lyapunov function comparison principle average dwell-time impulse differential inequality

Introduction

For systems with disturbance inputs, it is not enough to study its stability. Sontag (1989, 1998) first proposed the concept of continuous system input-to-state stability (ISS) and integral input to-state stability (iISS); iISS is a generalization of ISS. Both ISS and iISS are helpful in the analysis and design of nonlinear systems, which played an important role in the study of system stability with disturbance inputs. At present, ISS has become an important branch of the neighborhood of nonlinear system stability research. Generally speaking, ISS means that when the input of the system is bounded, the state of the system is also bounded; when the input of the system is very small, the state of the system is also small. ISS and iISS are widely used in system stability analysis and control of nonlinear systems with complex structures (Chen and Zhang, 2009; Dashkovskiy and Mironchenko, 2013; Fu and Zhu, 2018; Gao and Wang, 2019; Hespanha et al., 2008; Karafyllis and Jiang, 2013; Li and Deng, 2018; Liu et al., 2011; Ning et al., 2012; Peng and Deng, 2017; Peng and Zhang, 2010; Ren and Xiong, 2019; Wang et al., 2013; Wu et al., 2016, 2017; Zhang, 2016). Therefore, the ISS research of the system has received extensive attention. There have been many articles (Chen and Zhang, 2009; Dashkovskiy and Mironchenko, 2013; Hespanha et al., 2008; Karafyllis and Jiang, 2013; Liu et al., 2011; Ning et al., 2012; Wang et al., 2013) discussing deterministic differential equations without considering random disturbances. However, random interference can be seen everywhere in real life, stochastic differential equations (SDEs) are widely used in science and industry, such as finance, mathematical economy, neural networks, biology, and control (Jobert and Rogers, 2006; Khasminskii, 2012; Lei and Mackey, 2007; Luo and Mao, 2007; Mao et al., 2005). In actual situations, the system is inevitably affected by random disturbances (Fu and Zhu, 2018; Gao and Wang, 2019; Li and Deng, 2018; Peng and Deng, 2017; Peng and Zhang, 2010; Ren and Xiong, 2019; Wu et al., 2016, 2017; Zhang, 2016), which leads to instability. Therefore, it is meaningful to discuss the ISS/iISS of the stochastic differential system.

Impulse effect is a common phenomenon in nature. In real life, due to some reasons, the movement of some objects is not completely continuous, and may be suddenly disturbed in a very short period of time, causing rapid changes in the trajectory of the movement and the emergence of new features, this is the impulse phenomenon. The effect of impulse in the system may maintain the stability of the system, or it may destroy the stability of the system. Therefore, we need to discuss the interference of the impulse to the system. There have been many results (Fu et al., 2019; Guo et al., 2020; Kao et al., 2015; Wu et al., 2013; Zhu, 2014) on the stability of impulsive stochastic differential systems (ISDSs) and impulsive stochastic functional differential systems (ISFDSs). In recent years, there have been many discussions about the ISS/iISS of ISFDS. For example, in Fu and Zhu (2018) and Wu et al. (2017), using the average dwell-time (ADT) condition of the impulse and the Lyapunov functional method, it is discussed that the ISS of the hybrid ISDS without considering the effect of time delay. In Peng and Deng (2017), although the ISS of ISFDS is discussed, it only considers the destabilizing impulsive effect of the system. In Wu et al. (2016), the ISS of ISFDS under stabilizing impulse and destabilizing impulse is analyzed, by changing the impulsive ADT condition, and it requires that the linear coefficients of the upper bound of the differential operator of the Lyapunov functional are constants, but there are many systems of the differential operator of the Lyapunov functional that cannot be defined by constants. Therefore, this requires weaker conditions that extend to time-varying coefficients. We discuss ISFDS with Markovian switching; Markovian switching is an important model in hybrid systems and it has received a lot of attention (Fu and Zhu, 2018; Gao and Wang, 2019; Kao et al., 2015; Li and Deng, 2018; Liu et al., 2011; Luo and Mao, 2007; Peng and Zhang, 2010; Ren and Xiong, 2019; Wang et al., 2013; Wu et al., 2013; Zhu, 2014). For example, in the infectious disease model, due to different seasons, the individual incidence of a disease will also be different. Therefore, as the season changes, the system needs to switch in different states. So, it is necessary to discuss that the ISS of ISFDS with Markovian switching.

The main contributions of this paper are as follows. (1) This paper requires the linear coefficients of the upper bound of the differential operator of the Lyapunov functional to be a time-varying function, which makes the satisfied assumption weaker than the conditions in Wu et al. (2016), and including the case that the upper bound are constants. (2) In this paper, ADT condition is used to give a unified criterion that the hybrid system is ISS and under the stabilizing impulse and destabilizing impulse. In Peng and Deng (2017), it only considers the destabilizing impulsive effect of ISFDS. Therefore, the results obtained in this paper are more general and the analysis is more comprehensive. (3) This paper discusses that ISFDS with Markovian switching is pth moment input-to-state stability (p-ISS), pth moment integral input-to-state stability (p-iISS), and e^σt-weighted pth moment integral input-to-state stability (e^σt-p-ISS), this is more detailed than the system considered in Fu and Zhu (2018), Peng and Deng (2017), and Wu et al. (2016, 2017). Moreover, the system we considered includes a variety of simple systems. (4) In this paper, applying the impulsive differential inequality without considering the ADT condition, another sufficient condition for the system to be ISS is obtained.

The paper is organized as follows. Section “Preliminaries” introduces some necessary symbols and definitions. Section “Main results” gives two main theorems of the hybrid system that is p-ISS, p-iISS, and e^σt-p-ISS. Section “An example” gives a numerical example to show that the results are true. Section “Conclusion” summarizes the paper.

Preliminaries

In this paper, we introduce some notations. Let $R^{+} = [0, + \infty)$ , $Z_{+} = {1, 2, \dots}$ , $R = (- \infty, + \infty)$ , $| x | = \sqrt{x^{T} x}$ with $x \in ℝ^{n}$ , and $x^{T}$ represents the transposition of $x$ . In addition, for $τ > 0$ , $C^{2, 1} ([t_{0} - τ, \infty) \times ℝ^{n} \times S; ℝ^{+})$ denote function with the first variable is continuously once differentiable and the second variable is twice differentiable, $C (R^{+}, R) = {u (t) : R^{+} \to R | u (t) is a function}$ , $ℙ ℂ ([a, b]; ℝ^{n})$ denote the set of all continuous $ℝ^{n} - value$ function $ϕ$ defined on $[a, b]$ with norm $‖ ϕ ‖ = sup_{a \leq s \leq b} | ϕ (s) |$ . $ℙ ℂ_{ℱ_{t}}^{b} ([- τ, 0]; ℝ^{n}) (ℙ ℂ_{ℱ_{0}}^{b} ([- τ, 0]; ℝ^{n}))$ denote the family of all bounded $F_{t} (F_{0}) - measurable$ . $ϕ \in ℙ ℂ_{ℱ_{0}}^{b} ([- τ, 0]; ℝ^{n})$ , $p > 0$ , satisfy $‖ ϕ ‖_{L^{p}}^{p} = sup_{- τ \leq θ \leq 0} E {| ϕ (θ) |}^{p} < \infty$ , where $E$ denotes the expectation operator. Let $(Ω, F, {F}_{t \geq t_{0}}, P)$ represents a complete probability space with filtration ${F}_{t \geq t_{0}}$ satisfying the usual conditions. $w (t) = (w_{1} (t), w_{2} (t), \dots, w_{n} (t))^{T}$ is an m-dimensional Brownian motion on $(Ω, F, {F}_{t \geq t_{0}}, P)$ . Let ${r (t), t \geq t_{0}}$ be a right continuous Markovian chain on $(Ω, F, {F}_{t \geq t_{0}}, P)$ with its values in $S = {1, 2, \dots, N}$ , and its generator $T = (γ_{ij})_{n \times n}$ given by

P {r (t + Δ t) = j | r (t) = i} = {\begin{matrix} γ_{ij} Δ t + o (Δ t), i \neq j \\ 1 + γ_{ii} Δ t + o (Δ t), i = j \end{matrix}

where $Δ t > 0$ and $lim_{Δ t \to 0} o (Δ t) / Δ t = 0$ , $γ_{ij} \geq 0$ is the transition rate from $i$ to $j$ , $γ_{ii} = - \sum_{j \neq i} γ_{ij}$ . We assume that Markovian chain $r (t)$ is independent of Brownian motion $w (\cdot)$ . Let $K : = {α (\cdot) : R^{+} \to R^{+} | α (\cdot)$ is continuous and strictly increasing, $α (0) = 0, α (s) > 0$ for any $s > 0}$ ; $K L : = {β (\cdot, \cdot) : R^{+} \times R^{+} \to R^{+} | β (\cdot, t) \in K$ for each fixed $t \geq 0$ , and $β (s, t)$ decreases to $0$ as $t \to + \infty$ for each fixed $s \geq 0}$ ; $K_{\infty} : = {α (\cdot) \in K | α (\cdot) is unbounded}$ ; $V K : = {α (\cdot) \in K | α (\cdot) is convex}$ ; $V K_{\infty} : = {α (\cdot) \in K_{\infty} | α (\cdot) is convex}$ ; $C K : = {α (\cdot) \in K | α (\cdot) is concave}$ ; $C K_{\infty} : = {α (\cdot) \in K_{\infty} | α (\cdot) is concave}$ ; $α^{- 1} (\cdot)$ denotes the inverse of $α (\cdot)$ .

