Finite-time stability and applications of positive switched linear delayed impulsive systems

1.   Introduction

Positive systems represent dynamical systems whose states maintain nonnegative states on the condition that the original states are nonnegative ^[1,2]. Positive switched systems are complex dynamical systems that consist of several positive subsystems and a switching rule among them. Compared with switched systems, positive switched systems can model some complex systems, so it is of great practical value to study the stability and control problems of positive switched systems. It is commonly known that positive switched systems have a large number of applications in the areas of congestion control ^[3], chemical processes ^[4], aerospace engineering ^[5], multi-agent systems ^[6], etc. On the other hand, delays are frequently encountered in complex dynamical systems. Time delays bring a period of historical state information to the system dynamics; the solution states of the systems are not only affected by the current states, but also affected by the corresponding time delays. In other words, time delays physically represent the delays between the system's response and the external excitation. Time delays naturally occur in practical engineering systems, which are usually considered an important factor in degrading the performance of dynamic systems. Time delays affect the performance of the systems, which may make the systems unstable or even oscillate. Moreover, it is well known that even small delays may affect or even destroy the stability of systems, which make stability analysis difficult. The study of stability is always an important topic in control systems. Therefore, an increasing number of experts and scholars are taking notice of the stability of positive switched delayed systems ^[7,8,9,10]. For example, by constructing an appropriate co-positive Lyapunov-Krasovskii functional, Xiang and Xiang ^[7] investigated the exponential stability, $L_{1}$-gain performance and controller design problems for a class of positive switched systems with time-varying delay. Liu and Xiang ^[8] addressed the exponential $L_{1}$ output tracking control for positive switched linear systems with time-varying delays. Liu ^[9] studied the stability problem of delayed nonlinear positive switched systems. Sun ^[10] considered the stability of positive switched linear systems with time delay.

To our knowledge, the vast majority of existing results on the stability of positive switched systems concern Lyapunov stability, which indicates stability over an infinite time interval. It is well known that asymptotic stability can ensure that the stead-state performance of the systems is good, not necessarily have the corresponding transient performance, and may even make the transient performance of the systems very poor. For example, Zhao and Sun ^[11] addressed the absolute exponential stability for switching Lurie systems with time-varying delays using the switching time-varying Lyapunov function technique. Compared with Lyapunov stability, some systems do not need to stabilize over finite time interval but only need to stabilize over finite time interval. This stability of the system is named finite time stability (FTS), which means that given a bound on the original conditions, the system's solution states do not transcend a definite threshold during a concrete time range (see ^[12]). The transient performance of finite-time stability is better. Nevertheless, in some practical circumstances, some control systems usually do not need to be stable at infinite time; these control systems only need to be stable at a finite time interval. Finite-stability problems receive plenty of applications; see ^{[13,14,15,16,17,18]}. For example, Garcia et al. ^[13] studied the finite-time stabilization of linear time-varying continuous systems by Lyapunov differential matrix equations. Zhao et al. ^[14] considered the finite-time stability of linear time-varying singular systems with impulsive effects. Wang et al. ^[15] investigated the input-output finite-time stability for a class of networked control systems with network-induced delay. By a singular value decomposition approach, Thanh et al. ^[16] studied the finite-time stability of singular nonlinear switched time-delay systems. Wei et al. ^[17] studied the finite-time stability of linear discrete switched singular systems with finite-time unstable subsystems. Zhang et al. ^[18] studied the finite-time stability and stabilization of linear discrete time-varying stochastic systems with multiplicative noise.

There are some meaningful FTS results for switched systems ^{[19,20,21,22,23,24,25]}. Hou et al. ^[19] provided a sufficient FTS condition for switched linear systems with a new multiple Lyapunov functional. Xiang and Xiao ^[20] proposed a number of sufficient finite-time boundness and stability conditions as switched linear systems. Liu et al. ^[21] regarded the FTS for positive switched linear delayed systems (PSLDSs) with piecewise co-positive Lyapunov–Krasovskii functional. With the help of a time-varying co-positive Lyapunov functional, Chen and Yang ^[22] investigated the FTS as positive switched linear systems. After that, Xu et al. ^[23] established the FTS conditions of positive switched linear delayed systems by Lyapunov–Krosovskii functional. Zhang and Zhu ^[24] extended the finite-time input-to-state stability to switched stochastic time-varying nonlinear systems with time delays by the Razumikhin theorem, comparison principle, and average dwell-time approach. Recently, by using a type of linear time-varying co-positive Lyapunov functional, Huang et al. dealt with the FTS of a positive switched linear delayed system (PSLDS) as follows ^[25]:

