# Continuity-based robust stability condition of Lur'e systems

Mohamed Yassine Arkhis, Denis Efimov

► To cite this version:

Mohamed Yassine Arkhis, Denis Efimov. Continuity-based robust stability condition of Lur'e systems. IEEE CDC 2024 - IEEE Conference on Decision and Control, Dec 2024, Milan (Italie), Italy. hal-04895362

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https://inria.hal.science/hal-04895362v1

Submitted on 18 Jan 2025

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# Continuity-based robust stability condition of Lur'e systems

Mohamed Yassine Arkhis<fnref n="1" />, Denis Efimov<fnref n="1" />

**Abstract**—For forced continuous-time Lur'e systems with a Lipschitz nonlinearity, this note proposes sufficient conditions for (integral) input to state stability, based on real sector conditions and output global asymptotic stability. The proof is based on the converse Lyapunov function arguments for the output asymptotic stability and sector conditions.

## I. INTRODUCTION

Analysis of stability for dynamical systems and its robustness against exogenous disturbances is an important problem [8], which has many solutions. Among them, the input-to-state stability (ISS) property introduced in [15] and its subsequent generalisation, integral-input-to-state stability (iISS) [2], play a fundamental role in considering bounded and integrally bounded perturbations, respectively. These properties have numerous characterizations, and their necessary and sufficient conditions are formulated using the Lyapunov function method. As it is usual for the latter approach, it is rather challenging to find a Lyapunov function for nonlinear dynamics, especially under the presence of disturbances. Hence, consideration of the canonical representations of dynamical systems comes to the focus.

Forced Lur'e models form such an important canonical class of dynamical systems, because these models are characterized by feedback interconnections of linear and nonlinear components, which are common in practical control systems [18]. The special structure of Lur'e systems - which comprises a linear system with an additive nonlinearity in the feedback loop - has given rise to the specific Lur'e problem, also called the absolute stability problem [1], which consists of determining the conditions for the origin of the system

$$ \dot{x} = Ax + Bu, \quad u = \phi(y), \quad y = Cx, \quad \phi(0) = 0 $$

to be globally asymptotically stable for any $\phi : \mathbb{R}^q \to \mathbb{R}^m$ belonging to a sector (more precise definition will be given later), where $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $C \in \mathbb{R}^{q \times n}$ [12], which has received a large body of work, see for example [8], [16], [19]. In these works different sufficient and equivalent conditions of absolute stability have been established, see the survey [10], or description of the results in [17]. A recent work [13], addressed the notion of finite-time stability of Lur'e systems, in which sufficient conditions were established. Another recent work [5] considers the discontinuous nonlinearities and non-continuously differentiable Lyapunov functions. An intensive research on the absolute stability problem has inspired the (i)ISS analysis of Lur'e systems not as individual systems, but as an infinite family of systems characterized by some sector bound conditions

on the nonlinearity.

Conceptually, (i)ISS characterizes the behavior of a dynamical system being bounded in the presence of arbitrary (bounded) input perturbations, and also asymptotically stable in the absence of external inputs, so it is a natural extension of absolute stability of the unforced Lur'e system to the disturbed one.

To the best of our knowledge of the state of the art, no one managed to obtain a result about the relationship between absolute stability on a real sector, and ISS on it. Most of the recent works that addressed the problem of (i)ISS analysis of forced Lur'e systems (see for example [4], [6], [7], [14]) use the property $\exists 0 < P = P^\top \in \mathbb{R}^{n \times n}, \epsilon > 0$:

$$ \begin{aligned} A_k^\top P + P A_k &\le -\epsilon P, \\ PB &= C^\top. \end{aligned} $$

as a condition for (i)ISS. This assertion is equivalent to having absolute stability on a complex sector i.e., $\exists K \in \mathbb{C}^{m \times q}, \tilde{r} > 0$ such that $\forall \phi \in \{\psi : \mathbb{C}^q \to \mathbb{C}^m, \forall y \in \mathbb{C}^q: |\psi(y) - Ky| \le \tilde{r}|y|\}$ the origin of the associated system is globally asymptotically stable (KYP Lemma 6.3 in [10], Lemma 2.2 in [6] and Theorem 5.6.22, page 703 in [9]).

