# Integral version of Lyapunov-Razumikhin conditions

Jian Wang, Denis Efimov, Alexander Aleksandrov

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Jian Wang, Denis Efimov, Alexander Aleksandrov. Integral version of Lyapunov-Razumikhin conditions. Automatica, In press. hal-05225551

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Submitted on 27 Aug 2025

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# Integral version of Lyapunov-Razumikhin conditions

Jian Wang<sup>a</sup>, Denis Efimov<sup>b,c</sup>, Alexander Aleksandrov<sup>d,e</sup>

<sup>a</sup>Hangzhou Dianzi University, Hangzhou, China

<sup>b</sup>Inria, Univ. Lille, CNRS UMR 9189 - CRISTAL, F-59000 Lille, France

<sup>c</sup>ITMO University, 49 av. Kronverkskiy, 197101 Saint Petersburg, Russia

<sup>d</sup>Saint Petersburg State University, 7-9 Universitetskaya nab., 199034 Saint Petersburg, Russia

<sup>e</sup>Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg 199178, Russia

## Abstract

For a retarded nonlinear system, this paper proposes a modification of the Lyapunov-Razumikhin method introducing a distributed integral of Lyapunov function for comparison with its current value instead of the maximum over delayed interval. The developed conditions are illustrated on different analysis problems presenting integral Halanay-type inequalities.

## 1 Introduction

There are two common methods for analyzing the stability of time-delay systems, which are based on the search for a Lyapunov-Razumikhin function and a Lyapunov-Krasovskii functional [14, 18] (see also a recent survey [4]). Another popular tool is grounded on application of Halanay inequalities [16]. The Lyapunov-Krasovskii functional method is more generic, as it can be used to establish the necessary and sufficient conditions for asymptotic stability, input-to-state stability (ISS) [9, 25, 26], and even finite-time stability [11, 23], for different classes of time-delay systems. However, finding a suitable Lyapunov-Krasovskii functional can be challenging, especially for nonlinear systems, while this approach is well developed for linear models [14]. The Lyapunov-Razumikhin function method is simpler to apply, but in a general setting it is only sufficient for asymptotic stability or ISS [14, 18, 20, 29, 31]. Under additional constraints, the Lyapunov-Razumikhin conditions can also be necessary [8], and they can be used to evaluate the rates of convergence in the retarded dynamics [7]. Moreover, the Lyapunov-Razumikhin function method is often preferred for nonlinear systems because it is easier to find a suitable function than a Lyapunov-Krasovskii functional [10, 13, 29] (a Lyapunov function for the delay-free case can be tested to find delay-dependent conditions).

Both methods can be used to distinguish between delay-dependent and delay-independent stability scenarios [14], and they can be applied to small-gain analysis [6]. In addition, in the nonlinear setting both methods have comple-

mentary roles, when in one case the Lyapunov-Krasovskii approach is preferable, while under a mild modification of the assumptions, another tool provides a solution, see [1, 2, 3] for examples. Therefore, it is important to obtain extensions of both approaches to different scenarios, classes of models and stability concepts.

The aim of this work is to extend the Lyapunov-Razumikhin function (LRF) approach by considering the distributed integrals in comparison with the current value of the function. The results are developed to the ISS framework. The obtained outcomes will be illustrated on design of integral Halanay-type inequalities [16] (see also [21, 22, 30]) for stability analysis of time-delay systems with distributed delays. These new integral-based LRF conditions can be used to refine the domain of convergence in the systems with distributed delays, and they also require less computational power for verification.

The paper is organized as follows. Preliminaries are given in Section 2. The main results are formulated in Section 3. The Halanay-type inequalities are considered in Section 4. The final remarks and discussion are presented in Section 5.

## 2 Preliminaries

The real numbers are denoted by $\mathbb{R}$, $\mathbb{R}_+ = \{s \in \mathbb{R} : s \ge 0\}$. Euclidean norm for a vector $x \in \mathbb{R}^n$ is defined as $|x|$. We denote by $C_{[a,b]}$, $-\infty < a < b < +\infty$ the Banach space of continuous functions $\phi : [a,b] \to \mathbb{R}^n$ with the uniform
norm $||\phi|| = \sup_{a \le \zeta \le b} |\phi(\zeta)|$. For a Lebesgue measurable function of time $d: [a, b) \to \mathbb{R}^m$ define the norm $||d||_{[a,b]} = \text{ess sup}_{t \in [a,b]} |d(t)|$, then $||d||_{\infty} = ||d||_{[0, +\infty)}$ and the space of $d$ with $||d||_{[a,b)} < +\infty$ ($||d||_{\infty} < +\infty$) we further denote as $\mathcal{L}_{[a,b]}^m$ ($\mathcal{L}_{\infty}^m$).

For a locally Lipschitz continuous function $V: \mathbb{R}^n \to \mathbb{R}_+$ the upper directional Dini derivative is defined as follows:

$$D^+V(x)v = \limsup_{h \to 0^+} \frac{V(x+hv) - V(x)}{h}$$

for any $x \in \mathbb{R}^n$ and $v \in \mathbb{R}^n$.

A continuous function $\sigma: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{K}$ if it is strictly increasing and $\sigma(0) = 0$; it belongs to class $\mathcal{K}_\infty$ if it is also unbounded. A continuous function $\beta: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{KL}$ if $\beta(\cdot, \cdot, r) \in \mathcal{K}$ for all $r \in \mathbb{R}_+$ and $\beta(r, \cdot, \cdot)$ is a strictly decreasing to zero for any fixed $r > 0$.

## 2.1 Stability definitions

Consider a generic functional differential equation of retarded type [19]:

$$\dot{x}(t) = f(x_t, d(t)), \quad t \ge 0, \quad (1)$$

where $x(t) \in \mathbb{R}^n$ and $x_t \in C_{[-\tau, 0]}$ is the state function, $x_t(s) = x(t+s)$, $-\tau \le s \le 0$ and $\tau > 0$ is a finite delay; $d(t) \in \mathbb{R}^m$ is the external input, $d \in \mathcal{L}_\infty^m$; $f: C_{[-\tau, 0]} \times \mathbb{R}^m \to \mathbb{R}^n$ is a continuous function, $f(0, 0) = 0$, and it provides (at least local) existence and uniqueness of solutions in forward time for the system (1). Denote such a unique solution satisfying the initial condition $x_0 \in C_{[-\tau, 0]}$ with the input $d \in \mathcal{L}_\infty^m$ by $x(t, x_0, d)$, which is defined on some time interval $[-\tau, T)$ for $T \in (0, +\infty]$, and $x_t(s, x_0, d) = x(t+s, x_0, d)$ for $-\tau \le s \le 0$.