where $z(t)\in R^{n}$ represents the state vector, $\delta(t)$: $[0, \infty)\rightarrow \{1, 2, \cdots, q\}$ expresses the switching rule, which implies a piecewise continuous function, $q > 1$ expresses an integer, $M_{l}$ and $N_{l}$ symbolize system matrices of the $l$th subsystem for $l\in\{1, 2, \cdots, q\}$, $\rho > 0$ stands for time delay, $\nu(t)$: $[-\rho, 0]\rightarrow R^{n}$ shows the continuous vector valued original function.

However, in some practical applications, impulsive interference is inevitable. Impulsive behaviors are regarded as a dynamical course that expresses a state that changes abruptly at some time points ^[26,27,28]. Nowadays, numerous researchers focus on the stability of positive systems with impulsive interference, and several significant results have emerged in ^[29,30,31]. Specifically, by constructing a time-varying co-positive Lyapunov function and utilizing the average impulsive interval approach, Hu et al. ^[29] investigated the finite-time stability and stabilization problems of positive systems with impulses. Hu et al. ^[30] addressed the exponential stability and positive stabilization problems of impulsive positive systems with time delay. Based on the linear co-positive Lyapunov function method, Briat ^[31] obtained the stability and stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum, and range dwell-time. Therefore, it's beneficial to study the switched systems with impulsive effects.

Then, inspired by the method of ^[25], we will further study the FTS for a PSLDS with impulsive effects as follows:

where $g_{l}(z)$: $R^{n}\rightarrow R^{n}$ are impulses for $l\in\{1, 2, \cdots, q\}$, $t_{r}$, $r = 1, 2, \cdots$, those are not just the switching instants but the impulsive instants, and satisfying $0 < t_{r} < t_{r+1}$ and

When $t\in[t_{r}, t_{r+1})$, the $\delta(t_{r})$th subsystem is motivated, $r = 0, 1, 2, \cdots$. At the switching and impulsive instants, let $z(t) = z(t^{+})$ at $t = t_{r}$.

The differences between this paper and the related results lie mainly in the following two aspects: First, we introduce the impulsive effects of PSLDS. Next, we adjust the linear term coefficient of the linear vector function and add the vector constraints at every impulsive instant. The primary contributions are highlighted as follows:

(1) Compared with the existing results ^{[21,22,23,25]}, impulsive effects are introduced to the positive switched systems. By constructing a linear time-varying co-positive Lyapunov functional, we deduce several new finite-time stability criteria for the models in Huang et al. ^[25] under synchronous impulse control.

(2) Inspired by Chen and Yang ^[22] and Huang et al. ^[25], a linear time-varying co-positive Lyapunov functional is established. It should be emphasized that the linear term coefficient of the linear vector of the Lyapunov functional is adjusted to the difference between the weighting vector and the given vector. Then, at every impulsive and switching instant, we use impulsive vector constraints to replace the vector constraints at every switching instant.

(3) We apply the obtained results to the finite-time stability of linear time-varying systems with impulses. We design new linear programming algorithm using Lingo software to better demonstrate the feasibility of our results, and our programming process is relatively easy.

The framework of this paper is organized as follows. Indispensable definitions and preliminaries for this paper are provided in Section 2. Section 3 is dedicated to proving the main FTS criteria. In Section 4, we apply the obtained FTS criteria to the linear time-varying delayed systems with impulsive effects. Three examples are given in Section 5 to support our main theoretical results. A conclusion and some future directions are discussed in Section 6.