Consider the following hybrid ISFDS

{\begin{matrix} dx (t) = f (t, x (t), x_{t}, r (t), u_{c} (t)) dt + g (t, x (t), x_{t}, r (t), \\ u_{c} (t)) dw (t), t \geq t_{0}, t \neq t_{k} \\ Δ x (t_{k}) = x (t_{k}) - x (t_{k}^{-}) = I_{k} (t_{k}^{-}, x (t_{k}^{-}), r (t_{k}^{-}), u_{d} (t_{k}^{-})), \\ t = t_{k}, k \in Z_{+} \\ x (t_{0} + θ) = ϕ (θ), - τ \leq θ \leq 0 \end{matrix}

(1)

where $u_{c} (t) \in PC ([t_{0}, \infty); R^{m_{1}})$ is the external input, $u_{d} (t) \in PC ([t_{0}, \infty); R^{m_{2}})$ is the impulsive input. $f : R^{+} \times R^{n} \times {PC}_{F_{t}}^{b} ([- τ, 0]; R^{n}) \times S \times R^{m_{1}} \to R^{n}$ , $g : R^{+} \times R^{n} \times {PC}_{F_{t}}^{b} ([- τ, 0]; R^{n}) \times S \times R^{m_{1}} \to R^{n \times m}$ , $φ (θ) = x_{t} (θ) = x (t + θ), θ \in [- τ, 0]$ , $‖ φ ‖ = ‖ x_{t} ‖ = sup_{- τ \leq θ \leq 0} | x_{t} (θ) |$ . The initial function $ϕ \in {PC}_{F_{0}}^{b} ([- τ, 0]; R^{n})$ , $I_{k} : R^{+} \times R^{n} \times S \times R^{m_{2}} \to R^{n}, k \in Z_{+}$ , and the impulse times $t_{k}$ satisfy $0 \leq t_{1} < t_{2} < \dots < t_{k} < \dots$ , and $lim_{k \to \infty} t_{k} = \infty$ . Let $P C (R^{+}, R) = {u (t) : R^{+} \to R | u (t)$ is continuous for $t \neq t_{k}$ , and $u (0^{+})$ , $u (t_{k}^{-})$ , $u ({t_{k}}^{+})$ exist, $u (t_{k}) = u (t_{k}^{+})}$ , and $P C^{1} (R^{+}, R) = {u \in P C (R^{+}, R) | u' (t)$ is continuous for $t \neq t_{k}$ , and $u' (0^{+}), u' (t_{k}^{-}), u' ({t_{k}}^{+})$ exist, $u' (t_{k}) = u' (t_{k}^{+})}$ . We assume that $f, g$ , and $I$ satisfy the Lipchitz condition. So, system (1) has a unique solution $x (t) = x (t; t_{0}, ϕ, r_{0}), r_{0} \in S$ , $f (t, 0, 0, i, 0) = 0$ , $g (t, 0, 0, i, 0) = 0$ , $I_{k} = (t_{k}, 0, i, 0) = 0$ , $k \in Z_{+}, t \geq t_{0}$ . Then, system (1) has an initial solution $x (t) = 0$ .

Definition 1

System (1) is said to be:

p-ISS, if $ϖ \in K L$ and $χ_{c}, χ_{d} \in K_{\infty}$ , such that for every $ϕ \in {PC}_{F_{0}}^{b} ([- τ, 0]; R^{n})$ and input $(u_{c}, u_{d})$ , the solution of equation (1) satisfies

\begin{matrix} E ({| x (t) |}^{p}) \leq ϖ (‖ ϕ ‖_{L^{p}}^{p}, t - t_{0}) + χ_{c} ({‖ u_{c} ‖}_{[t_{0}, t]}) \\ + χ_{d} (max_{t_{k} \in (t_{0}, t]} | u_{d} (t_{k}^{-}) |), t \geq t_{0} \end{matrix}

(2)

p-iISS, if $ϖ \in K L$ and $φ, χ_{c}, χ_{d} \in K_{\infty}$ , such that

\begin{matrix} φ (E ({| x (t) |}^{p})) \leq ϖ (‖ ϕ ‖_{L^{p}}^{p}, t - t_{0}) + \int_{t_{0}}^{t} χ_{c} (| u_{c} (s) |) ds \\ + \sum_{t_{k} \in (t_{0}, t]} χ_{d} (| u_{d} (t_{k}^{-}) |), t \geq t_{0} \end{matrix}

(3)

e^σt-p-ISS, if $φ_{1}, φ_{2}, χ_{c}, χ_{d} \in K_{\infty}$ , and a constant $σ > 0$ , such that

\begin{matrix} e^{σ (t - t_{0})} φ_{1} (E ({| x (t) |}^{p})) \leq φ_{2} (‖ ϕ ‖_{L^{p}}^{p}) \\ + sup_{t_{0} \leq s \leq t} {e^{σ (s - t_{0})} χ_{c} (| u_{c} (s) |)} \\ + max_{t_{k} \in (t_{0}, t]} {e^{σ (t_{k} - t_{0})} χ_{d} (| u_{d} (t_{k}^{-}) |)}, t \geq t_{0} \end{matrix}

(4)

Definition 2

For an impulsive sequence ${t_{k}, k \in Z_{+}}$ , let $N (t, s)$ be the number of impulsive times of $t_{k} \in (s, t]$ , if (Wu et al., 2016)

\frac{t - s}{T_{a}} - N_{0} \leq N (t, s) \leq \frac{t - s}{T_{a}} + N_{0}

(5)

for $T_{a} > 0$ , $N_{0} > 0$ , then $T_{a}$ is called ADT and $N_{0}$ is called the elasticity number. The formula (5) is called the impulsive ADT condition.

Definition 3

The function $V \in C^{2, 1} ([t_{0} - τ, \infty) \times R^{n} \times S; R^{+})$ , we define an operator $L V$ for system (1) as follows (Fu and Zhu, 2018)

\begin{matrix} L V (t, x_{t}, i) = V_{t} (t, x_{t}, i) + V_{x} (t, x_{t}, i) f (t, x (t), x_{t}, i, u_{c} (t)) \\ + \frac{1}{2} trace [g^{T} (t, x (t), x_{t}, i, u_{c} (t)) V_{xx} (t, x_{t}, i) g (t, x (t), x_{t}, i, u_{c} (t))] \\ + \sum_{j = 1}^{N} γ_{ij} V (t, x_{t}, j) \end{matrix}

where

\begin{matrix} V_{t} (t, x_{t}, i) = \frac{\partial V (t, x_{t}, i)}{\partial t}, V_{x} (t, x_{t}, i) \\ = (\frac{\partial V (t, x_{t}, i)}{\partial x_{1}}, \dots, \frac{\partial V (t, x_{t}, i)}{\partial x_{n}}) \\ V_{xx} (t, x_{t}, i) = {(\frac{\partial^{2} V (t, x_{t}, i)}{\partial x_{k} \partial x_{ℓ}})}_{n \times n} \\ = [\begin{matrix} {[V_{xx} (t, x_{t}, i)]}_{11} & \dots & {[V_{xx} (t, x_{t}, i)]}_{1 ℓ} & \dots & {[V_{xx} (t, x_{t}, i)]}_{1 n} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ {[V_{xx} (t, x_{t}, i)]}_{k 1} & \dots & {[V_{xx} (t, x_{t}, i)]}_{k ℓ} & \dots & [V_{xx} (t, x_{t}, i))]_{kn} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ {[V_{xx} (t, x_{t}, i)]}_{n 1} & \dots & {[V_{xx} (t, x_{t}, i)]}_{n ℓ} & \dots & {[V_{xx} (t, x_{t}, i)]}_{nn} \end{matrix}] \end{matrix}

with $\partial^{2} V (t, x_{t}, i) / \partial x_{k} \partial x_{ℓ}$ being the element in row $k$ and column $ℓ$ of $V_{xx} (t, x_{t}, i)$ as