Let $\Omega$ be an open neighborhood of zero in $C_{[-\tau, 0]}$.

**Definition 1** [14, 18, 23] At the origin the system (1) with $d \equiv 0$ is said to be

(a) stable if there is $\sigma \in \mathcal{K}$ such that for any $x_0 \in \Omega$, the solutions are defined and $|x(t, x_0, 0)| \le \sigma(||x_0||)$ for all $t \ge 0$;

(b) uniformly asymptotically stable if it is stable and for any $\epsilon, \delta > 0$ there is $T_{\epsilon, \delta} > 0$ such that $|x(t, x_0, 0)| \le \epsilon$ for all $t \ge T_{\epsilon, \delta}$ for any $x_0 \in \Omega$ with $||x_0|| \le \delta$;

(c) finite-time stable if it is stable and for any $x_0 \in \Omega$ there exists $0 \le T^{x_0} < +\infty$ such that $x(t, x_0, 0) = 0$ for all $t \ge T^{x_0}$. The functional $T_0(x_0) = \inf\{T^{x_0} \ge 0 : x(t, x_0, 0) = 0 \forall t \ge T^{x_0}\}$ is called the settling time of the system (1);

(d) nearly fixed-time stable if it is stable and for any $\rho > 0$ there exists $0 < T_\rho < +\infty$ such that $|x(t, x_0, 0)| \le \rho$ for all $t \ge T_\rho$ and all $x_0 \in \Omega$.

The set $\Omega$ is called the domain of stability.

If $\Omega = C_{[-\tau, 0]}$, then the corresponding properties are called global stability/uniform asymptotic stability/finite-time stability/nearly fixed-time stability.

The definition of uniform global asymptotic stability (UGAS) can be given in terms of existence of a function $\beta \in \mathcal{KL}$ such that $|x(t, x_0, 0)| \le \beta(||x_0||, t)$ for all $x_0 \in C_{[-\tau, 0]}$ and $t \ge 0$ (for autonomous time-delay systems the attractivity can be non-uniform [5]).

The property of nearly fixed-time stability is conceptually different from the fixed-time stability notion studied in [27, 28], since in the former case $\sup_{\rho>0} T_\rho \le +\infty$, i.e., at the origin a nearly fixed-time stable system (1) may be just asymptotically stable. The difference between nearly fixed-time stable and asymptotically stable systems becomes important for an unbounded set $\Omega$.

## 2.2 ISS for time-delay systems

This framework is used to analyze and quantify behavior of the system (1) with $d \ne 0$.

**Definition 2** [15, 24] The system (1) is called practically ISS, if there exist $\beta \in \mathcal{KL}$, $\gamma \in \mathcal{K}$ and $q \ge 0$ such that

$$|x(t, x_0, d)| \le \beta(||x_0||, t) + \gamma(||d||_\infty) + q, \quad \forall t \ge 0$$

for all $x_0 \in C_{[-\tau, 0]}$ and all $d \in \mathcal{L}_\infty^m$.

If $q = 0$ then (1) is called ISS.

If the above estimate is satisfied for all $x_0 \in C_{[-\tau, 0]}$ and all $d \in \mathcal{L}_\infty^m$ with $\max\{||x_0||, ||d||_\infty\} \le \rho$ for some $\rho > 0$, then (1) is called locally (practically) ISS.

If $d \equiv 0$, then ISS property is reduced to UGAS.

The LRF approach evaluating ISS property of the system (1) can be formulated as follows [29]:

**Definition 3** A locally Lipschitz continuous function $V: \mathbb{R}^n \to \mathbb{R}_+$ is called practical ISS-LRF if

(i) for some $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$, and all $x \in \mathbb{R}^n$:

$$\alpha_1(|x|) \le V(x) \le \alpha_2(|x|);$$

(ii) for some $\alpha, \gamma_x, \gamma_d \in \mathcal{K}$ and $r \ge 0$, with $\gamma_x(s) < s$ for all $s > 0$:

$$\max \left\{ \gamma_x \left( \max_{\theta \in [-\tau, 0]} V(\varphi(\theta)) \right), \gamma_d(|d|), r \right\} \le V(\varphi(0)) \Rightarrow D^+V(\varphi(0)) f(\varphi, d) \le -\alpha(|\varphi(0)|)$$

for all $\varphi \in C_{[-\tau, 0]}$ and all $d \in \mathbb{R}^m$.

If $r = 0$, then such a function $V$ is called ISS-LRF.
**Theorem 1** If for the system (1) there exists a (practical) ISS-LRF, then it is (practically) ISS.

If the conditions of Theorem 1 are verified only for $\max\{\|\varphi\|, |d|\} \le \rho$ with some $\rho > 0$ (and $\gamma_x(s) < s$ for $s \in (0, \rho)$), then it implies local (practical) ISS.

Therefore, to check the UGAS property for the system (1) with $d \equiv 0$ we need to find a locally Lipschitz continuous function $V: \mathbb{R}^n \to \mathbb{R}_+$ such that

$$ 
\begin{aligned}
\alpha_1(|x|) &\le V(x) \le \alpha_2(|x|), \quad \forall x \in \mathbb{R}^n; \\
V(\varphi(0)) &\ge \gamma_x \left( \max_{\theta \in [-\tau, 0]} V(\varphi(\theta)) \right) \Rightarrow \\
D^+V(\varphi(0)) &f(\varphi, 0) \le -\alpha(|\varphi(0)|), \quad \forall \varphi \in C_{[-\tau, 0]}
\end{aligned} 
\quad (2) $$

for some $\alpha, \gamma_x \in \mathcal{K}$ with $\gamma_x(s) < s$ for all $s > 0$ (for stability it is enough to have $\gamma_x(s) \le s$ for $s \in \mathbb{R}_+$). The respective extensions to finite-time or nearly fixed-time stability at the origin can be found in [7] (where the special forms of $\gamma_x$ and $\alpha$ are given that provide the accelerated convergence).