Aiming on filling this gap, and differently to the state of the art works on the ISS property of Lur'e systems such as [14], we do not heavily leverage the Lur'e structure, and we do not assume absolute stability on a complex sector. Instead, we capitalize on the Lipschitz continuity property of the nonlinearity to derive sufficient conditions for (i)ISS: it is shown that under the assumption that the system has a globally asymptotically stable, locally Lipschitz, stabilizing output feedback, the system is (i)ISS. This set of conditions was inspired by the equivalence between the absolute stability property, and the existence of a stabilizing output feedback in the sector with output asymptotic stability [11].

## II. PRELIMINARIES

### Notation

We denote by $\mathbb{R}$ the set of real numbers, $\mathbb{R}_+ = \{s \in \mathbb{R} : s \ge 0\}$. For $x \in \mathbb{R}$ we denote by $\lfloor x \rfloor$ the biggest integer smaller than $x$. Euclidean norm for a vector $x \in \mathbb{R}^n$ is defined as $|x|$, for a matrix, the symbol $| \cdot |$ denotes the induced norm. The definitions of standard stability notions (as asymptotic stability, for instance) can be found in [10].

### A. *Sectors and comparison functions*

We call real sector centered in $K \in \mathbb{R}^{m \times q}$ of radius $r > 0$ as the set defined by $S_K(K, r) := \{\phi : \mathbb{R}^q \to \mathbb{R}^m \text{ such that } \forall y \in \mathbb{R}^q : |\phi(y) - Ky| \le r|y|\}$ [14].

For $K \in \mathbb{R}^{m \times q}, r > 0$, we denote the ball centered in $K$ of

<footnote n="1">¹Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France.</footnote>
radius $r$ by $B_R(K, r) := \{\Delta \in \mathbb{R}^{m \times q} \text{ such that } |\Delta - K| \le r\}$.

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 radially unbounded. A continuous function $\sigma: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{L}$ if it is strictly decreasing to zero. A continuous function $\beta: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{KL}$ if $\beta(\cdot, r) \in \mathcal{K}$ and $\beta(r, \cdot) \in \mathcal{L}$ for each fixed $r > 0$.

## B. Derivatives

For a continuous function $V: \mathbb{R}^n \to \mathbb{R}$, the upper Dini derivative at $x \in \mathbb{R}^n$ in the direction of $d \in \mathbb{R}^n$ is defined by

$$DV(x)d := \limsup_{\epsilon \to 0^+} \frac{V(x + \epsilon d) - V(x)}{\epsilon}.$$

For a system

$$\dot{x}(t) = f(x(t), t), t \ge 0,$$

where $x(t) \in \mathbb{R}^n$ and $f: \mathbb{R}^n \times \mathbb{R}_+ \to \mathbb{R}^n$ is locally Lipschitz continuous with respect to the first variable, the derivative along the trajectory $x(t)$ for $V(t) := V(x(t))$, is denoted by $\dot{V}: \mathbb{R}_+ \to \mathbb{R}$. When $V$ is locally Lipschitz, we get $\forall t \ge 0: \dot{V}(t) = DV(x(t))f(x(t), t)$ ([3], page 8).

## C. Output stability

**Definition 1:** A system

$$\begin{cases} \dot{x} = f(x) \\ y = h(x) \end{cases}$$

where $x \in \mathbb{R}^n$, $f: \mathbb{R}^n \to \mathbb{R}^n$ is locally Lipschitz continuous and $h: \mathbb{R}^n \to \mathbb{R}^q$ is a continuous function, is called:

1. Output globally asymptotically stable (O-GAS) if there exists $\beta \in \mathcal{KL}$ such that,

$$\forall t \ge 0: |y(t)| \le \beta(|x(0)|, t), \forall x(0) \in \mathbb{R}^n.$$

2. Output globally exponentially stable (O-GES) if there exist $k > 0, \lambda > 0$ such that,

$$\forall t \ge 0: |y(t)| \le k|x(0)|e^{-\lambda t}, \forall x(0) \in \mathbb{R}^n.$$

## D. Absolute stability of Lur'e systems

Consider the system

$$(L): \begin{cases} \dot{x} = Ax + Bu, \\ u = \phi(y) \\ y = Cx \end{cases}$$

where $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{q \times n}, \phi: \mathbb{R}^q \to \mathbb{R}^m$ is a locally-Lipschitz nonlinearity such that $\phi(0) = 0$.