2.   Preliminaries

Some necessary notations or symbols are given here. $R^{n}$ represents the group of $n$-dimensional real vector spaces. $R^{n\times n}$ represents $n\times n$-dimensional real matrices space. Two vectors $p, q\in R^{n}$, $p\succ0$ implies its entry $p_{l}\succ0$ for $l\in\{1, 2, \cdots, n\}$, $p\succeq q$ (or $q\preceq p$) if $p_{l}\succeq q_{l}$ (or $p_{l}\preceq q_{l}$) for $l\in\{1, 2, \cdots, n\}$. $p\in R^{n}$ expresses a positive vector, provided that $p\succ0$. A vector $p\in R^{n}$ and a matrix $M\in R^{n\times n}$, $p^{\top}$ and $M^{\top}$ characterize the transpose of $p$ and $M$, respectively. Provided that all off-diagonal elements of matrix $N$ are nonnegative, then $N$ is called a Metzler matrix. If all elements of matrix $C$ are nonnegative, then $C$ is called positive.

Set $t_{0} = 0$ and $t_{q} = S$. The switching and impulsive moments are $0 < t_{1} < \cdot\cdot\cdot < t_{r} < \cdot\cdot\cdot < t_{q-1} < S$, where $q > 1$ is an integer.

The next definitions, assumptions, and lemmas are important for the formation of the crucial FTS criteria.

Definition 2.1. (^[25]) Provided that $\nu(t)\succeq0, t\in[-\rho, 0]$, for any switching signal $\delta(t)$, the state solution trajectory $z(t)$ fulfills $z(t)\succeq0$, for any $t\geq0$, then PSLDS (1.2) is positive.

Definition 2.2. (^[25]) Given positive scalars $S$, $b_{1} < b_{2}$ and a positive vector $p\in R^{n}$, PSLDS (1.2) is called finite-time stable about $(S, p, b_{1}, b_{2})$ provided that

for $t\in[0, S]$.

Definition 2.3. (^[25]) $N_{\delta}(0, t)$ defines the number of switching points in the time interval $(0, S)$. Provided that there is a positive number $T_{b} > 0$ in order that

holds, $T_{b}$ represents the average dwell time (ADT) of $\delta(t)$ in the time interval $[0, S]$.

An assumption is proposed as follows:

$(H_{1})$ Impulsive function $g_{fl}(z)\succeq0$ for $z\succeq0$, and there exist a group of positive matrices $D_{l}\in R^{n\times n}$, satisfying $g_{fl}(z)\preceq D_{l}z$ for $z\in R^{n}$, where $l, f \in\{1, 2, \cdots, q\}$, $\sigma(t^{+}_{r}) = f$, $\sigma(t^{-}_{r}) = l$, $l\neq f$.

Lemma 2.1. (^[25]) PSLDS (1.2) is positive if and only if $M_{l}$ represents a Metzler matrix and $N_{l}\succeq0$, for any $l\in\{1, 2, \cdots, q\}$.

3.   Main results

Firstly, we propose FTS criteria for PSLDS (1.2) under arbitrary switching.

Theorem 3.1. Suppose that $(H_{1})$ holds. Given positive scalars $S$, $b_{1} < b_{2}$, a positive vector $p\in R^{n}$, PSLDS (1.2) is finite-time stable about $(S, p, b_{1}, b_{2})$ under arbitrary switching, provided that there are a positive vector $\psi\in R^{n}$, a free weighting vector $\chi\in R^{n}$, parameters $\delta_{1} > 0$, $\delta_{2} > 0$, and $\omega\geq1$, in order to satisfy the following inequalities:

hold, where $l, f\in\{1, 2, \cdots, q\}$.

Proof. For any $\nu(t)\succeq0$, $t\in[-\rho, 0]$, it is obvious that $z(t)\succeq0$, $t\geq0$. Construct the linear time-varying co-positive Lyapunov functional as follows:

where

for $t\geq S$, and $\delta(S^{-})$ characterizes the left limitation of $\delta(t)$ at $t = S$. For any $t\in[0, S]$, let

which are related to $t$.