{[V_{xx} (t, x_{t}, i)]}_{k ℓ} = \frac{\partial^{2} V (t, x_{t}, i)}{\partial x_{k} \partial x_{ℓ}}, k, ℓ \in {1, \dots, n}

Lemma 1

Let the function $m (t) \in ℙ ℂ^{1} (ℝ^{+}, ℝ)$ satisfies the following inequalities (Bainov and Simeonov, 1993)

\begin{matrix} m' (t) \leq p (t) m (t) + q (t), t \geq t_{0}, t \neq t_{k} \\ m (t_{k}) \leq d_{k} m (t_{k}^{-}) + b_{k}, t_{k} > t_{0} \end{matrix}

where $p, q \in PC (R^{+}, R), d_{k} \geq 0$ and $b_{k}$ are the constants. Then

\begin{matrix} m (t) \leq m (t_{0}) \underset{t_{0} < t_{k} < t}{Π} d_{k} e^{\int_{t_{0}}^{t} p (s) ds} + \sum_{t_{0} < t_{k} < t} (\underset{t_{k} < t_{j} < t}{Π} d_{j} e^{\int_{t_{k}}^{t} p (s) ds}) b_{k} \\ + \int_{t_{0}}^{t} \underset{s < t_{k} < t}{Π} d_{k} e^{\int_{s}^{t} p (σ) d σ} q (s) ds, t \geq t_{0} \end{matrix}

Lemma 2

Let $X (u)$ be a process satisfying SDE on the time interval [s,T], $V \in C^{2, 1}$ , $τ_{U}$ be the random variable equal to the time at which the sample function of the process $X (u)$ first leaves the bounded neighborhood $U$ , and let $τ_{U} (t) = min (τ_{U}, t)$ . Moreover, suppose that $P$ holds for $P {X (s) \in U} = 1$ . Then, the following equality holds for (Lemma 3.2 in Khasminskii, 2012)

E [V (τ_{U} (t), X (τ_{U} (t))) - V (s, X (s))] = E \int_{s}^{τ_{U} (t)} L V (u, X (u)) du

Main results

The following uses Lyapunov function approach, the comparison principle, impulsive ADT condition, and impulsive differential inequality, we discuss the ISS for ISFDS with Markovian switching.

Theorem 1

Assume that there are functions $V (t, x_{t}, r (t)) = V_{1} (t, x (t), r (t)) + V_{2} (t, x_{t}, r (t))$ , $V_{1} \in C^{2, 1} ([t_{0}, \infty) \times R^{n} \times S; R^{+})$ and $V_{2} \in C^{2, 1} ([t_{0}, \infty) \times {PC}_{F_{t}}^{b} ([- τ, 0]; R^{n}) \times S; R^{+})$ and for any $x \in PC ([t_{0} - τ, \infty); R^{n})$ , $E V_{2} (t, x_{t}, r (t))$ is continuous for $t \geq t_{0}$ . Suppose there are functions $χ_{1}$ , $χ_{2}$ , $ν_{1}$ , $ν_{2}$ , and $χ_{1}, χ_{2} \in K_{\infty}$ , $ν_{1} \in V K_{\infty}$ , $ν_{2} \in C K_{\infty}$ , $λ (t) \in PC ([t_{0} - τ, \infty); R)$ , $T_{a}$ satisfies ADT condition, $p > 0$ and there are constants $η > 0, β \geq 0, δ > 0$ such that

\begin{matrix} (H_{1}) ν_{1} (| x (t) |^{p}) \leq V (t, x_{t}, r (t)) \leq ν_{2} ({‖ x_{t} ‖}^{p}), t \geq t_{0}, x \in R^{n}, \\ x_{t} \in {PC}_{F_{t}}^{b} ([- τ, 0]; R^{n}), r (t) \in S \\ (H_{2}) E L V (t, x_{t}, r (t)) \leq λ (t) E V (t, x_{t}, r (t)) + χ_{1} (| u_{c} (t) |) \\ for all t \in [t_{k - 1}, t_{k}), k \in Z_{+} \\ (H_{3}) E V_{1} (t_{k}, x (t_{k}), r (t_{k})) \leq η E V_{1} (t_{k}^{-}, x (t_{k}^{-}), r (t_{k}^{-})) \\ + χ_{2} (| u_{d} (t_{k}^{-}) |), k \in Z_{+}, r (t_{k}) \in S, r (t_{k}^{-}) \in S \\ (H_{4}) \int_{s}^{t} (\frac{\ln \bar{η}}{T_{a}} + λ (v)) dv \leq β - δ (t - s), t > s \geq t_{0}, inf_{k \in R_{+}} \\ {t_{k} - t_{k - 1}} > 0, \bar{η} = max {1, η} \end{matrix}

Then, the system (1) is p-ISS, p-iISS, and e^σt-p-ISS, where $σ \in (0, δ)$ . In addition, if $0 < η \leq 1$ , the ADT condition can be omitted.

Proof

Let $x (t) = x (t; t_{0}, ϕ, r_{0})$ denote the solution of equation (1), $r_{0} \in S$ , $V (t, x_{t}, r (t)) = V (t)$ , $r (t) \in S$ , $V_{1} (t, x (t), r (t)) = V_{1} (t)$ , and $V_{2} (t, x_{t}, r (t)) = V_{2} (t)$ for any $t \geq t_{0}$ and $ϕ \in {PC}_{F_{0}}^{b} ([- τ, 0]; R^{n})$ . By Ito’s differential formula, we have

\begin{matrix} dV (t) = L V (t, x_{t}, r (t)) dt + V_{x} (t, x_{t}, r (t)) g (t, x (t), x_{t}, r (t), u_{c} (t)) \\ dw (t), t \in [t_{k - 1}, t_{k}), k \in Z_{+} \end{matrix}

Let $Δ t > 0$ and $t + Δ t \in (t_{k - 1}, t_{k})$ , then by Fubini’s theorem and Lemma 3.2 in Khasminskii (2012) we get

E V (t + Δ t) - E V (t) = \int_{t}^{t + Δ t} E L V (s, x_{s}, r (s)) ds

for any $t \in [t_{k - 1}, t_{k}), k \in Z_{+}$ , let $Δ t \to 0^{+}$ , we have

\begin{matrix} D^{+} E V (t) : = \underset{Δ t \to 0^{+}}{lim sup} \frac{E V (t + Δ t) - E V (t)}{Δ t} \\ = \underset{Δ t \to 0^{+}}{lim sup} \frac{1}{Δ t} \int_{t}^{t + Δ t} E L V (s, x_{s}, r (s)) ds \end{matrix}

by the continuity of $L V (t, x_{t}, r (t))$ and definition of the above formula $D^{+}$ , we obtain

D^{+} E V (t) = E L V (t, x_{t}, r (t))

(6)

By $(H_{1})$ , we have

E V (t_{0}) \leq E [ν_{2} (| ϕ (θ) |^{p})] \leq ν_{2} (E (| ϕ (θ) |^{p})) \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}), θ \in [- τ, 0]

First, we will prove that for $t \in [t_{0}, t_{1})$

E V (t) \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{\int_{t_{0}}^{t} λ (s) ds} + \int_{t_{0}}^{t} [χ_{1} (| u_{c} (s) |) e^{\int_{s}^{t} λ (v) dv}] ds

(7)

Consider the comparison differential equation, $ε$ is a positive number

{\begin{matrix} f' (t) = λ (t) f (t) + χ_{1} (| u_{c} (t) |) + ε, t \in [t_{0}, t_{1}) \\ f (t_{0}) = ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) + ε \end{matrix}

(8)

Solving the ordinary differential equation (8), the equation is solved as

\begin{matrix} f_{ε} (t) = (ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) + ε) e^{\int_{t_{0}}^{t} λ (s) ds} + \int_{t_{0}}^{t} \\ {[χ_{1} (| u_{c} (s) |) + ε] e^{\int_{s}^{t} λ (v) dv}} ds, t \in [t_{0}, t_{1}) \end{matrix}

(9)

Now, we prove that

EV (t) < f_{ε} (t), t \in [t_{0}, t_{1})

(10)

Supposing equation (10) is not true, then there exists some $t \in (t_{0}, t_{1})$ such that $E V (t) \geq f_{ε} (t)$ . Let

t^{*} = inf {t \in (t_{0}, t_{1}) : E V (t) \geq f_{ε} (t)}

Due to the continuity of $E V (t)$ , and $E V (t_{0}) < f_{ε} (t_{0})$ , then $E V (t) < f_{ε} (t)$ for all $t \in [t_{0}, t^{*})$ , $E V (t^{*}) = f_{ε} (t^{*})$ , and $E V (t) \geq f_{ε} (t)$ for all $t \in (t^{*}, t^{*} + ς)$ , $ς$ is a sufficiently small positive number. Therefore, we get

\frac{E V (t) - E V (t^{*})}{t - t^{*}} \geq \frac{f_{ε} (t) - f_{ε} (t^{*})}{t - t^{*}}, t \in (t^{*}, t^{*} + ς)

then let $t \to t^{*}$ , we obtain

\begin{matrix} D^{+} E V (t^{*}) \geq f'_{ε} (t^{*}) = λ (t^{*}) f_{ε} (t^{*}) + χ_{1} (| u_{c} (t^{*}) |) + ε > λ (t^{*}) E V \\ (t^{*}) + χ (_{1} | u_{c} (t^{*}) |) \end{matrix}

(11)

From $(H_{2})$ and equation (6), we obtain

D^{+} E V (t^{*}) = E L V (t^{*}, x_{t^{*}}, r (t^{*})) \leq λ (t^{*}) E V (t^{*}) + χ_{1} (| u_{c} (t^{*}) |)

This is contradictory to formula (11), so equation (10) holds. Let $ε \to 0$ in equation (10), we get equation (7).