**Definition 2:** The system (L) is said to be absolutely stable on the sector $S_R(K, r)$, for some $K \in \mathbb{R}^{m \times q}$ and $r > 0$, if for all $\phi \in S_R(K, r)$, the origin of the associated system (L) is globally asymptotically stable.

**Theorem 1:** [11] Let $K \in \mathbb{R}^{m \times q}$ and $r > 0$. The system (L) is absolutely stable on $S_R(K, r)$ if and only if $A + BKC$ is Hurwitz, and (L) is O-GAS for any nonlinearity in $S_R(K, r)$.

## E. ISS and integral-ISS

Consider the system

$$(E): \dot{x} = f(x, d),$$

where $f: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ is locally Lipschitz in the state $x \in \mathbb{R}^n$ and the input $d \in \mathbb{R}^m$.

**Definition 3:** [15] The system (E) is called ISS if there exist functions $\gamma \in \mathcal{K}$ and $\beta \in \mathcal{KL}$ such that for any initial state $x(0) \in \mathbb{R}^n$, and any essentially bounded input $d$, the solution $x(\cdot)$ exists for all $t \ge 0$ and satisfies:

$$|x(t)| \le \beta(|x(0)|, t) + \gamma(\underset{0 \le s \le t}{\text{ess sup}} |d(s)|).$$

**Definition 4:** [2] The system (E) is called iISS if there exist functions $\alpha \in \mathcal{K}_\infty, \beta \in \mathcal{KL}, \gamma \in \mathcal{K}$ such that for any initial state $x(0) \in \mathbb{R}^n$, and any input $d$, the solution $x(\cdot)$ exists for all $t \ge 0$ and satisfies:

$$\alpha(|x(t)|) \le \beta(|x(0)|, t) + \int_{0}^{t} \gamma(|d(s)|)ds.$$

The following theorem represents the sufficient part of the conventional results [2], [15], restricted to locally Lipschitz Lyapunov functions (the proof is sketched for completeness):

**Theorem 2:** Suppose there exists a locally Lipschitz function $V: \mathbb{R}^n \to \mathbb{R}_+$ for which there exist $\alpha_1, \alpha_2, \gamma \in \mathcal{K}_\infty$ and a real valued function $\alpha_3: \mathbb{R}_+ \to \mathbb{R}_+$ such that for all $x \in \mathbb{R}^n, d \in \mathbb{R}^m$:

$$\alpha_1(|x|) \le V(x) \le \alpha_2(|x|) \quad (1)$$

$$DV(x)f(x, d) \le -\alpha_3(|x|) + \gamma(|d|). \quad (2)$$

1) If $\alpha_3$ is positive definite, then the system (E) is iISS.

2) If $\alpha_3 \in \mathcal{K}_\infty$, then the system (E) is ISS.

**Remark 1:** V is called an (i)ISS Lyapunov function of (E).

*Proof:* 1) In this case we can use Lemmas 3.4 and 4.4 from [10] and the same idea of the proof of 2 $\implies$ 1 in Theorem 1 from [2].

2) Here we can use 1.1. Theorem from [3] and the respective steps of the proof of Theorem 4.19 from [10]. ■

## III. PROBLEM STATEMENT

The paper addresses the problem of (i)ISS analysis for a forced Lur'e system:

$$(S) \begin{cases} \dot{x} = f(x, d) := Ax + B\phi(y) + d, \\ y = Cx \end{cases}$$

where $x \in \mathbb{R}^n$ is the state, $d \in \mathbb{R}^n$ is the disturbance, $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{q \times n}$, and $\phi: \mathbb{R}^q \to \mathbb{R}^m$ is a locally Lipschitz nonlinearity such that $\phi(0) = 0$.

Our goal is to derive (i)ISS criteria for the system (S) from some type of asymptotic stability of the origin of the system (S) based on the fact that the nonlinearity satisfies some sort of continuity and it is in a real sector.
IV. MAIN RESULT

Recall that according to Theorem 1, the absolute stability of (S) is equivalent to O-GAS on the sector and the matrix $A+BKC$ being Hurwitz. Our first set of conditions is inspired by this equivalence:

**Theorem 3:** Given some $K \in \mathbb{R}^{m \times q}$ and $\tilde{r} > 0$, with $A+BKC$ being Hurwitz, consider the sector $S_{\mathbb{R}}(K, \tilde{r})$ composed by global Lipschitz functions. If the associated system (S) is O-GES for $d = 0$ and any such $\phi \in S_{\mathbb{R}}(K, \tilde{r})$, then the system (S) is ISS.