Taking the derivative of $V(t, z(t))$ about $t$ along the solution trajectory of PSLDS (1.2), we obtain

where

Then, we prove $P_{lf}(t)\preceq0$ for $l, f\in\{1, 2, \cdots, q\}$ and $t\in[0, S]$. Because

in other words, $P_{lf}(t)$ is monotone with $t$ in the time interval $[0, S]$, for $l, f\in\{1, 2, \cdots, q\}$. Furthermore, conditions (3.1) and (3.2) show that $P_{lf}(0)\preceq0$ and $P_{lf}(S)\preceq0$, respectively. After that, it concludes from the monotonicity of $P_{lf}(t)$ that $P_{lf}(t)\preceq0$, for $l, f\in\{1, 2, \cdots, q\}$ and $t\in[0, S]$, so $\dot{V}(t, z)\leq0$ for $t\in[0, S]$. Condition (3.5) indicates that

for $t\in[0, \rho]$ and $l\in\{1, 2, \cdots, q\}$. Furthermore, condition (3.5) also shows that

At every switching and impulsive instant $t_{r}$, due to the definition of $V(t, z(t))$, we acquire

By condition (3.6), we obtain

Therefore,

From (3.9), we gain

where $t = t_{r}, \ r = 0, 1, 2, 3, \cdots, q.$ By (3.10), we get

where $N(0, S)$ expresses the number of impulses in time interval $(0, S)$.

(ⅰ) At every switching and impulsive instant $t_{r}$, we obtain

If

then

Since (3.5), (3.8) and (3.11), we have

Thus, if

we conclude from (3.12) to (3.13) that

(ⅱ) When $t\in[0, S]$, $t\neq t_{r}$, conditions (3.3) and (3.4) indicate that $\eta^{\top}(t)\succeq p^{\top}$, for $t\in[0, S]$, $t\neq t_{r}$ and

for $t\in[0, S+\rho]$, $t\neq t_{r}$ and $l\in\{1, 2, \cdots, q\}$. Accordingly, provided that

from (3.7) and (3.8), we get the conclusion that

Consequently, PSLDS (1.2) is finite-time stable about $(S, p, b_{1}, b_{2})$. □

Remark 3.1. In the course of proving Theorem 3.1, compared to ^[25], we define that the linear term coefficient of the linear vector function $\eta(t)$ is about $\psi$ and the weighting vector $\chi$ (not required $\chi\succ0$), which makes the results more accurate.

Remark 3.2. Condition (3.6) is a vector constraint at every impulsive instant. If there exists a common positive vector $\psi$ and a free weighting vector $\chi$, the conditions (3.1)–(3.5) hold. If we cannot find the common vectors $\psi$ and $\chi$, we further construct multiple time-varying linear co-positive Lyapunov functionals to deduce the next FTS criteria for PSLDS (1.2).

Second, we will further investigate the FTS of PSLDS (1.2) under ADT switching.

Theorem 3.2. Suppose that $(H_{1})$ holds. Given positive scalars $S$, $b_{1} < b_{2}$, and a positive vector $p\in R^{n}$, PSLDS (1.2) is finite-time stable about $(S, p, b_{1}, b_{2})$ under arbitrary switching, if there exist positive vectors $\psi_{l}\in R^{n}$, free vectors $\chi_{l}\in R^{n}$, parameters $\omega\geq1$, $\varepsilon\in R$, $\delta_{1} > 0$, $\delta_{2} > 0$ in order to satisfy the following inequalities:

hold, where $l, f\in\{1, 2, \cdots, q\}$, $\varepsilon_{+} = \max\{\varepsilon, 0\}$ and

when $\varepsilon = 0$.

Proof. For $\nu(t)\succeq0$, $t\in[-\rho, 0]$, we get that $z(t)\succeq0$ for $t\geq0$. Construct a piecewise linear time-varying co-positive Lyapunov functional as follows:

$t\in[t_{r}, t_{r+1})$, where

$t\in[0, S+\rho]$, and

for $t\in[S, S+\rho]$. The proof will be separated into the next three steps.

(Ⅰ). For any $t\in[t_{r}, t_{r+1})$, $r = 0, 1, \cdots, q-1$, we prove that

Denote $\delta(t) = f$, when $t\in[t_{r}, t_{r+1})$, $r = 0, 1, \cdots, q-1$. Then, the derivative of $V_{f}(t, z(t))$ about $t\in[t_{r}, t_{r+1})$ along the solution trajectory of PSLDS (1.2) shows

Therefore, we get:

where

Since

then $Q_{f}(t)$ is monotone over an interval $[0, S]$, it concludes that $Q_{f}(t)\leq0$ for $t\in[0, S]$ if and only if $Q_{f}(0)\leq0$ and $Q_{f}(S)\leq0$. After that, conditions (3.14) and (3.15) indicate that $Q_{f}(t)\leq0$, for $t\in[t_{r}, t_{r+1})$, and hence

Thus, inequality (3.21) is held.