Set

\begin{matrix} Q (k, t, ε) = \int_{t_{k - 1}}^{t} (χ_{1} (| u_{c} (s) |) + ε) e^{\int_{s}^{t} λ (v) dv} ds + (ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) + ε) \\ {\bar{η}}^{k - 1} e^{\int_{t_{0}}^{t} λ (s) ds} \\ + \sum_{i = 1}^{k - 1} {\bar{η}}^{k - 1 - i} (χ_{2} (| u_{d} (t_{i}^{-}) |) + ε) e^{\int_{t_{i}}^{t} λ (s) ds} \\ + \sum_{i = 1}^{k - 1} {\bar{η}}^{k - i} \int_{t_{i - 1}}^{t_{i}} [(χ_{1} (| u_{c} (s) |) + ε) e^{\int_{s}^{t} λ (v) dv}] ds, k \in Z_{+}, ε > 0 \end{matrix}

Next, we will prove that for all $t \in [t_{k - 1}, t_{k})$ and $k \in Z_{+}$

E V (t) < Q (k, t, ε)

(12)

In fact, by equation (10), for $k = 1$ , equation (12) is true. We assume that equation (12) is true for $k = m, m \in Z_{+}$ , then it is sufficient to prove that for all $t \in [t_{m}, t_{m + 1})$ formula (12) is also true. Due to for any $x \in PC ([t_{0} - τ, \infty); R^{n})$ , $E V_{2} (t, x_{t}, r (t))$ is continuous for $t \geq t_{0}$ , we have

\begin{matrix} E V (t_{m}) - E V (t_{m}^{-}) = E V_{1} (t_{m}) + E V_{2} (t_{m}) - E V_{1} (t_{m}^{-}) \\ - E V_{2} (t_{m}^{-}) = E V_{1} (t_{m}) - E V_{1} (t_{m}^{-}) \end{matrix}

So, by $(H_{3})$ , we get

\begin{matrix} E V (t_{m}) = E V (t_{m}^{-}) + E V_{1} (t_{m}) - E V_{1} (t_{m}^{-}) \\ \leq E V (t_{m}^{-}) + (η - 1) E V_{1} (t_{m}^{-}) + χ_{2} (| u_{d} (t_{m}^{-}) |) \end{matrix}

If $0 < η \leq 1$ , then

E V (t_{m}) \leq E V (t_{m}^{-}) + χ_{2} (| u_{d} (t_{m}^{-}) |)

If $η > 1$ , then

\begin{matrix} E V (t_{m}) \leq E V (t_{m}^{-}) + (η - 1) [E V_{1} (t_{m}^{-}) + E V_{2} (t_{m}^{-})] + χ_{2} (| u_{d} (t_{m}^{-}) |) \\ = η E V (t_{m}^{-}) + χ_{2} (| u_{d} (t_{m}^{-}) |) \end{matrix}

Therefore

\begin{matrix} E V (t_{m}) \leq \bar{η} E V (t_{m}^{-}) + χ_{2} (| u_{d} (t_{m}^{-}) |) \\ \leq \bar{η} Q (m, t_{m}, ε) + χ_{2} (| u_{d} (t_{m}^{-}) |) \end{matrix}

Given the following comparison equation

{\begin{matrix} f' (t) = λ (t) f (t) + χ_{1} (| u_{c} (t) |) + ε, t \in [t_{m}, t_{m + 1}) \\ f (t_{m}) = \bar{η} Q (m, t_{m}, ε) + χ_{2} (| u_{d} (t_{m}^{-}) |) + ε \end{matrix}

(13)

The solution of equation (13) is

\begin{matrix} f_{ε} (t) = [\bar{η} Q (m, t_{m}, ε) + χ_{2} (| u_{d} (t_{m}^{-}) |) + ε] e^{\int_{t_{m}}^{t} λ (s) ds} \\ + \int_{t_{m}}^{t} [(χ_{1} (| u_{c} (s) |) + ε) e^{\int_{s}^{t} λ (v) dv}] ds \\ = Q (m + 1, t, ε) \end{matrix}

We prove that

E V (t) < Q (m + 1, t, ε), t \in [t_{m}, t_{m + 1})

(14)

Supposing equation (14) is not true, then there exists some $t \in (t_{m}, t_{m + 1})$ such that

E V (t) \geq Q (m + 1, t, ε) = f_{ε} (t)

Let

\hat{t} = inf {t \in (t_{m}, t_{m + 1}) : E V (t) \geq f_{ε} (t)}

since $E V (t)$ is continuous, and $E V (t_{m}) < f_{ε} (t_{m})$ , then $E V (t) < f_{ε} (t)$ for all $t \in [t_{m}, \hat{t})$ , and $E V (\hat{t}) = f_{ε} (\hat{t})$ , $E V (t) \geq f_{ε} (t)$ for all $t \in (\hat{t}, \hat{t} + ς)$ . So

\frac{E V (t) - E V (\hat{t})}{t - \hat{t}} \geq \frac{f_{ε} (t) - f_{ε} (\hat{t})}{t - \hat{t}}, t \in (\hat{t}, \hat{t} + ς)

then let $t \to \hat{t}$

\begin{matrix} D^{+} E V (\hat{t}) \geq f'_{ε} (\hat{t}) = λ (\hat{t}) f_{ε} (\hat{t}) + χ_{1} (| u_{c} (\hat{t}) |) + ε > λ (\hat{t}) E V (\hat{t}) \\ + χ_{1} (| u_{c} (\hat{t}) |) \end{matrix}

(15)

By condition $(H_{2})$ and equation (6), we get

D^{+} E V (\hat{t}) \leq λ (\hat{t}) E V (\hat{t}) + χ_{1} (| u_{c} (\hat{t}) |)

This is contradictory to equation (15), so the formula (14) is correct. According to the method of mathematical induction, we obtain

E V (t) < Q (k, t, ε), t \in [t_{k - 1}, t_{k}), k \in Z_{+}

Let $ε \to 0$ , for all $t \in [t_{k - 1}, t_{k}), k \in Z_{+}$ , we have

\begin{matrix} E V (t) \leq \int_{t_{k - 1}}^{t} [χ_{1} (| u_{c} (s) |) e^{\int_{s}^{t} λ (v) dv}] ds + ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) {\bar{η}}^{k - 1} e^{\int_{t_{0}}^{t} λ (s) ds} \\ + \sum_{i = 1}^{k - 1} {\bar{η}}^{k - 1 - i} χ_{2} (| u_{d} (t_{i}^{-}) |) e^{\int_{t_{i}}^{t} λ (s) ds} + \sum_{i = 1}^{k - 1} {\bar{η}}^{k - i} \int_{t_{i - 1}}^{t_{i}} \\ [(χ_{1} (| u_{c} (t) |)) e^{\int_{s}^{t} λ (v) dv}] ds \\ = ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) {\bar{η}}^{N (t, t_{0})} e^{\int_{t_{0}}^{t} λ (s) ds} + \int_{t_{0}}^{t} {\bar{η}}^{N (t, s)} χ_{1} (| u_{c} (s) |) e^{\int_{s}^{t} λ (v) dv} ds \\ + \sum_{t_{0} < t_{i} \leq t} {\bar{η}}^{N (t, t_{i})} χ_{2} (| u_{d} (t_{i}^{-}) |) e^{\int_{t_{i}}^{t} λ (s) ds} \end{matrix}

(16)

We will consider the following two cases of $η > 1$ and $0 < η \leq 1$ .