### 3 Main result

The main idea of our contribution is presented in the following result:

**Theorem 2** If for the system (1) with $d \equiv 0$ there exists a locally Lipschitz continuous function $V: \mathbb{R}^n \to \mathbb{R}_+$ such that

$$ 
\begin{aligned}
\alpha_1(|x|) &\le V(x) \le \alpha_2(|x|), \quad \forall x \in \mathbb{R}^n; \\
V(\varphi(0)) &\ge \gamma_1 \left( \int_{-\tau}^{0} \gamma_2(V(\varphi(\theta))) d\theta \right) \Rightarrow \\
D^+V(\varphi(0)) &f(\varphi, 0) \le -\alpha(|\varphi(0)|), \quad \forall \varphi \in C_{[-\tau, 0]}
\end{aligned} 
\quad (3) $$

for some $\alpha: \mathbb{R}_+ \to \mathbb{R}_+$, $\gamma_1, \gamma_2 \in \mathcal{K}$ with $\gamma_1(\tau\gamma_2(s)) \le s$ for all $s \in \mathbb{R}_+$, then it is globally stable at the origin. If $\alpha \in \mathcal{K}$ and $\gamma_1(\tau\gamma_2(s)) < s$ for all $s > 0$, then it is UGAS at the origin.

**PROOF.** For the proof, we will consider the case of UGAS property only, since the stability result can be obtained by the respective adaptation of the arguments.

Take $x_0 \in C_{[-\tau, 0]}$ and consider the corresponding unique solution $x(t) = x(t, x_0, 0)$ of the system (1) with $d \equiv 0$ that is defined on an interval of time $t \in \mathcal{T} = [-\tau, T)$ for some $T \in (0, +\infty]$. We can decompose the set $\mathcal{T} = \mathcal{T}_1 \cup \mathcal{T}_2$ on two subsets such that

$$ 
\begin{aligned}
V(x(t)) &\ge \gamma_1 \left( \int_{-\tau}^{0} \gamma_2(V(x(t+\theta))) d\theta \right), \quad \forall t \in \mathcal{T}_1; \\
V(x(t)) &< \gamma_1 \left( \int_{-\tau}^{0} \gamma_2(V(x(t+\theta))) d\theta \right), \quad \forall t \in \mathcal{T}_2.
\end{aligned} 
$$

Further denote for brevity $V(t) = V(x(t))$, then for any interval $[t_s^1, t_e^1] \subset \mathcal{T}_1$ we have by (3) that

$$ \dot{V}(t) := D^+V(x(t)) f(x_t, 0) \le -\alpha(|x(t)|) \le -\alpha \alpha_2^{-1}(V(t)) $$

for all $t \in [t_s^1, t_e^1]$, which due to the conventional comparison theorems [17] implies in the time domain:

$$ V(t) \le \beta_1(V(t_s^1), t - t_s^1) $$

for all $t \in [t_s^1, t_e^1]$ and some $\beta_1 \in \mathcal{KL}$ with the property $\beta_1(s, 0) = s$ (in other words, the LRF $V(t)$ is strictly decreasing for $t \in \mathcal{T}_1$). Now consider a non-empty interval $(t_s^2, t_e^2) \subset \mathcal{T}_2$, where we assume that $t_s^2$ is the last time instant when (3) holds and next it is violated. Then, by the integral mean value theorem, for any $t \in (t_s^2, t_e^2)$ there is a time instant $\psi_t \in [t - \tau, t]$ such that

$$ 
\begin{aligned}
\int_{-\tau}^{0} \gamma_2(V(x(t+\theta))) d\theta &= \gamma_2(V(x(\psi_t))) \int_{-\tau}^{0} d\theta \\
&= \tau\gamma_2(V(x(\psi_t))).
\end{aligned} 
$$

Therefore,

$$ V(t) < \gamma_1(\tau\gamma_2(V(\psi_t))) = \rho_\tau(V(\psi_t)) $$

with $\rho_\tau \in \mathcal{K}$.

Next, if $\psi_t \in (t_s^2, t_e^2)$, then there is another time instant $\psi_{\psi_t} \in [\psi_t - \tau, \psi_t]$ such that

$$ V(\psi_t) < \rho_\tau(V(\psi_{\psi_t})), $$

hence,

$$ V(t) < \rho_\tau(\rho_\tau(V(\psi_{\psi_t}))). $$

If $\psi_t \notin (t_s^2, t_e^2)$, then $\psi_t = t_s^2 + \chi_t$ for $\chi_t \in [-\tau, 0]$ and

$$ V(t) < \rho_\tau(V(\psi_t)) = \rho_\tau(V(t_s^2 + \chi_t)). $$

In general case, recursively applying the same computations we obtain

$$ V(t) < \rho_\tau(\cdots \rho_\tau(V(t_s^2 + \chi_t))) $$

where $\chi_t \in [-\tau, 0]$, i.e., while staying in the interval $[t_s^2, t_e^2]$ we use integral mean value theorem (the inverse of (3)) until we return back to the beginning of the interval.

Note that the map $\rho_\tau(s) = \gamma_1(\tau\gamma_2(s)) < s$ is a contraction by the conditions of the theorem. Denote $\rho_\tau^{[k]}(s) = \overbrace{\rho_\tau \circ \cdots \circ \rho_\tau}^{k}(s)$ as the multiple recursive application $k$ times of $\rho_\tau$, then the inequality above can be written as

$$ V(t) < \rho_\tau^{[k]}(V(t_s^2 + \chi_t)) $$

for some integer $k_t \ge 1$, which admits a more accurate lower bound $\frac{t-t_s^2}{\tau} + 1 \le k_t$, consequently,

$$ 
\begin{aligned}
V(t) < \rho_\tau^{[\frac{t-t_s^2}{\tau} + 1]} (V(t_s^2 + \chi_t)) \\
\le \rho_\tau^{[\frac{t-t_s^2}{\tau} + 1]} \left( \max_{\theta \in [-\tau, 0]} V(t_s^2 + \theta) \right),
\end{aligned} 
$$
and there exists some $\beta_2 \in \mathcal{KL}$ such that