(Ⅱ). We indicate that for any switching signal $\delta(t)$,

holds, where $r = 1, 2, \cdots, q-1$. At every switching and impulsive instant $t_{r}$, on the basis of the definition of $V(t, z(t))$, we obtain that

From condition (3.16), we obtain

where $\delta(t^{+}_{l-1}) = l, \; \delta(t^{-}_{l-1}) = l-1.$ Accordingly,

From (3.23), we get

(Ⅲ). According to (3.21) and (3.22), the following relations are established:

(1) When $t\in[t_{0}, t_{1})$,

(2) When $t = t_{1}$,

(3) When $t\in[t_{1}, t_{2})$,

(4) When $t = t_{2}$,

By repeating the above process, we obtain

Owing to

and $\omega\geq1$, it shows that

Conditions (3.17) and (3.18) indicate that

condition (3.19) demonstrates that

Combining the above inequality, (3.20), (3.24) and (3.25), we get that

for $t\in[0, S]$. Accordingly, PSLDS (1.2) is finite-time stable about $(S, p, b_{1}, b_{2})$. □

Remark 3.3. Condition (3.16) is a vector constraint at every impulsive instant. Compared to ^[25], we use impulsive constraints to replace the vector constraints at switching instants in proofing the process (Ⅱ).

Remark 3.4. In this section, the delays represent constant delays, and when the delays become time-varying delays, we will next consider establishing an appropriate Lyapunov function to obtain the FTS criteria.

4.   Applications in FTS of linear time-varying delayed systems

By the same approach, we investigate the linear time-varying delayed system with impulsive effects as follows:

where the definitions of $z$, $\rho$ and $\nu$ are similar to PSLDS (1.2),

and

represent piecewise continuous matrix functions. Separate the time interval $[0, S]$ into $Q$ subintervals $[t_{r-1}, t_{r}]$ for $r = 1, 2, \cdots, Q$, where

for $r = 0, 1, \cdots, Q$. We present the next two hypotheses:

($H_{2}$) There exist matrices

in order that

for $t\in[t_{r-1}, t_{r}]$, $r = \{1, 2, \cdots, Q\}$, $l, f\in\{1, 2, \cdots, q\}$, $l\neq f$.

($H_{3}$) There exist matrices

in order that

and

for $t\in[t_{r-1}, t_{r}]$, $r = \{1, 2, \cdots, Q\}$, $l, f\in\{1, 2, \cdots, q\}$, $l\neq f$.

Note that system (4.1) is not confined to being positive. For $t\in[t_{l-1}, t_{l})$, $l\in\{1, 2, \cdots, q\}$, construct the following time-varying Lyapunov functional:

where

Then, we get the next FTS conditions for system (4.1).

Theorem 4.1. Suppose that $(H_{1})$–$(H_{3})$ hold. Given positive scalars $S$, $b_{1} < b_{2}$ and a positive vector $p\in R^{n}$, system (4.1) is finite-time stable about $(S, p, b_{1}, b_{2})$. If there exist positive vectors $\psi_{l}\in R^{n}$, free vectors $\chi_{l}\in R^{n}$, parameters $\omega\geq1$, $\varepsilon\in R$, $\delta_{1} > 0$, $\delta_{2} > 0$, in order that the following inequalities:

hold, where $l\in\{1, 2, \cdots, q\}$, $\varepsilon_{+} = \max\{\varepsilon, 0\}$, and

when $\varepsilon = 0$.

Proof. The proof is separated into three steps:

(Ⅰ). For $t\in[t_{l-1}, t_{l})$, $l\in\{1, 2, \cdots, q\}$, we first demonstrate that

$V_{l}(t, z(t))$ represents the right derivative of $D^{+}V_{l}(t, z(t))$. By the $l$-th entry of the state $z(t)$, it concludes that

Because

if $z_{l}(t)\neq0$ and

if $z_{l}(t) = 0$, it shows

Therefore, we acquire

Conditions (4.2) and (4.3) indicate that

so inequality (4.10) holds for $\forall$ $t\in[t_{l-1}, t_{l})$.