Case 1

If $η > 1$ , then $\bar{η} = η$ . By equation (5), for any $t > s \geq t_{0}$ , we have

{\bar{η}}^{N (t, s)} \leq e^{(\frac{t - s}{T_{a}} + N_{0}) \ln \bar{η}} = {\bar{η}}^{N_{0}} e^{\frac{(t - s) \ln \bar{η}}{T_{a}}} = η^{N_{0}} e^{\frac{(t - s) \ln η}{T_{a}}}

(17)

Then, by equations (16) and (17) and $(H_{4})$ , we obtain

\begin{matrix} E V (t) \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) η^{N_{0}} \exp {\frac{(t - t_{0}) \ln η}{T_{a}} + \int_{t_{0}}^{t} λ (s) ds} \\ + \int_{t_{0}}^{t} η^{N_{0}} χ (_{1} | u_{c} (s) |) \exp {(\frac{(t - s) \ln η}{T_{a}} + \int_{s}^{t} λ (v) dv) ds} \\ + \sum_{t_{0} < t_{i} \leq t} η^{N_{0}} χ_{2} (| u_{d} (t_{i}^{-}) |) \exp {\frac{(t - t_{i}) \ln η}{T_{a}} + \int_{t_{i}}^{t} λ (s) ds} \\ \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) η^{N_{0}} \exp {\int_{t_{0}}^{t} (\frac{\ln η}{T_{a}} + λ (s)) ds} \\ + \int_{t_{0}}^{t} η^{N_{0}} χ_{1} (| u_{c} (s) |) \exp {\int_{s}^{t} (\frac{\ln η}{T_{a}} + λ (v) dv) ds} \\ + \sum_{t_{0} < t_{i} \leq t} η^{N_{0}} χ_{2} (| u_{d} (t_{i}^{-}) |) \exp {\int_{t_{i}}^{t} (\frac{\ln η}{T_{a}} + λ (s)) ds} \\ \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) η^{N_{0}} e^{β} e^{- δ (t - t_{0})} + η^{N_{0}} e^{β} \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) e^{- δ (t - s)} ds \\ + η^{N_{0}} e^{β} \sum_{t_{0} < t_{i} \leq t} χ_{2} (| u_{d} (t_{i}^{-}) |) e^{- δ (t - t_{i})} \end{matrix}

(18)

From $(H_{1})$ , we obtain

\begin{matrix} ν_{1} (E (| x (t) |^{p})) \leq E V (t) \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) η^{N_{0}} e^{β} e^{- δ (t - t_{0})} + η^{N_{0}} e^{β} \int_{t_{0}}^{t} χ_{1} \\ (| u_{c} (s) |) ds + η^{N_{0}} e^{β} \sum_{t_{0} < t_{i} \leq t} χ_{2} (| u_{d} (t_{i}^{-}) |) \end{matrix}

(19)

So, we get equation (3), then the system (1) is p-iISS.

From equation (18) and $(H_{1})$ , we also get

\begin{matrix} ν_{1} (E (| x (t) |^{p})) \leq η^{N_{0}} e^{β} ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ (t - t_{0})} + \frac{1}{δ} η^{N_{0}} e^{β} χ_{1} ({‖ u_{c} ‖}_{[t_{0}, t]}) \\ (1 - e^{- δ (t - t_{0})}) \\ + \sum_{t_{0} < t_{i} \leq t} e^{- δ (t - t_{i})} η^{N_{0}} e^{β} χ_{2} (max_{t_{k} \in (t_{0}, t]} {| u_{d} (t_{k}^{-}) |}) \\ \leq η^{N_{0}} e^{β} ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ (t - t_{0})} + \frac{1}{δ} η^{N_{0}} e^{β} χ_{1} ({‖ u_{c} ‖}_{[t_{0}, t]}) \\ + \frac{η^{N_{0}} e^{β}}{1 - e^{- δ ξ}} χ_{2} (max_{t_{k} \in (t_{0}, t]} {| u_{d} (t_{k}^{-}) |}) \end{matrix}

(20)

where $ξ = inf_{k \in Z_{+}} {t_{k} - t_{k - 1}} > 0 .$ Let $ϖ (r, s) = ν_{1}^{- 1} (3 η^{N_{0}} e^{β} ν_{2} (r) e^{- δ s})$ , $γ_{c} (r) = ν_{1}^{- 1} ((3 / δ) η^{N_{0}} e^{β} χ_{1} (r))$ , $γ_{d} (r) = ν_{1}^{- 1} ((3 η^{N_{0}} e^{β} / 1 - e^{- δ ξ}) χ_{2} (r))$ . So, we get equation (2), then the system (1) is p-ISS.

Let $δ = δ_{1} + σ$ , where $δ_{1}$ and $σ$ are the positive constants, by equation (18) and $(H_{1})$ , we have

\begin{matrix} e^{σ (t - t_{0})} ν_{1} (E (| x (t) |^{p})) \leq η^{N_{0}} e^{β} ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ_{1} (t - t_{0})} \\ + η^{N_{0}} e^{β} \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) e^{- δ_{1} (t - s)} e^{σ (s - t_{0})} ds \\ + η^{N_{0}} e^{β} \sum_{t_{0} < t_{i} \leq t} e^{σ (t_{i} - t_{0})} χ_{2} (| u_{d} (t_{i}^{-}) |) e^{- δ_{1} (t - t_{i})} \\ \leq η^{N_{0}} e^{β} ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) + \frac{η^{N_{0}} e^{β}}{δ_{1}} sup_{t_{0} \leq s \leq t} {e^{σ (s - t_{0})} (χ_{1} | u_{c} (s) |)} \\ + \frac{η^{N_{0}} e^{β}}{1 - e^{- δ_{1} ξ}} max_{t_{k} \in (t_{0}, t]} {e^{σ (t_{k} - t_{0})} χ_{2} (| u_{d} (t_{k}^{-}) |)} \end{matrix}

(21)

So, we get equation (4), then the system (1) is e^σt-p-ISS where $σ \in (0, δ)$ .

Case 2

If $0 < η \leq 1$ , then $\bar{η} = 1$ , by $(H_{4})$ , there are constants $β \geq 0$ and $δ > 0$ , we get

\int_{t_{0}}^{t} λ (s) ds \leq β - (δ + \frac{\ln \bar{η}}{T_{a}}) (t - t_{0}) = β - δ (t - t_{0})

(22)

So, by equations (16) and (22), we have

\begin{matrix} E V (t) \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{\int_{t_{0}}^{t} λ (s) ds} + \int_{t_{0}}^{t} [χ_{1} (| u_{c} (s) |) e^{\int_{s}^{t} λ (v) dv}] ds \\ + \sum_{t_{0} < t_{i} \leq t} χ_{2} (| u_{d} (t_{i}^{-}) |) e^{\int_{t_{i}}^{t} λ (s) ds} \\ \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{β} e^{- δ (t - t_{0})} + e^{β} \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) e^{- δ (t - s)} ds \\ + e^{β} \sum_{t_{0} < t_{i} \leq t} χ_{2} (| u_{d} (t_{i}^{-}) |) e^{- δ (t - t_{i})} \end{matrix}

(23)

By equation (23) and $(H_{1})$ , we have

\begin{matrix} ν_{1} (E (| x (t) |^{p})) \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{β} e^{- δ (t - t_{0})} + e^{β} \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) ds \\ + e^{β} \sum_{t_{0} < t_{i} \leq t} χ_{2} (| u_{d} (t_{i}^{-}) |) \end{matrix}

(24)

So, we get equation (3), then the system (1) is p-iISS.

From equation (23) and $(H_{1})$ , we also get

\begin{matrix} ν_{1} (E (| x (t) |^{p})) n \leq e^{β} ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ (t - t_{0})} + \frac{e^{β}}{δ} χ_{1} ({‖ u_{c} ‖}_{[t_{0}, t]}) \\ + \frac{e^{β}}{1 - e^{- δ ξ}} χ_{2} (max_{t_{k} \in (t_{0}, t]} {| u_{d} (t_{k}^{-}) |}) \end{matrix}

(25)

where $ϖ (r, s) = ν_{1}^{- 1} (3 e^{β} ν_{2} (r) e^{- δ s})$ , $γ_{c} (r) = ν_{1}^{- 1} ((3 e^{β} / δ) χ_{1} (r))$ , and $γ_{d} (r) = ν_{1}^{- 1} ((3 e^{β} / 1 - e^{- δ ξ}) χ_{2} (r))$ . So, we get equation (2), then the system (1) is p-ISS.