$$ 
\begin{aligned}
V(t) &\le \beta_2(V(t_s^2 + \chi_t), t - t_s^2) \\
&\le \beta_2(\max_{\theta \in [-\tau, 0]} V(t_s^2 + \theta), t - t_s^2)
\end{aligned}
$$

for all $t \in (t_s^2, t_e^2)$ with the property $\beta_2(s, 0) = s$. Therefore, $V(t)$ stays bounded for any $t \in \mathcal{T}$ and, hence, $\mathcal{T} = \mathbb{R}_+$. Moreover, for any $t \in \mathbb{R}_+$ there is a sequence of time instants $[-\tau, 0] \ni \chi_0 = t_0 < t_1 < \dots < t_{n_t} = t$ for some integer $n_t \ge 1$ such that $V(t_{i+1}) \le \rho(V(t_i))$ for $i = 0, \dots, n_t - 1$ and $\rho(s) = \max\{\rho_\tau(s), \beta_1(s, \tau)\}$. Moreover, $V(t) \le \max\{\beta_1(\max_{\theta \in [-\tau, 0]} V(t_i + \theta), t - t_i), \beta_2(\max_{\theta \in [-\tau, 0]} V(t_i + \theta), t - t_i)\}$ for all $t \in [t_i, t_{i+1})$, consequently, there exists a function $\beta$ from the class $\mathcal{KL}$ such that

$$ V(t) \le \beta(\max_{\theta \in [-\tau, 0]} V(\theta), t) $$

for all $t \ge 0$.

**Remark 1** Comparing the conditions (3) with the conventional ones in (2), we can observe that

$$ \gamma_1 \left( \int_{-\tau}^{0} \gamma_2 \left( V(\varphi(\theta)) \right) d\theta \right) \le \gamma_1 \left( \tau \gamma_2 \left( \max_{\theta \in [-\tau, 0]} V(\varphi(\theta)) \right) \right) $$

and the relation (2) for $\gamma_x(s) = \gamma_1(\tau\gamma_2(s))$ implies (3). In some cases it is possible to show that for (we denote $V(t) = V(x(t))$)

$$ \gamma_1(\tau\gamma_2(\max_{\theta \in [-\tau, 0]} V(t + \theta))) > V(t) \ge \gamma_1 \left( \int_{-\tau}^{0} \gamma_2(V(t + \theta)) d\theta \right) $$

the estimate for $\dot{V}(t)$ can be negative, while for $V(t) \ge \gamma_1(\tau\gamma_2(\max_{\theta \in [-\tau, 0]} V(t + \theta)))$ it is just zero, as for instance in the following example:

$$ \dot{V}(t) \le - \frac{V(t)}{1 + \frac{1}{\tau} \int_{-\tau}^{0} V(t + \theta) d\theta} + \frac{\frac{1}{\tau} \int_{-\tau}^{0} V(t + \theta) d\theta}{1 + \max_{\theta \in [-\tau, 0]} V(t + \theta)} $$

for $t \ge 0$. To apply the standard condition, we first need to make the substitution $\frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta)d\theta \le \max_{\theta \in [-\tau, 0]} V(t+\theta)$ leading to the estimate:

$$ \dot{V}(t) \le - \frac{V(t) - \max_{\theta \in [-\tau, 0]} V(t + \theta)}{1 + \max_{\theta \in [-\tau, 0]} V(t + \theta)}. $$

If the stability condition (2) is satisfied, i.e., $V(t) \ge \max_{\theta \in [-\tau, 0]} V(t+\theta)$ with $\gamma_x(s) = s$, then the right-hand side of the inequality above becomes zero. Otherwise, if the counterpart from (3) is verified, i.e., $V(t) \ge \frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta)d\theta$ with $\gamma_1(s) = \frac{s}{\tau}$ and $\gamma_2(s) = s$ implying $\gamma_1(\tau\gamma_2(s)) = s$, then the right-hand side is non-positive (it equals zero for $V(t) = \max_{\theta \in [-\tau, 0]} V(t+\theta)$ only):

$$ \dot{V}(t) \le - \frac{V(t)}{1 + V(t)} + \frac{V(t)}{1 + \max_{\theta \in [-\tau, 0]} V(t + \theta)}, $$

but in both cases, the system is stable at the origin. However, the new conditions may provide a better estimate of the convergence domain if an integral of a LRF appears in the estimate for the time derivative of $V$.

The integral LRF conditions (3) can be combined with the conventional ones from (2), which may simplify the estimation of $\dot{V}$ (the case of stability property is skipped for brevity):

**Corollary 1** If for the system (1) with $d \equiv 0$ there exists a locally Lipschitz continuous function $V: \mathbb{R}^n \to \mathbb{R}_+$ such that

$$ 
\begin{aligned}
\alpha_1(|x|) &\le V(x) \le \alpha_2(|x|), \quad \forall x \in \mathbb{R}^n; \\
V(\varphi(0)) &\ge \max\left\{\gamma_1\left(\int_{-\tau}^{0} \gamma_2(V(\varphi(\theta))) d\theta\right), \\ &\qquad \gamma_x\left(\max_{\theta \in [-\tau, 0]} V(\varphi(\theta))\right)\right\} \Rightarrow \\ &D^+V(\varphi(0)) f(\varphi, 0) \le -\alpha(|\varphi(0)|), \quad \forall \varphi \in C_{[-\tau, 0]}
\end{aligned}
$$

for some $\alpha, \gamma_1, \gamma_2, \gamma_x \in \mathcal{K}$ with $\gamma_1(\tau\gamma_2(s)) < s$ and $\gamma_x(s) < s$ for all $s > 0$, then it is UGAS at the origin.