Thus, we assume that the system is absolutely stable and additionally strengthening O-GAS property to O-GES and imposing a stricter requirement on regularity of $\phi$ (the latter property is naturally satisfied in many popular examples of nonlinearities, as saturation or deadzone, for instance).

*Proof:* Let $\phi \in S_{\mathbb{R}}(K, \tilde{r})$ be globally Lipschitz such that the associated system (S) is O-GES for $d = 0$, then $f$ admits the same property. Let $k > 0$ be the Lipschitz constant of $f$. **1st Part.** We construct the first part of our candidate Lyapunov function.

$A_k := A + BKC$ is Hurwitz, therefore, for $\alpha > 0$ there exists a unique matrix $0 < P = P^\top \in \mathbb{R}^{n \times n}$ such that: $A_k^\top P + P A_k = -3\alpha I_n$.

Denote $\phi_k(y) := \phi(y) - Ky$ for $y \in \mathbb{R}^q$, and consider the candidate Lyapunov function $V(x) := x^\top P x$, $x \in \mathbb{R}^n$.
Clearly, $\exists \alpha_1, \alpha_2 > 0$ such that $\forall x \in \mathbb{R}^n$:

$$ \alpha_1 |x|^2 \leq V(x) \leq \alpha_2 |x|^2. $$

Next, $\forall x \in \mathbb{R}^n, d \in \mathbb{R}^n$:

$$ 
\begin{aligned}
\nabla V(x)f(x, d) &= 2x^\top P \dot{x} \\
&= 2x^\top P(A_k + B\phi(Cx) - BKCx + d) \\
&= -3\alpha x^\top x + 2x^\top PB\phi_k(Cx) + 2x^\top Pd,
\end{aligned}
 $$

Let $\rho > 0$ be such that $\rho P BB^\top P \le \alpha I_n$. Using Young's inequality we obtain:

$$ 2x^\top P B \phi_k(Cx) \le \frac{1}{\rho} |\phi_k(Cx)|^2 + \rho x^\top P B B^\top P x, $$

and:

$$ 2x^\top P d \le \alpha |x|^2 + \frac{1}{\alpha} |P d|^2. $$

Consequently,

$$ \nabla V(x)f(x, d) \le \frac{1}{\rho} |\phi_k(Cx)|^2 + \frac{1}{\alpha} |Pd|^2 - \alpha |x|^2. $$

We proved that $\exists \omega, \sigma > 0$ such that:

$$ \nabla V(x)f(x, d) \le \omega |\phi_k(Cx)|^2 + \sigma |d|^2 - \alpha |x|^2. \quad (3) $$

Now we change $V$ to obtain a more general power $r > 2$ on the bounding terms. For an arbitrary $r > 2$, denote $p = \frac{r-2}{2}$ and define

$$ U_r(x) := \frac{1}{p+1} V(x)^{p+1}. \quad (4) $$

We have $\forall x \in \mathbb{R}^n$:

$$ \frac{2}{r} \alpha_1^{\frac{r}{2}} |x|^r \le U_r(x) \le \frac{2}{r} \alpha_2^{\frac{r}{2}} |x|^r, \quad (5) $$

from inequality (3) we get,

$$ \nabla U_r(x)f(x, d) \le V(x)^p (\omega |\phi_k(Cx)|^2 + \sigma |d|^2 - \alpha |x|^2). $$

Young's inequality implies $\forall x \in \mathbb{R}^n, \alpha_3 > 0$:

$$ 
\begin{aligned}
V(x)^p |\phi_k(Cx)|^2 &\le \frac{2}{r} \left(\frac{1}{\alpha_3}\right)^{\frac{r}{2}} |\phi_k(Cx)|^r + \frac{r-2}{r} \alpha_3^{\frac{r}{r-2}} \\
&\qquad V(x)^{p+1}, \\
\sigma V(x)^p |d|^2 &\le \frac{p}{p+1} \alpha_3^{\frac{p+1}{p}} V(x)^{p+1} + \frac{1}{p+1} \left(\frac{\sigma}{\alpha_3}\right)^{p+1} |d|^{2(p+1)},
\end{aligned}
 $$