Let $δ = δ_{1} + σ$ , $δ_{1}$ and $σ$ are the positive numbers, by equation (23) and $(H_{1})$ , we have

\begin{matrix} e^{σ (t - t_{0})} ν_{1} (E (| x (t) |^{p})) \leq e^{β} ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ_{1} (t - t_{0})} \\ + e^{β} \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) e^{- δ_{1} (t - s)} e^{σ (s - t_{0})} ds \\ + e^{β} \sum_{t_{0} < t_{i} \leq t} e^{σ (t_{i} - t_{0})} χ_{2} (| u_{d} (t_{i}^{-}) |) e^{- δ_{1} (t - t_{i})} \\ \leq e^{β} ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) + \frac{e^{β}}{δ_{1}} sup_{t_{0} \leq s \leq t} {e^{σ (s - t_{0})} χ_{1} (| u_{c} (s) |)} \\ + \frac{e^{β}}{1 - e^{- δ_{1} ξ}} max_{t_{k} \in (t_{0}, t]} {e^{σ (t_{k} - t_{0})} χ_{2} (| u_{d} (t_{k}^{-}) |)} \end{matrix}

(26)

So, we get equation (4), then the system (1) is e^σt-p-ISS, where $σ \in (0, δ)$ .

In summary, we proof that the system (1) is p-ISS, p-iISS, and e^σt-p-ISS, where $σ \in (0, δ)$ . In addition, we can find that the ADT condition is redundant for $0 < η \leq 1$ . End of this proof.

Remark 1

The condition $(H_{1})$ and $(H_{2})$ in Theorem 1 indicates that if the system (1) without impulse, then it can be ISS, the function of the impulse at this time is to keep ISS of the system. When the impulse parameter $η \in (0, 1)$ , the impulsive signal of the system (1) is stabilizing, when $η \in (1, + \infty)$ , the impulsive signal is destabilizing. In both cases, we need conditions $(H_{3})$ and $(H_{4})$ to stabilize the system. In addition, only the case of $η \in [1, + \infty)$ is considered in Peng and Deng (2017). Therefore, we consider more comprehensively in this paper, and the uniform criterion that the system is ISS under different impulse disturbances is given.

Remark 2

In condition $(H_{2})$ of Theorem 1, if there exist constants $κ_{1} \geq 0, κ_{2} \geq 0$ satisfy $\int_{s}^{t} λ (v) dv \leq - κ_{1} (t - s) + κ_{2}$ , $\forall t > s \geq t_{0}$ , then we have

\begin{matrix} \int_{s}^{t} (\frac{\ln \bar{η}}{T_{a}} + λ (v)) dv = \frac{\ln \bar{η}}{T_{a}} (t - s) + \int_{s}^{t} λ (v) dv \\ \leq κ_{2} - (κ_{1} - \frac{\ln \bar{η}}{T_{a}}) (t - s) \end{matrix}

By $(H_{4})$ , we just have to satisfy $δ = κ_{1} - \ln \bar{η} / T_{a} > 0$ . This indicates if ISFDS with Markovian switching is ISS and the impulsive disturbance is destabilizing $(η > 1)$ , then the system (1) is ISS with respect to a lower bound of $T_{a}$ .

Remark 3

If the input state of system (1) is 0, that is, $u_{c} = 0$ , $u_{d} = 0$ , then system (1) is pth moment asymptotically exponentially stable. In Fu and Zhu (2018), the paper discussed p-th moment globally asymptotic stable in probability and p-th moment stochastic input-to-state stable for SDEs with Markovian switching, the system does not consider time delay. However, we considered the ISFDS with Markovian switching, obviously the system we are considering is more detailed and more applicable.

Remark 4

The input state of the differential equation and the input state of the impulse in the system considered in this paper are different, which makes the system more general. The discussion in Wu et al. (2016) is only a special case of this paper, when the input state of the differential equation and the impulse input state are the same. In addition, we also consider Markovian switching in system, which is not considered in Peng and Deng (2017) and Wu et al. (2016).

Remark 5

Under conditions of Theorem 1, if $ν_{1} (r) = α_{1} \cdot r$ , $ν_{2} (r) = α_{2} \cdot r$ , $α_{1}, α_{2}$ are the two positive numbers, and $u_{c} = u_{d} = 0$ , then by equations (20) and (25), we get the system (1) is pth moment exponentially stable, and the condition $(H_{4})$ of $ξ = inf_{k \in R_{+}} {t_{k} - t_{k - 1}} > 0$ can be omitted.

Remark 6

In system (1), if $g (t, x (t), x_{t}, r (t), u_{c} (t)) = 0$ , the system has been analyzed in Wang et al. (2013), but when it discussed that the system is ISS, it required the linear coefficient of the upper bound of the Lyapunov functional derivative to be a negative constant. In the discussion of this paper, the sufficient conditions of Theorem 1 only require $λ (t)$ is a time-varying function, this includes the situation in Wang et al. (2013). The condition in Wang et al. (2013) is more strict and its applicability is relatively limited. Therefore, the applicability of this paper is more extensive.

Remark 7

Consider the system without Markovian switching in the system (1)

{\begin{matrix} dx (t) = f (t, x (t), x_{t}, u_{c} (t)) dt + g (t, x (t), x_{t}, u_{c} (t)) dw (t), \\ t \geq t_{0}, t \neq t_{k} \\ Δ x (t_{k}) = x (t_{k}) - x (t_{k}^{-}) = I_{k} (t_{k}^{-}, x ({t_{k}}^{-}), u_{d} ({t_{k}}^{-})), t = t_{k}, k \in Z_{+} \\ x (t_{0} + θ) = ϕ (θ), - τ \leq θ \leq 0 \end{matrix}

(27)

The system (27) is the same as the system discussed in Peng and Deng (2017). In Peng and Deng (2017), only the sufficient condition that the system is ISS under the influence of the destabilizing impulse is analyzed, and the situation when the impulse is a stable disturbance is ignored. In this paper, the unified criterion of ISS is given under the disturbance of stabilizing impulse or destabilizing impulse. In equation (27), Markovian switching is not considered. When there is no Markovian switching, the stability of the system may not be as good as that of the original system (1), this also shows that the Markovian switching can be make the system more stable.

Theorem 2

Assume that there are functions $V (t, x_{t}, r (t)) = V_{1} (t, x (t), r (t)) + V_{2} (t, x_{t}, r (t))$ , and $V_{1} \in C^{2, 1} ([t_{0}, \infty) \times R^{n} \times S; R^{+})$ , $V_{2} \in C^{2, 1} ([t_{0}, \infty) \times P C_{F_{t}}^{b} ([- τ, 0]; R^{n}) \times S; R^{+})$ and for any $x \in PC ([t_{0} - τ, \infty; R^{n})$ , $E V_{2} (t, x_{t}, r (t))$ is continuous for $t \geq t_{0}$ , and the first derivative of the function $V$ with respect to $t$ is continuous. $χ_{1}, χ_{2} \in K_{\infty}$ , $ν_{1} \in V K_{\infty}$ , $ν_{2} \in C K_{\infty}$ , $λ (t) \in PC ([t_{0}, \infty); R)$ , and constants $p > 0$ , $σ > 0$ , $δ > 0$ , $b_{k} > 0$ , $M \geq 1$ such that

\begin{matrix} (H_{1}) ν_{1} (| x (t) |^{p}) \leq V (t, x_{t}, r (t)) \leq ν_{2} ({‖ x_{t} ‖}^{p}), t \geq t_{0}, x \in R^{n}, \\ x_{t} \in {PC}_{F_{t}}^{b} ([- τ, 0]; R^{n}), r (t) \in S \\ (H_{2}) E L V (t, x_{t}, r (t)) \leq λ (t) E V (t, x_{t}, r (t)) + χ_{1} (| u_{c} (t) |) \\ for all t \in [t_{k - 1}, t_{k}), k \in Z_{+} \\ (H_{3}) E V_{1} (t_{k}, x (t_{k}), r (t_{k})) \leq b_{k} E V_{1} (t_{k}^{-}, x (t_{k}^{-}), r (t_{k}^{-})) \\ + χ_{2} (| u_{d} (t_{k}^{-}) |), k \in Z_{+}, r (t_{k}), r (t_{k}^{-}) \in S \\ (H_{4}) \underset{s < t_{k} < t}{Π} {\bar{b}}_{k} e^{\int_{s}^{t} λ (s) ds} \leq M e^{- δ (t - s)}, {\bar{b}}_{k} = max {1, b_{k}} \end{matrix}

Then, the system (1) is p-ISS, p-iISS, and e^σt-p-ISS, where $σ \in (0, δ)$ .