**PROOF.** In this case $\mathcal{T} = \mathcal{T}_1 \cup \mathcal{T}_2 \cup \mathcal{T}_3$, where

$$ 
\begin{aligned}
V(x(t)) &\ge \max\left\{\gamma_1\left(\int_{-\tau}^{0} \gamma_2(V(\varphi(\theta))) d\theta\right), \\ &\qquad \gamma_x\left(\max_{\theta \in [-\tau, 0]} V(\varphi(\theta))\right)\right\}, \quad \forall t \in \mathcal{T}_1;
\end{aligned}
$$

$$ V(x(t)) < \gamma_1 \left( \int_{-\tau}^{0} \gamma_2 \left( V(x(t+\theta)) \right) d\theta \right), \quad \forall t \in \mathcal{T}_2; $$

$$ V(x(t)) < \gamma_x \left( \max_{\theta \in [-\tau, 0]} V(\varphi(\theta)) \right), \quad \forall t \in \mathcal{T}_3, $$

where the properties of the behavior of $V(t)$ are the same as before for $t \in \mathcal{T}_1$ and $t \in \mathcal{T}_2$. For $t \in (t_s^2, t_e^2) \subset \mathcal{T}_3$,

$$ V(t) < \gamma_x \left( V(\tilde{\psi}_t) \right) $$

for some time instant $\tilde{\psi}_t \in [t-\tau, t]$. The rest of analysis is identical to the case $t \in \mathcal{T}_2$ by replacing $\rho_\tau$ with $\gamma_x$.

Introducing the restrictions similar to ones proposed in [7], these results can be developed to the case of finite-time and nearly fixed-time stability:

**Corollary 2** If for the system (1) with $d \equiv 0$ there exists a locally Lipschitz continuous function $V: \mathbb{R}^n \to \mathbb{R}_+$ such that

$$ 
\begin{aligned}
\alpha_1(|x|) &\le V(x) \le \alpha_2(|x|), \quad \forall x \in \mathbb{R}^n; \\
V^{1-\mu}(\varphi(0)) + c\tau &\ge \gamma_1 \left( \int_{-\tau}^{0} \gamma_2(V(\varphi(\theta))) d\theta \right) \Rightarrow \\ &D^+V(\varphi(0)) f(\varphi, 0) \le -\alpha V^\mu(\varphi(0)), \quad \forall \varphi \in C_{[-\tau, 0]}
\end{aligned}
$$

for some $\alpha, c > 0$, $\mu \in (0, 1)$, $\gamma_1, \gamma_2 \in \mathcal{K}$ with $\gamma_1(\tau\gamma_2(s)) \le s^{1-\mu}$ for all $s \in \mathbb{R}_+$, then it is globally finite-time stable at the origin, and the following estimate is satisfied:

$$ 
|x(t, x_0, 0)| \le \alpha_1^{-1} \circ \max\{0, \alpha(\|x_0\|)^{1-\mu} \\ - \min\{\alpha(1-\mu), c\}t\}^{\frac{1}{1-\mu}} $$
for all $t \ge 0$ and the initial conditions $x_0 \in C_{[-\tau,0]}$.

**PROOF.** For any $x_0 \in C_{[-\tau,0]}$ consider the corresponding solution $x(t) = x(t, x_0, 0)$ of the system (1) with $d \equiv 0$ that is defined on an interval of time $t \in \mathcal{T} = [-\tau, T)$ for some $T \in (0, +\infty]$. We can decompose the set $\mathcal{T} = \mathcal{T}_1 \cup \mathcal{T}_2$ on two subsets such that

$$V^{1-\mu}(t) + c\tau \ge \gamma_1 \left( \int_{-\tau}^{0} \gamma_2 (V(t+\theta)) d\theta \right), \forall t \in \mathcal{T}_1;$$

$$V^{1-\mu}(t) + c\tau < \gamma_1 \left( \int_{-\tau}^{0} \gamma_2 (V(t+\theta)) d\theta \right), \forall t \in \mathcal{T}_2.$$

For any interval $[t_s^1, t_e^1] \subset \mathcal{T}_1$ we have by conditions of the corollary that

$$\dot{V}(t) \le -\alpha V^{\mu}(t)$$

for all $t \in [t_s^1, t_e^1]$, which implies in the time domain:

$$V^{1-\mu}(t) \le \max\{0, V^{1-\mu}(t_s^1) - \alpha(1-\mu)(t - t_s^1)\}$$

for all $t \in [t_s^1, t_e^1]$. Now consider a non-empty interval $(t_s^2, t_e^2) \subset \mathcal{T}_2$, then as above, by the integral mean value theorem,

$$\int_{-\tau}^{0} \gamma_2 (V(x(t+\theta))) d\theta = \tau \gamma_2 (V(x(\psi_t)))$$

for any $t \in (t_s^2, t_e^2)$, where $\psi_t \in [t - \tau, t]$ is a suitable time instant, and

$$\begin{aligned} V^{1-\mu}(t) &< \max\{0, \gamma_1 (\tau \gamma_2 (V(\psi_t))) - c\tau\} \\ &\le \max\{0, V^{1-\mu}(\psi_t) - c\tau\}. \end{aligned} \quad (4)$$

Recursively applying the same arguments we obtain (recall that $\int_{-\tau}^{0} \gamma_2 (V(x(\psi_t + \theta))) d\theta = \tau \gamma_2 (V(x(\psi_t)))$

$$\begin{aligned} V(t) &< \max\{0, V^{1-\mu}(\psi_{\psi_t}) - c\tau - c\tau\} \\ &< \max\{0, V^{1-\mu}(t_s^2 + \chi_t) - c\tau - \dots - c\tau\} \\ &\le \max \left\{ 0, V^{1-\mu}(t_s^2 + \chi_t) - \left( \frac{t-t_s^2}{\tau} + 1 \right) c\tau \right\} \\ &\le \max\{0, V^{1-\mu}(t_s^2 + \chi_t) - c(t - t_s^2)\} \end{aligned}$$

where $\chi_t \in [-\tau, 0]$ and we used the lower bound $\frac{t-t_s^2}{\tau} + 1$ for the number of applications of the relation (4) on the interval $(t_s^2, t)$. Obviously, combination of the estimates derived for $t \in \mathcal{T}_1$ and $t \in \mathcal{T}_2$ leads to the property:

$$V^{1-\mu}(t) \le \max\{0, \max_{\theta \in [-\tau,0]} V^{1-\mu}(\theta) - \min\{\alpha(1-\mu), c\}t\}$$

for all $t \in \mathbb{R}_+$, which implies the required finite-time stability at the origin due to the properties of $V$.

The obtained estimate implies that once $x(t') = 0$ for some $t' \in \mathbb{R}_+$, then $x(t) = 0$ for all $t \ge t'$, which corresponds to the observations given in [12] (see also the necessary and sufficient conditions of finite-time stability formulated in terms of Lyapunov-Krasovskii framework in [11, Theorem 7.2, Example 7.1]).