we obtain $\forall \alpha_3 > 0, x \in \mathbb{R}^n, d \in \mathbb{R}^n$:

$$ 
\begin{aligned}
\nabla U_r(x)f(x, d) &\le \frac{2\omega}{r} \alpha_3^{-\frac{r}{2}} |\phi_k(Cx)|^r + \frac{|d|^{2(p+1)}}{p+1} \left(\frac{\sigma}{\alpha_3}\right)^{p+1} \\
&\quad - \alpha V(x)^p |x|^2 + \left(\frac{p}{p+1} \alpha_3^{\frac{p+1}{p}} + \omega \frac{r-2}{r} \alpha_3^{-\frac{r}{r-2}}\right) V(x)^{p+1}.
\end{aligned}
 $$

Denote $\Gamma(\alpha_3) = \frac{p}{p+1} \alpha_3^{\frac{p+1}{p}} + \omega \frac{r-2}{r} \alpha_3^{-\frac{r}{r-2}}$. $V(x) \le \alpha_2 |x|^2$, therefore, $\forall \alpha_3 > 0, x \in \mathbb{R}^n, d \in \mathbb{R}^n$:

$$ \nabla U_r(x)f(x, d) \le \frac{2\omega}{r} \alpha_3^{-\frac{r}{2}} |\phi_k(Cx)|^r + \frac{1}{p+1} \left(\frac{\sigma}{\alpha_3}\right)^{p+1} |d|^{2(p+1)} - \left(\frac{\alpha}{\alpha_2} - \Gamma(\alpha_3)\right) V(x)^{p+1}. $$

Since $\frac{\alpha_2}{\alpha_2} > 0$ and $\Gamma(\alpha_3) \xrightarrow{\alpha_3 \to 0} 0$, let $\alpha_3 > 0$ be such that $\frac{\alpha_2}{\alpha_2} - \Gamma(\alpha_3) > 0$.

By replacing $p$ with its value $\frac{r-2}{2}$, we conclude that $\exists \gamma_1, \omega_1, \sigma_1 > 0$ such that $\forall x \in \mathbb{R}^n, d \in \mathbb{R}^n$:

$$ \nabla U_r(x)f(x, d) \le \omega_1 |\phi_k(Cx)|^r + \sigma_1 |d|^r - \gamma_1 V(x)^{\frac{r}{2}}. $$

Recalling that $\alpha_1 |x|^2 \le V(x)$, we conclude the 1st part.

**Conclusion of 1st Part.** For $r > 2$ and $U_r$ defined as in (4), $\exists \gamma_1, \omega_1, \sigma_1 > 0$ such that $\forall x, d \in \mathbb{R}^n$:

$$ \nabla U_r(x)f(x, d) \le \omega_1 |\phi_k(Cx)|^r + \sigma_1 |d|^r - \gamma_1 |x|^r. \quad (6) $$

2nd Part. We construct the second part of our candidate Lyapunov function.

For $x_0 \in \mathbb{R}^n$, denote $t \mapsto x(t, x_0)$ the solution of (S) with $d = 0$. The system (S) is O-GES, hence, $\exists \lambda, \kappa > 0$ such that $\forall x_0 \in \mathbb{R}^n, t \ge 0$:

$$ |Cx(t, x_0)| \le \kappa |x_0| e^{-\lambda t}. $$

Let $r > 2$ be an integer such that $\lambda(r-1) > k$. Let $\gamma > 0$ and define:

$$ W_{\gamma}(x_0) := \gamma \int_{0}^{\infty} |Cx(t, x_0)|^r dt, \quad (7) $$

which is well defined and continuous, since $(t, x_0) \in \mathbb{R}_+ \times \mathbb{R}^n \mapsto |Cx(t, x_0)|^r$ is continuous, and $\forall x_0 \in \mathbb{R}^n, t \ge 0$:

$$ |Cx(t, x_0)|^r \le \kappa^r |x_0|^r e^{-\lambda rt}. $$

We have $\forall x_0 \in \mathbb{R}^n, t \ge 0$:

$$ 0 \le W_{\gamma}(x_0) \le \frac{\gamma \kappa^r}{r \lambda} |x_0|^r. \quad (8) $$
For $x_0 \in \mathbb{R}^n, t \ge 0$:

$$
\begin{align*}
W_{\gamma}(x(t, x_0)) - W_{\gamma}(x_0) &= \gamma \int_0^{\infty} |Cx(t+s, x_0)|^r ds - W_{\gamma}(x_0) \\
&= \gamma \int_t^{\infty} |Cx(s, x_0)|^r ds - W_{\gamma}(x_0) \\
&= -\gamma \int_0^t |Cx(s, x_0)|^r ds.
\end{align*}
$$