Proof

Let $V (t, x_{t}, r (t)) = V (t)$ , $V_{1} (t, x (t), r (t)) = V_{1} (t)$ , and $V_{2} (t, x_{t}, r (t)) = V_{2} (t)$ , $r (t) \in S$ . By equation (6) and $(H_{2})$ , we get

D^{+} E V (t) \leq λ (t) E V (t) + χ_{1} (| u_{c} (t) |)

(28)

Due to for any $x \in PC ([t_{0} - τ, \infty); R^{n})$ , $E V_{2} (t, x_{t}, r (t))$ is continuous for $t \geq t_{0}$ , we have

\begin{matrix} E V (t_{k}) - E V (t_{k}^{-}) = E V_{1} (t_{k}) + E V_{2} (t_{k}) - E V_{1} (t_{k}^{-}) \\ - E V_{2} (t_{k}^{-}) = E V_{1} (t_{k}) - E V_{1} (t_{k}^{-}) \end{matrix}

So, by $(H_{3})$ , we get

\begin{matrix} E V (t_{k}) = E V (t_{k}^{-}) + E V_{1} (t_{k}) - E V_{1} (t_{k}^{-}) \\ \leq E V (t_{k}^{-}) + (b_{k} - 1) E V_{1} (t_{k}^{-}) + χ_{2} (| u_{d} (t_{k}^{-}) |) \end{matrix}

If $0 < b_{k} \leq 1$ , then

E V (t_{k}) \leq E V (t_{k}^{-}) + χ_{2} (| u_{d} (t_{k}^{-}) |)

If $b_{k} > 1$ , then

\begin{matrix} E V (t_{k}) \leq E V (t_{k}^{-}) + (b_{k} - 1) [E V_{1} (t_{k}^{-}) + E V_{2} (t_{k}^{-})] + χ_{2} (| u_{d} (t_{k}^{-}) |) \\ = b_{k} E V (t_{k}^{-}) + χ_{2} (| u_{d} (t_{k}^{-}) |) \end{matrix}

Therefore

E V (t_{k}) \leq {\bar{b}}_{k} E V (t_{k}^{-}) + χ_{2} (| u_{d} (t_{k}^{-}) |)

(29)

By equations (28) and (29), Lemma 1 and $(H_{4})$ , we obtain

\begin{matrix} E V (t) \leq ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) \underset{t_{0} < t_{k} < t}{Π} {\bar{b}}_{k} e^{\int_{t_{0}}^{t} λ (s) ds} \\ + \int_{t_{0}}^{t} \underset{s < t_{k} < t}{Π} {\bar{b}}_{k} e^{\int_{s}^{t} λ (σ) d σ} χ_{1} (| u_{c} (s) |) ds \\ + \sum_{t_{0} < t_{k} < t} (\underset{t_{k} < t_{j} < t}{Π} {\bar{b}}_{j} e^{\int_{t_{k}}^{t} λ (s) ds}) χ_{2} (| u_{d} (t_{k}^{-}) |) \\ \leq M ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ (t - t_{0})} + M \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) e^{- δ (t - s)} ds \\ + M \sum_{t_{0} < t_{k} < t} χ_{2} (| u_{d} (t_{k}^{-}) |) e^{- δ (t - t_{k})} \\ \leq M ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ (t - t_{0})} + M \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) e^{- δ (t - s)} ds \\ + M \sum_{t_{0} < t_{k} \leq t} χ_{2} (| u_{d} (t_{k}^{-}) |) e^{- δ (t - t_{k})}, t \geq t_{0} \end{matrix}

(30)

From $(H_{1})$ , we obtain

\begin{matrix} ν_{1} (E (| x (t) |^{p})) \leq E V (t) \leq M ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ (t - t_{0})} \\ + M \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) ds \\ + M \sum_{t_{0} < t_{k} \leq t} χ_{2} (| u_{d} (t_{k}^{-}) |) \end{matrix}

(31)

So, we get equation (3), then the system (1) is p-iISS.

From equation (30) and $(H_{1})$ , we also get

\begin{matrix} ν_{1} (E (| x (t) |^{p})) \leq M ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ (t - t_{0})} + \frac{M}{δ} χ_{1} ({‖ u_{c} ‖}_{[t_{0}, t]}) \\ (1 - e^{- δ (t - t_{0})}) \\ + \sum_{t_{0} < t_{k} \leq t} M e^{- δ (t - t_{k})} χ_{2} (max_{t_{i} \in (t_{0}, t]} {| u_{d} (t_{i}^{-}) |}) \\ \leq M ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ (t - t_{0})} + \frac{M}{δ} χ_{1} ({‖ u_{c} ‖}_{[t_{0}, t]}) \\ + \frac{M}{1 - e^{- δ ξ}} χ_{2} (max_{t_{i} \in (t_{0}, t]} {| u_{d} (t_{i}^{-}) |}) \end{matrix}

(32)

Let $ϖ (r, s) = ν_{1}^{- 1} (3 M ν_{2} (r) e^{- δ s})$ , $γ_{c} (r) = ν_{1}^{- 1} ((3 M / δ) χ_{1} (r))$ , and $γ_{d} (r) = ν_{1}^{- 1} ((3 M / 1 - e^{- δ ξ}) χ_{2} (r))$ . So, we get equation (2), then the system (1) is p-ISS.

Let $δ = δ_{1} + σ$ , where $δ_{1}$ and $σ$ are the positive constants, by equation (30) and $(H_{1})$ , we have

\begin{matrix} e^{σ (t - t_{0})} ν_{1} (E (| x (t) |^{p})) \leq M ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) e^{- δ_{1} (t - t_{0})} \\ + M \int_{t_{0}}^{t} χ_{1} (| u_{c} (s) |) e^{- δ_{1} (t - s)} e^{σ (s - t_{0})} ds \\ + M \sum_{t_{0} < t_{k} \leq t} e^{σ (t_{k} - t_{0})} χ_{2} (| u_{d} (t_{k}^{-}) |) e^{- δ_{1} (t - t_{k})} \\ \leq M ν_{2} (‖ ϕ ‖_{L^{p}}^{p}) + \frac{M}{δ_{1}} sup_{t_{0} \leq s \leq t} {e^{σ (s - t_{0})} χ_{1} (| u_{c} (s) |)} \\ + \frac{M}{1 - e^{- δ_{1} ξ}} max_{t_{i} \in (t_{0}, t]} {e^{σ (t_{i} - t_{0})} χ_{2} (| u_{d} (t_{i}^{-}) |)} \end{matrix}

(33)

So, we get equation (4), then the system (1) is e^σt-p-ISS, where $σ \in (0, δ)$ .

By equations (31)–(33), the proof is completed.

Remark 8

In the condition $(H_{3})$ of Theorem 2, the linear coefficient of the upper bound of the Lyapunov function at the impulse point is required to be the sequence $b_{k}$ . When the sequence $b_{k}$ is always a constant, it happens to be the situation analyzed in the previous theorem; this generalizes the results of Theorem 1. In addition, by applying the conclusion of impulsive differential inequality, $b_{k}$ and $λ (t)$ are restricted in condition $(H_{4})$ without considering the impulse ADT condition, indicating that the system (1) is ISS.

An example

Consider the following ISFDS with Markovian switching

{\begin{matrix} d x (t) = F (t, x (t), x (t - τ), r (t), u_{c} (t)) d t \\ + G (t, x (t - τ), r (t), u_{c} (t)) d w (t), t \geq t_{0}, t = t_{k} \\ Δ x (t_{k}) = x (t_{k}) - x (t_{k}^{-}) = σ x (t_{k}^{-}) + u_{d} (t_{k}^{-}), k \in ℤ_{+} \\ x (t_{0} + θ) = ϕ (θ), - τ \leq θ \leq 0 \end{matrix}

(34)

where $w (t) = (w_{1} (t), w_{2} (t))^{T}, x (t) = (x_{1} (t), x_{2} (t))^{T}$ , $r (t)$ is a Markovian chain takes values in $S = {1, 2}$ with generator

T = {(γ_{ij})}_{2 \times 2} = [\begin{matrix} - 2 & 2 \\ 1 & - 1 \end{matrix}]