(Ⅱ). At every impulsive instant $t_{l-1}$, from the definition of $V_{l}(t, z(t))$, we acquire

From condition (4.4), we get

where $\delta(t^{+}_{l-1}) = l, \; \delta(t^{-}_{l-1}) = l-1.$ Hence,

According to (4.11), we obtain

(Ⅲ). Combining (4.10) and (4.12), by mathematical induction, we obtain

where $t\in[t_{l-1}, t_{l}), l\in\{1, 2, \cdots, q\}.$ Since $\omega\geq1$, by conditions (4.5)–(4.9), we gain

for $t\in[0, S].$ Accordingly, system (4.1) is finite-time stable about $(S, p, b_{1}, b_{2})$. □

Remark 4.1. Condition (4.4) is a vector constraint at every impulsive instant. Compared to^[25], we add the impulsive vector constraints at the segmented time points in proofing the process (Ⅱ).

Remark 4.2. We regard the effect of the synchronous impulses on the stability of the systems in this paper, and change the conditions at the switching point of the synchronous impulses. In other words, we limit the impulses jump so that the impulses jump is not so large that the system is unstable. In fact, there may be an asynchronous impulses jump at the switching moment. For the case of asynchronous impulses jumping, the difficulty is how to limit the condition constraints under the influence of asynchronous impulses, which will be the next consideration.

Remark 4.3. Chen et al. ^[32] studied the mean square exponential stability analysis for itô stochastic systems with aperiodic sampling and multiple time delays by the Razumikhin-type theorems method and looped-functionals method. Chen et al. ^[33] considered the sampled-data synchronization of stochastic Markovian jump neural networks with time-varying delay by constructing a mode-dependent one-sided loop-based Lyapunov functional method. And Chen et al. ^[34] investigated the stability analysis and controller design issues for aperiodic sampled-data networked control systems with time-varying delays. The method in this paper cannot directly deduce the sampled-data control of the systems. The main difficulty is how to choose an appropriate Lyapunov functional to stabilize sampled-data control systems. We will next use another effective method to study the sampled-data synchronization of PSLDS with impulsive effects. Furthermore, we will study the controller synthesis of aperiodic sampled data of PSLDS with impulsive effects in the future.

5.   Simulation examples

The theoretical results are verified by three simulation examples.

Example 5.1. Consider PSLDS (1.2), where $\rho = 1$,

Impulsive matrices take the following form:

For given $p = (0.7, 0.3, 0.4)^{\top}$, $S = 50s$, $\varepsilon = 0$, $\omega = 1.01$, $b_{1} = 0.92$, and $b_{2} = 651$, by solving inequalities from (3.14) to (3.20), we can gain $\delta_{1} = 15$, $\delta_{2} = 16$, $T_{b} = 4.13$ s, and

Therefore, by applying Theorem 3.2, for any switching signal with ADT $T_{b} = 4.13$ s, PSLDS (1.2) is finite-time stable about $(S, p, b_{1}, b_{2})$.

Figure 1 depicts the switching signal $\delta(t)$ for PSLDS (1.2) designed as

Figure 2 implies that the impulsive signal $p(t)$ is designed as $p(t) = 1$, when $t = [2, 9, 14, 19, 23, 25, 29, 34, 39, 43, 47]$, where $p(t) = 1$ indicates that the impulse jumps occur at these moments.

Choosing the initial value $z(0) = (0.6, 1.1, 0.4)^{\top}$, Figure 3 signifies that each solution of PSLDS (1.2) is finite-time stable. What's more, Figure 4 fulfills the response of PSLDS (1.2) from $0$ to $50$ s. Furthermore, PSLDS (1.2) contains the impulses, so the results in ^[25] can-not be applied to the example. The solution and response states of PSLDS (1.2) are all discontinuous due to the impulse jumping effects.

Example 5.2. Let $z_{1}$, $z_{2}$, $z_{3}$ represent the positions of three vehicles, respectively, whose dynamics satisfy PSLDS (1.2), where $\rho = 2$,

Impulsive matrices take the following form:

We now discuss the positional relationship of three vehicles related to the hyperplane

with

For given $S = 90$ s, $\varepsilon = 0$, $\omega = 1.2$, $b_{1} = 1.3$, and $b_{2} = 10001$, by solving inequalities from (3.14) to (3.20), we can gain $\delta_{1} = 12.4$, $\delta_{2} = 13.5$, $T_{b} = 4.73$ s, and

Therefore, by applying Theorem 3.2, for any switching signal with ADT $T_{b} = 4.73$ s, PSLDS (1.2) is finite-time stable about $(S, p, b_{1}, b_{2})$.