**Corollary 3** If for the system (1) with $d \equiv 0$ there exists a locally Lipschitz continuous function $V: \mathbb{R}^n \to \mathbb{R}_+$ such that

$$\begin{aligned} \alpha_1(|x|) &\le V(x) \le \alpha_2(|x|), \forall x \in \mathbb{R}^n; \\ V^{\mu-1}(\varphi(0)) &\ge \frac{1}{\gamma_1 \left( \int_{-\tau}^{0} \gamma_2(V(\varphi(\theta))) d\theta \right) + c\tau} \Rightarrow \\ D^+V(\varphi(0)) f(\varphi, 0) &\le -\alpha V^{\mu}(\varphi(0)), \forall \varphi \in C_{[-\tau,0]} \end{aligned}$$

for some $\alpha, c > 0$, $\mu > 1$, $\gamma_1, \gamma_2 \in \mathcal{K}$ with $\gamma_1(\tau\gamma_2(s)) \le s^{\mu-1}$ for all $s \in \mathbb{R}_+$, then it is globally nearly fixed-time stable at the origin, and the following estimate is satisfied:

$$|x(t, x_0, 0)| \le \alpha_1^{-1} \left( \frac{1}{(\alpha_2^{1-\mu} (\|x_0\|) + \min\{\alpha(\mu-1), c\}t)^{\frac{1}{\mu-1}}} \right)$$

for all $t \ge 0$ and the initial conditions $x_0 \in C_{[-\tau,0]}$.

**PROOF.** The proof follows the argumentation used in the previous corollary considering

$$V^{\mu-1}(t) \ge \frac{1}{\gamma_1 \left( \int_{-\tau}^{0} \gamma_2(V(t+\theta)) d\theta \right) + c\tau}, \forall t \in \mathcal{T}_1;$$

$$V^{\mu-1}(t) < \frac{1}{\gamma_1 \left( \int_{-\tau}^{0} \gamma_2(V(t+\theta)) d\theta \right) + c\tau}, \forall t \in \mathcal{T}_2.$$

Then, for any interval $[t_s^1, t_e^1] \subset \mathcal{T}_1$ we have

$$V^{\mu-1}(t) \le \frac{1}{V^{1-\mu}(t_s^1) + \alpha(\mu-1)(t - t_s^1)}$$

for all $t \in [t_s^1, t_e^1]$; and for any $t \in (t_s^2, t_e^2) \subset \mathcal{T}_2$ we obtain that

$$\begin{aligned} V^{\mu-1}(t) &< \frac{1}{\frac{1}{\gamma_1(\tau\gamma_2(V(\psi_t)))} + c\tau} \\ &\le \frac{1}{V^{1-\mu}(\psi_t) + c\tau} \end{aligned} \quad (5)$$

for some $\psi_t \in [t - \tau, t]$. All other steps are similar.

Using these ideas, the result of Theorem 1 admits an important extension to a distributed delay with a nonnegative kernel in the presence of external inputs:

**Theorem 3** If for the system (1) there exists a locally Lipschitz continuous function $V: \mathbb{R}^n \to \mathbb{R}_+$ such that

$$\begin{aligned} \alpha_1(|x|) &\le V(x) \le \alpha_2(|x|), \forall x \in \mathbb{R}^n; \\ V(\varphi(0)) &\ge \max\left\{\gamma_1 \left( \int_{-\tau}^{0} \mathbb{K}(\theta)\gamma_2(V(\varphi(\theta))) d\theta \right), \right. \\ &\qquad \left. \gamma_d(\|d\|_{\infty}), r \right\} \Rightarrow D^+V(\varphi(0)) f(\varphi, \mathbf{d}) \le -\alpha(|\varphi(0)|), \\ &\forall \varphi \in C_{[-\tau,0]}, \forall \mathbf{d} \in \{\mathbf{d} \in \mathbb{R}^m : |\mathbf{d}| \le \|d\|_{\infty}\} \end{aligned}$$
for some $\alpha, \gamma_1, \gamma_2 \in \mathcal{K}$, an integrable map $\mathbb{K}: [-\tau, 0] \to \mathbb{R}_+$ and $r \ge 0$ with $\gamma_1(\kappa\gamma_2(s)) < s$ for all $s > 0$, where $\kappa = \int_{-\tau}^{0} \mathbb{K}(\theta)d\theta$, then it is practically ISS. If $r = 0$, then the system (1) admits ISS property.

**PROOF.** The proof of the theorem repeats the main arguments of the proof of Theorem 2 except that now $\mathcal{T} = \mathcal{T}_1 \cup \mathcal{T}_2 \cup \mathcal{T}_3$ such that

$$ 
\begin{aligned}
V(t) &\ge \max\left\{ \gamma_1 \left( \int_{-\tau}^{0} \mathbb{K}(\theta)\gamma_2(V(t+\theta)) d\theta \right), \\ 
&\qquad \gamma_d(\|d\|_{\infty}), r \right\}, \quad \forall t \in \mathcal{T}_1; \\ 
V(t) &< \gamma_1 \left( \int_{-\tau}^{0} \mathbb{K}(\theta)\gamma_2(V(t+\theta)) d\theta \right), \quad \forall t \in \mathcal{T}_2; \\ 
V(t) &< \max\{\gamma_d(\|d\|_{\infty}), r\}, \quad \forall t \in \mathcal{T}_3,
\end{aligned}
$$

where the properties of the behavior of $V(t)$ are the same as previously for $t \in \mathcal{T}_1$ and $t \in \mathcal{T}_2$ (in the latter case the integral mean value theorem takes the form $\int_{-\tau}^{0} \mathbb{K}(\theta)\gamma_2(V(t+\theta)) d\theta = \gamma_2(V(\psi_t)) \int_{-\tau}^{0} \mathbb{K}(\theta)d\theta = \kappa\gamma_2(V(\psi_t))$ for some $\psi_t \in [t - \tau, t]$ and the map $\rho_\tau(s) = \gamma_1(\kappa\gamma_2(s)) < s$ is again a contraction), while $V(t)$ is bounded by $\max\{\gamma_d(\|d\|_{\infty}), r\}$ for $t \in \mathcal{T}_3$. Since $\beta_i(s, 0) = s$ for $s = 1, 2$, there exists $\beta \in \mathcal{KL}$ such that

$$ V(t) \le \max \left\{ \beta \left( \max_{\theta \in [-\tau, 0]} V(\theta), t \right), \gamma_d (\|d\|_{\infty}), r \right\} $$

for all $t \ge 0$, which leads to the desired results.