Let $t_{0} = 0$ , $τ = 0.25$ , $u_{c} (t) = 1 + \sin t$ , $u_{d} (t) = (\sin t, \cos t)^{T}$ , $ϕ (θ) = (- 5, 8)^{T}$ , $- 0.25 \leq θ \leq 0$

\begin{matrix} F_{1} (t, x (t), x (t - τ), 1, u_{c} (t)) \\ = [\begin{matrix} \frac{(- | \sin t | \int_{t - τ}^{t} | x (s) |^{2} ds) x_{1}}{2 (x_{1}^{2} + x_{2}^{2})} - (1 + | \sin t |) x_{1} - 2 | \cos t | x_{2} \\ + | \sin t | u_{c}^{3} (t) \\ \frac{(- | \sin t | \int_{t - τ}^{t} | x (s) |^{2} ds) x_{2}}{2 (x_{1}^{2} + x_{2}^{2})} - (1 + | \sin t |) x_{2} + 2 | \cos t | x_{1} \\ + | \sin t | u_{c} (t) \end{matrix}] \\ G_{1} (t, x (t - τ), 1, u_{c} (t)) = [\begin{matrix} y_{1} & - y_{2} \\ y_{2} & y_{1} \end{matrix}] \end{matrix}

and $y (t) = (y_{1} (t), y_{2} (t))^{T} = (x_{1} (t - τ), x_{2} (t - τ))^{T}$

\begin{matrix} F_{2} (t, x (t), x (t - τ), 2, u_{c} (t)) \\ = [\begin{matrix} \frac{(- | \sin t | \int_{t - τ}^{t} | x (s) |^{2} ds) x_{1}}{2 (x_{1}^{2} + x_{2}^{2})} - (4 + | \sin t |) x_{1} + 2 | \cos t | x_{2} \\ + | \sin t | u_{c}^{3} (t) \\ \frac{(- | \sin t | \int_{t - τ}^{t} | x (s) |^{2} ds) x_{2}}{2 (x_{1}^{2} + x_{2}^{2})} - (4 + | \sin t) | x_{2} - 2 | \cos t | x_{1} \\ + | \sin t | u_{c} (t) \end{matrix}] \\ G_{2} (t, x (t - τ), 2, u_{c} (t)) = [\begin{matrix} 2 y_{1} & 2 y_{2} \\ - 2 y_{2} & 2 y_{1} \end{matrix}] \end{matrix}

Let $V (t, x_{t}, i) = x^{T} x + k_{i} \int_{t - τ}^{t} | x (s) |^{2} ds, i = 1, 2,$ where $k_{1} = 2, k_{2} = 8$ and $V_{1} (t, x, i) = x^{T} x$ , $V_{2} (t, x_{t}, i) = k_{i} \int_{t - τ}^{t} | x (s) |^{2} ds$ , $i = 1, 2$ . $ν_{1} (r) = ν_{2} (r) = (1 + τ) r$ . So, this obviously satisfies condition $(H_{1})$ of Theorem 1. In addition

\begin{matrix} E L V (t, x_{t}, i) = k_{i} E (| x |^{2}) - k_{i} E (| y |^{2}) + 2 E (x^{T} F_{i}) + E [trace (G_{i}^{T} G_{i})] + \sum_{j = 1}^{N} γ_{ij} E V (t, x, j) \\ = E [(- | \sin t | \int_{t - τ}^{t} x (s) |^{2} ds) - 2 | \sin t | | x |^{2} + 2 x_{1} | \sin t | u_{c}^{3} (t) + 2 x_{2} | \sin t | u_{c} (t)] \\ \leq E [(- | \sin t | \int_{t - τ}^{t} x (s) |^{2} ds) - 2 | \sin t | | x |^{2} + | \sin t (| x_{1}^{2} + u_{c}^{6} (t)) + | \sin t | (x_{2}^{2} + u_{c}^{2} (t))] \\ \leq E [(- | \sin t | \int_{t - τ}^{t} x (s) |^{2} ds) - | \sin t | | x |^{2}] + u_{c}^{6} (t) + u_{c}^{2} (t) \\ \leq (\frac{1}{2} | \cot t | - | \sin t) | E [| x |^{2} + \int_{t - τ}^{t} | x (s) |^{2} ds]] + u_{c}^{6} (t) + u_{c}^{2} (t) \end{matrix}

Let $λ (t) = (1 / 2) | \cot t | - | \sin t |$ , $χ_{1} (r) = r^{6} + r^{2}$ . So, the condition $(H_{2})$ of Theorem 1 holds. In addition

\int_{s}^{t} λ (v) dv \leq - \frac{1}{π} (t - s) + 2, \forall t > s \geq t_{0}

(35)

Let $η = 2 (1 + σ)^{2}$ , $χ_{2} (r) = 2 r^{2},$ then $χ_{2} (u_{d} (t)) = 2$

\begin{matrix} E V_{1} (t_{k}, x (t_{k}), r (t_{k})) = E (| x (t_{k}) |^{2}) \\ = E (| (1 + σ) x (t_{k}^{-}) + u_{d} (t_{k}^{-}) |^{2}) \\ \leq E (2 {(1 + σ)}^{2} | x (t_{k}^{-}) |^{2} + 2 | u_{d} (t_{k}^{-}) |^{2}) \\ = [2 {(1 + σ)}^{2}] E (| x (t_{k}^{-}) |^{2}) + 2 | u_{d} (t_{k}^{-}) |^{2} \\ = η E V_{1} (t_{k}^{-}, x (t_{k}^{-}), r (t_{k}^{-})) + χ_{2} (| u_{d} (t_{k}^{-}) |) \end{matrix}

So, the condition $(H_{3})$ of Theorem 1 holds. We give the impulsive sequences Lu et al. (2010) as

t_{k} - t_{k - 1} = {\begin{matrix} ϵ & mod (k, N_{0}) \neq 0 & (36 a) \\ N_{0} (T_{a} - ϵ) + ϵ & mod (k, N_{0}) = 0 & (36 b) \end{matrix}

where $T_{a} \geq ϵ > 0$ , and $inf_{k \in Z_{+}} {t_{k} - t_{k - 1}} = ϵ$ , $sup_{k \in Z_{+}} {t_{k} - t_{k - 1}} = N_{0} (T_{a} - ϵ) + ϵ$ . Let $ϵ = 0.3, T_{a} = 2.3$ , and $N_{0} = 4$ , this impulsive sequence is shown in Figure 1.

Case 1

$σ = 0.121, \bar{η} = η = 2.029282 \in (1, \infty)$ , Then, by equation (35), we get

\begin{matrix} \int_{s}^{t} \frac{\ln \bar{η}}{T_{a}} + λ (v) dv = \int_{s}^{t} \frac{\ln η}{T_{a}} + λ (v) dv \leq (\frac{\ln η}{T_{a}} - \frac{1}{π}) (t - s) \\ + 2 = 2 - (\frac{1}{π} - \frac{\ln η}{T_{a}}) (t - s) \end{matrix}

where $β = 2$ , $δ = (1 / π) - (\ln η / T_{a}) > 0$ , and $inf_{k \in Z_{+}} {t_{k} - t_{k - 1}} = ϵ = 0.3 > 0$ . So, the condition $(H_{4})$ of Theorem 1 holds. From the above, all the conditions of Theorem 1 are satisfied, so the system (34) is p-ISS, p-iISS, and e^σt-p-ISS, where $σ \in (0, δ)$ . Figure 2 gives the trajectory of the state $x (t)$ of system (34) with $σ = 0.121$ , it shows that system (34) is ISS when the system satisfies all the conditions of Theorem 1. Figure 3 shows the numerical simulation trajectory of state $r (t) = {1}$ with impulsive disturbance of Figure 1 when there is no Markovian switching in the system (34) with $σ = 0.121$ . Figure 4 shows the numerical simulation trajectory of state $r (t) = {2}$ with impulsive disturbance of Figure 1 when there is no Markovian switching in the system (34) with $σ = 0.121$ . When there is no Markovian switching in Figures 3 and Figure 4 and $σ = 0.121$ .

Case 2

$σ = - 0.6, η = 2 (1 + σ)^{2} = 0.32 \in (0, 1]$ . Then, $\bar{η} = 1$ . By equation (35), we get

\int_{s}^{t} \frac{\ln \bar{η}}{T_{a}} + λ (v) dv = \int_{s}^{t} λ (v) dv \leq - \frac{1}{π} (t - s) + 2

where $β = 2$ , $δ = (1 / π)$ . So, the condition $(H_{4})$ of Theorem 1 holds. From the above, all the conditions of Theorem 1 are satisfied, so the system (34) is p-ISS, p-iISS, and e^σt-p-ISS, where $σ \in (0, δ)$ . Figure 5 gives the trajectory of the state $x (t)$ of system (34) with $σ = - 0.6$ , it shows that system (34) is ISS when $0 < η \leq 1$ .

In summary, under the conditions of stabilizing impulse $(0 < η \leq 1)$ and destabilizing impulse $(η > 1)$ , the system can be ISS as long as condition $(H_{4})$ is satisfied.

Conclusion

A unified criterion for ISS of ISFDS with Markovian switching under the perturbation of the stabilizing impulse $(0 < η \leq 1)$ and destabilizing impulse $(η > 1)$ is considered, using Lyapunov function approach, comparison principle, and impulsive ADT method. The upper bound of the derivative of the Lyapunov function given in this paper is a time-varying function, which advances and improves the existing results. In addition, the same results were obtained by applying impulse differential inequality. Finally, an example is given to verify that the results are correct.

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 by the NNSF of China (12071105) and the Key Projects of Science Research in University of Anhui Province (KJ2021A1049).

ORCID iD

Jianli Li

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