Figure 5 depicts the switching signal $\delta(t)$ for PSLDS (1.2) designed as

Figure 6 implies the impulsive signal $p(t)$ is designed as $p(t) = 1$, when

where $p(t) = 1$ indicates that the impulses jumps occur at these moments.

Choosing the initial value

Figure 7 signifies that each solution of PSLDS (1.2) is finite-time stable.

What is more, Figure 8 fulfills the response of PSLDS (1.2) from $0$ to $90$ s. Furthermore, because PSLDS (1.2) contains impulses, the results in ^[25] are invalid. The solution and response states of PSLDS (1.2) are all discontinuous due to the impulse jumping effects.

Example 5.3. Consider system (4.1), where $\rho = 3$. The time interval is divided into three subintervals,

Choose $Q = 3$. According to assumptions $(H_{2})$ and $(H_{3})$, we have

and

Impulsive matrices take the following form:

For given

$S = 90$ s, $\varepsilon = 0$, $\omega = 1.25$, $b_{1} = 1.5$, $b_{2} = 90$, by solving conditions (4.2) to (4.9), we obtain $\delta_{1} = 11.5$, $\delta_{2} = 12.6$, and

Therefore, by Theorem 4.1, system (4.1) is finite-time stable about $(S, p, b_{1}, b_{2})$.

The impulsive signal is in Figure 9, which shows the impulsive signal $p(t) = 1$ when $t = [30, 60]$, where $p(t) = 1$ indicates that the impulse jumps occur at these moments. Choosing the original value

the solution trajectory and system response from $0$ to 90 s of system (4.1) are exhibited in Figures 10 and 11, respectively. What is more, there exist impulses for system (4.1), so the results in ^[25] are invalid. The solution and response states of system (4.1) are all discontinuous due to the impulse jumping effects.

We present Algorithms 1 and 2 to verify Theorem 3.2. Then a similar algorithm can be easily obtained for Theorem 4.1 in Section 4.

Remark 5.1. It is noticeable that the recent results in ^[25] can not be used for Examples 5.1–5.3 due to the impulses. To some extent, the proposed results in this paper are more effective than previous ones.

Remark 5.2. It should be emphasized that the designed linear programming process is relatively simple, which makes it easy to program and solve. By using Lingo software and setting the appropriate precision, we get a more accurate result.

Remark 5.3. For the convenience of simulations, we give small $b_{1}$ and large $b_{2}$, which can satisfy the inequality condition. In fact, the corresponding parameters of $b_{1}$ and $b_{2}$ are not unique; they are related to the parameters $\delta_{1}$ and $\delta_{2}$. Therefore, we can adjust the size of the parameters $b_{1}$ and $b_{2}$ to satisfy the condition, and the parameter $b_{2}$ is not sufficiently large in advance.

6.   Conclusions

In this paper, we add the impulsive effects of PSLDS. Then, we provide FTS conditions for the systems with an appropriate Lyapunov functional. After that, we apply the main FTS criteria to general linear time-varying delayed systems with impulsive effects. At last, we provide three examples to verify the results, which include the specific algorithms progress. And our programming process is simple and easy to verify.

There are many limitations to our work. It is worth noting that the impulsive and switching instants may not be synchronous. We deal with the impulsive effects of PSLDS in this paper, which are synchronous with switching instants. However, the impulsive signals may not be instantaneous with switching signals. Then, we will gradually explore the case of FTS for the systems (1.2) and (4.1) with asynchronous switching impulsive signals and non-instantaneous impulses in the future. The method in this paper cannot be directly extended to the asynchronous impulses case. The main difficulty is how to deal with the Lyapunov functional value estimation under the coupling effects of asynchronous switches and state jumps, which will be left for our future study.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors thank the reviewers for their valuable comments and suggestions. This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2021MA016).

Conflict of interest

All authors declare no conflicts of interest in this paper.