The findings of this theorem can be straightforwardly combined with ones from the above corollaries to get the conditions of finite/fixed-time ISS properties, etc. For instance, for $d \equiv 0$ we recover the LRF-based integral conditions for UGAS property with the kernel $\mathbb{K}$ (which is assumed to be constant in Theorem 2).

The conditions for local (asymptotic) stability property and local ISS can be obtained by restricting the domain of validity in (3), i.e., replacing the set $C_{[-\tau, 0]}$ by its bounded subset $\Omega$ containing the zero solution, and introducing a uniform bound on $d$. Note that in such a case the estimation of the domain of convergence can be less restrictive for the new conditions than for the conventional ones. For an illustration of this claim, consider

$$ 
\begin{aligned}
\dot{V}(t) \le & -\frac{\frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta) d\theta}{1 + \left(\frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta) d\theta\right)^2} \\ 
& + \frac{\left(\frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta) d\theta\right)^2}{1 + \max_{\theta \in [-\tau, 0]} V(t+\theta)}
\end{aligned}
$$

for all $t \ge 0$. Under the standard conditions, if $V(t) \ge \gamma_x(\max_{\theta \in [-\tau, 0]} V(t+\theta))$ for $\gamma_x(s) = \min(s, s^2)$ we obtain

$$ \dot{V}(t) \le -\frac{\frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta) d\theta}{1 + V(t)} + \frac{\left(\frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta) d\theta\right)^2}{1 + \max_{\theta \in [-\tau, 0]} V(t+\theta)} $$

that is non-positive for all $\max_{\theta \in [-\tau, 0]} V(t+\theta) \le 1$. For the new conditions with $V(t) \ge \gamma_x(\frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta)d\theta)$ the same inequality is verified for $\frac{1}{\tau} \int_{-\tau}^{0} V(t+\theta)d\theta \le 1$ which contains the set $\max_{\theta \in [-\tau, 0]} V(t+\theta) \le 1$.

**Remark 2** Verification of the condition (3) requires the same memory as the condition (2), but much less computational power. Indeed, to check the latter one needs on each step to perform the search of the maximum of a function on the interval $[t - \tau, t]$ (the maximal element of an array on a grid corresponding to the interval), while the former needs an integration of a scalar variable (a couple of multiplications and summations) with one subtraction:

$$ 
\begin{aligned}
\int_{-\tau}^{0} \gamma_2(V(t+\theta)) d\theta &= w(t) - w(t - \tau), \\ 
\dot{w}(t) &= \gamma_2(V(t)), \quad w(0) = 0.
\end{aligned}
$$

# 4 Halanay-type inequalities

In this section several examples of application of the proposed extensions of the LRF method are given, which are formulated in the form of differential inequalities for scalar variables, e.g., LRFs. On these examples we also can compare the conventional LRF method and the new extensions presented in this paper.

To put these investigations in the context, consider a non-linear dynamical system with distributed delay:

$$ \dot{x}(t) = F(x(t)) + \int_{t-\tau}^{t} G(x(\theta))d\theta, $$

where $x(t) \in \mathbb{R}^n$ is the instantaneous state, $F: \mathbb{R}^n \to \mathbb{R}^n$ and $G: \mathbb{R}^n \to \mathbb{R}^n$ are continuous maps, which are standard homogeneous of degrees $\omega \in \mathbb{R}_+$ and $\eta \in \mathbb{R}_+$, respectively:

$$ F(\lambda x) = \lambda^{\omega} F(x), \quad G(\lambda x) = \lambda^{\eta} G(x), \quad \forall \lambda > 0, \quad x \in \mathbb{R}^n. $$

Assume that the initial conditions are in $C_{[-\tau, 0]}$, and that the nominal system $\dot{x}(t) = F(x(t))$ is UGAS at the origin. Then, there exists a continuously differentiable and standard homogeneous of degree $\chi > 1$ Lyapunov function $V: \mathbb{R}^n \to \mathbb{R}_+$ such that

$$ 
\begin{aligned}
a_1 |x|^{\chi} &\le V(x) \le a_2 |x|^{\chi}, \quad V(\lambda x) = \lambda^{\chi} V(x), \\ 
\frac{\partial V(x)}{\partial x} F(x) &\le -a_3 V^{1+\frac{\omega-1}{\chi}}(x)
\end{aligned}
$$

for all $x \in \mathbb{R}^n$ and $\lambda > 0$, with some $a_1, a_2, a_3 > 0$. Note that by the homogeneity property there exist $b_1, b_2 > 0$ such that

$$ \left| \frac{\partial V(x)}{\partial x} \right| \le b_1 |x|^{\chi-1}, \quad |G(x)| \le b_2 |x|^{\eta} $$
for all $x \in \mathbb{R}^n$. Therefore, considering $V$ as a LRF candidate we obtain for $V(t) = V(x(t))$ that

$$ 
\begin{aligned}
\dot{V}(t) & \le -a_3 V^{1+\frac{\omega-1}{\chi}}(t) + b_1 |x(t)|^{\chi-1} \int_{t-\tau}^{t} b_2 |x(\theta)|^{\eta} d\theta \\
& \le -a_3 V^{1+\frac{\omega-1}{\chi}}(t) + \frac{b_1 b_2}{a_1^{1+\frac{\eta-1}{\chi}}} V^{1-\chi^{-1}}(t) \int_{t-\tau}^{t} V^{\frac{\eta}{\chi}}(\theta) d\theta
\end{aligned}
$$

for all $V(t) \in \mathbb{R}_+$ and $t \ge 0$. Now, introducing the Lyapunov-Razumikhin conditions we need to find the restrictions on the degrees of $F$ and $G$, the coefficients and the delay value providing the negative definiteness of $\dot{V}(t)$.