Consequently, $\forall x \in \mathbb{R}^n$:

$$
DW_{\gamma}(x)f(x, 0) \leq -\gamma|x|^{r}. \quad (9)
$$

Now we prove that $W_\gamma$ is locally Lipschitz.

For any $x_1, x_2 \in \mathbb{R}^n$:

$$
\begin{align*}
|W_{\gamma}(x_1) - W_{\gamma}(x_2)| &= \gamma \left| \int_{0}^{\infty} (|Cx(s, x_1)|^r - |Cx(s, x_2)|^r) ds \right| \\
&\leq \gamma \int_{0}^{\infty} |C(x(s, x_1) - x(s, x_2))| r \kappa^{r-1} (\max(|x_1|, |x_2|))^{r-1} \\
&\phantom{\leq} e^{-\lambda(r-1)s} ds \\
&\leq \gamma r |C| \kappa^{r-1} (\max(|x_1|, |x_2|))^{r-1} \int_{0}^{\infty} e^{-(\lambda(r-1)-k)s} ds |x_1 - x_2|,
\end{align*}
$$

Consequently,

$$
|W_{\gamma}(x_1) - W_{\gamma}(x_2)| \le \chi(\max(|x_1|, |x_2|))^{r-1} |x_1 - x_2|, \quad (10)
$$

$$
\text{where } \chi = \gamma r |C| \kappa^{r-1} \int_{0}^{\infty} e^{-(\lambda(r-1)-k)s} ds.
$$

In the first inequality we used the O-GES bound and the relation:

$$
a^n - b^n \le n|a - b| \max(a, b)^{n-1}. \quad (11)
$$

In the second inequality we used:

$$
x(t, x_1) - x(t, x_2) = \int_{0}^{t} (f(x(s, x_1), 0) - f(x(s, x_2), 0))ds,
$$

and Gronwall's inequality to obtain:

$$
|C(x(s, x_1) - x(s, x_2))| \leq |C||x_1 - x_2|e^{ks}.
$$

Further, for any $x, d \in \mathbb{R}^n$:

$$
\begin{align*}
DW_{\gamma}(x)f(x, d) &= \limsup_{\epsilon \to 0^+} \frac{W_{\gamma}(x+\epsilon f(x,d))-W_{\gamma}(x)}{\epsilon} = \limsup_{\epsilon \to 0^+} \left( \frac{W_{\gamma}(x+\epsilon f(x,d))-W_{\gamma}(x+\epsilon f(x,0))}{\epsilon} + \frac{W_{\gamma}(x+\epsilon f(x,0))-W_{\gamma}(x)}{\epsilon} \right) \\
&\leq \limsup_{\epsilon \to 0^+} \frac{W_{\gamma}(x+\epsilon f(x,d))-W_{\gamma}(x+\epsilon f(x,0))}{\epsilon} + DW_{\gamma}(x)f(x, 0)
\end{align*}
$$

From (9), (10) we obtain:

$$
DW_{\gamma}(x)f(x, d) \leq -\gamma|x|^{r} + \chi|x|^{r-1}|d|
$$

Young's inequality implies $\forall x \in \mathbb{R}^n, \alpha_4 > 0$:

$$
\chi|x|^{r-1}|d| < \frac{r-1}{r} \alpha_4^{\frac{r}{r-1}} |x|^r + \frac{1}{r} \left( \frac{\chi}{\alpha_4} \right)^r |d|^r.
$$

Therefore, we obtain $\forall \alpha_4 > 0, x \in \mathbb{R}^n, d \in \mathbb{R}^n$:

$$
DW_{\gamma}(x)f(x, d) \leq -\gamma|x|^{r} + \frac{1}{r} \left( \frac{\chi}{\alpha_{4}} \right)^{r} |d|^{r} + \frac{r-1}{r} \alpha_{4}^{\frac{r}{r-1}} |x|^{r}.
$$