To begin with, the simplest example is as follows. Let a variable $v : [-\tau, +\infty) \to \mathbb{R}_+$ satisfy the inequality

$$ \dot{v}(t) \le -a(v(t)) + g(v(t))c \left( \int_{t-\tau}^{t} b(v(\theta))d\theta \right) $$

for all $t \ge 0$, where $a, b, c, g \in \mathcal{K}$ (i.e., here $v(t) = V(t)$ with $a(s) = a_3 s^{1+\frac{\omega-1}{\chi}}$, $b(s) = s^{\frac{\eta}{\chi}}$, $c(s) = \frac{b_1 b_2}{a_1^{1+\frac{\eta-1}{\chi}}} s$

and $g(s) = s^{1-\chi^{-1}}$. According to Theorem 2, $v(t) \le \beta(\max_{\theta \in [-\tau, 0]} v(\theta), t)$ for all $t \ge 0$ with some $\beta \in \mathcal{K}\mathcal{L}$ provided that

$$ a(s) > g(s)s, \ s > c(\tau b(s)), \ \forall s > 0, $$

which can be verified for the homogeneous system provided that $\omega = \eta = \chi$ and the delay is sufficiently small. Using the conventional condition (2), the same conclusion can be obtained under the restriction on existence of a function $\rho \in \mathcal{K}_\infty$ such that

$$ a(s) > g(s)c(\tau b \circ \rho^{-1}(s)), \ \rho(s) < s, \ \forall s > 0 $$

whose formulation is technically more involved.

**Remark 3** Recalling Remark 2, the proposed nonlinear Halanay-type inequality can be used to assess stability and convergence of a second order dynamics:

$$ 
\begin{aligned}
\dot{v}(t) & \le -a(v(t)) + g(v(t))c(w(t) - w(t - \tau)), \ t \ge 0, \\
\dot{w}(t) & \le b(v(t)), \ t \ge -\tau, \ w(-\tau) = 0,
\end{aligned}
$$

where all functions and variables save their meaning, and $w : [-\tau, +\infty) \to \mathbb{R}_+$ is a new state component. Therefore, contrarily to the conventional Lyapunov-Razumikhin conditions and related Halanay inequalities, the integral ones admit a simple and natural generalization for analysis of systems of a higher order.

Now consider an illustration for Corollary 2. Let dynamics of variable $v : [-\tau, +\infty) \to \mathbb{R}_+$ verify the inequality

$$ \dot{v}(t) \le -av^{\mu}(t) + bv^{\nu}(t) \int_{t-\tau}^{t} v^{\lambda}(\theta)d\theta $$

for all $t \ge 0$, where $a, b > 0$, $\mu \in (0, 1)$, $\nu \ge \mu$ and $\lambda > 0$, then there exists $\bar{\tau} > 0$ such that for all $\tau \in (0, \bar{\tau}]$ the variable $v(t)$ is locally finite-time stable at zero. If the power inequalities are strict, then $\bar{\tau}$ can be chosen arbitrary influencing the domain of convergence only. Returning to

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 1. The results for simulation for finite-time case</figcaption></figure>

the motivating example with a homogeneous system, the conditions read as follows: $\omega \le 0$ and $\eta \ge 1 - \omega$, which contradict the initial continuity restriction $\omega > 0$ (it is a well-known fact that homogeneous time-delay systems cannot be finite-time stable at the origin).

**Example 1** For an illustration, consider the case with

$$ a = 1, \ b = 2, \ \mu = \nu = 0.5, \ \lambda = 2 $$

and two delay values $\tau = 0.5$ and $\tau = 0.05$. Interpreting the above inequality as an equality, the results of simulation of $v(t)$ versus time are shown in Fig. 1 for three different initial conditions, and in a logarithmic scale (the explicit Euler method was used with the discretization step 0.01). As we can conclude from these results, $v(t)$ converges to zero in a finite time dependent on initial conditions.

For the Corollary 3, consider behavior of a variable $v : [-\tau, +\infty) \to \mathbb{R}_+$ satisfying

$$ \dot{v}(t) \le -av^{\mu}(t) + \frac{v^{\nu}(t) \int_{t-\tau}^{t} v^{\lambda}(\theta)d\theta}{1 + b \int_{t-\tau}^{t} v^{\lambda}(\theta)d\theta} $$

for all $t \ge 0$, where $a, b > 0$, $\mu > 1$, $0 \le \nu \le 1$ and $0 \le \lambda \le \mu - 1$, then there exists $\bar{\tau} > 0$ such that for all $\tau \in (0, \bar{\tau}]$ the variable $v(t)$ is practically nearly fixed-time stable (i.e., there is a bounded set containing the zero solution, such that all trajectories reach this set in a finite time independent on initial conditions). Note that in [7] only conditions for the case $\nu = 1$ and $\lambda = \mu - 1$ were obtained.

**Example 2** For an illustration, select

$$ a = 1, \ b = 2, \ \mu = 2, \ \nu = 0.5, \ \lambda = 0.5 $$
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. The results for simulation for fixed-time case</figcaption></figure>

with an additive perturbation

$$d(t) = 0.25 \sin(5t)$$

and two delay values $\tau = 0.5$ and $\tau = 0.05$. Again, interpreting the above inequality as an equality (and adding $d(t)$ to the right-hand side), the results of simulation of $v(t)$ versus time are presented in Fig. 2 for three different initial conditions with a significant variation of amplitude, and in a logarithmic scale (the explicit Euler method with the discretization step 0.01 was applied). As we can conclude from these results, $v(t)$ converges to a vicinity of zero, demonstrating ISS behavior, while the time of convergence to this neighborhood is independent in initial conditions (the system is nearly fixed-time stable).

## 5 Conclusions

Considering retarded nonlinear systems, several modifications of the LRF method were presented, where the supremum over an interval of time for a LRF is replaced with its integral weighted by a nonnegative kernel. The conditions of finite-time/nearly fixed-time stability and ISS were formulated. The obtained conditions were applied to derive new integral Halanay-type inequalities.

Enlarging the class of applicable models and relaxing the technical constraints can be considered as directions of further researches.

## Acknowledgment

The research of A. Aleksandrov was partially supported by the Ministry of Science and Higher Education of the Russian Federation (project no. 124041500008 – 1).

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