**Conclusion of 2nd Part.** For any $\gamma > 0$ and $W_\gamma$ defined as in (7), $\forall \alpha_5 > 0, \exists h > 0$ such that $\forall x \in \mathbb{R}^n, d \in \mathbb{R}^n$:

$$
DW_{\gamma}(x)f(x, d) \leq -\gamma|x|^{r} + h|d|^{r} + \alpha_{5}|x|^{r}. \quad (12)
$$

Since $\phi \in S_R(K, \tilde{r})$, then, $\forall x \in \mathbb{R}^n: |\phi_k(Cx)| \le \tilde{r}|Cx|$. Denote $\omega_2 := \omega_1 \tilde{r}^r$, and let $\gamma \ge \omega_2, 0 < \alpha_5 < \gamma_1$. Consider

the candidate Lyapunov function

$$
V_{r,\gamma}(x) := U_r(x) + W_{\gamma}(x).
$$

From (5), (8) we get $\forall x \in \mathbb{R}^n$:

$$
\frac{2}{r} \alpha_1^{\frac{r}{r-1}} |x|^r \leq V_{r,\gamma}(x) \leq \frac{2}{r} \alpha_2^{\frac{r}{r-1}} |x|^r + \frac{\gamma \kappa^r}{r \lambda} |x|^r,
$$

and from (6), (12) we get $\forall x, d \in \mathbb{R}^n$:

$$
DV_{r,\gamma}(x)f(x, d) \le -(\gamma - \omega_2)|Cx|^r - (\gamma_1 - \alpha)|x|^r + (\sigma_1 + h)|d|^r
$$

**Conclusion:** $\exists \mu_1 > 0, \mu_2 > 0$ such that $\forall x \in \mathbb{R}^n, d \in \mathbb{R}^n$:

$$
DV_{r,\gamma}(x)f(x, d) \leq -\mu_1|x|^r + \mu_2|d|^r.
$$

We proved the existence of an ISS-Lyapunov function for the system (S), therefore, (S) is ISS. ■

**Remark 2:** Since $\phi(0) = 0$, a sector condition can be obtained from the global Lipschitz property, hence, this result means that the system (S) with a globally Lipschitz nonlinearity being O-GES, is ISS.

**Corollary 1:** If $\phi$ is globally Lipschitz continuous, and the associated system (S) is GES, then the system (S) is ISS.

**Proof:** The result can be substantiated repeating the steps of the proof of Theorem 3 by replacing O-GES with GES. ■

Assuming that $\phi$ is also continuously differentiable, this result can be found in Lemma 4.6 of [10] (for the case of non-autonomous system (S)).

**Remark 3:** Two observations are in order:

1) We only need the global-Lipschitz property to get the Lipschitz continuity of the solutions in the initial conditions, which can be replaced by the requirement on the output to satisfy $\forall x_1, x_2 \in \mathbb{R}^n, t \ge 0$:

$$
|Cx(t, x_1) - Cx(t, x_2)| \leq |x_1 - x_2|\zeta(t),
$$

for some $\zeta \in \mathcal{K}$.

2) In such a case the O-GES property can be avoided since the solution growth rate $\zeta$ is not necessarily the one we get from Gronwall's inequality and it doesn't necessarily increase exponentially.

This guides us towards the following relaxations.

**Theorem 4:** Given some $K \in \mathbb{R}^{m \times q}$ and $\tilde{r} > 0$, with $A + BKC$ being Hurwitz, for any $\phi \in S_R(K, \tilde{r})$, let the associated system (S) with $d = 0$ satisfy:

1) $\exists \sigma \in \mathcal{L}, \kappa > 0$ such that $\forall x_0 \in \mathbb{R}^n, t \ge 0$:

$$
|Cx(t, x_0)| \leq \kappa |x_0| \sigma(t),
$$

2) $\exists \zeta : \mathbb{R}_+ \to \mathbb{R}_+$ and $L \ge 1$ such that $\forall x_1, x_2 \in \mathbb{R}^n, t \ge 0$:

$$
|Cx(t, x_1) - Cx(t, x_2)| \leq |x_1 - x_2|(L + |x_1| + |x_2|)\zeta(t),
$$

3) $\exists r_1, r_2 > 0$ such that $t \mapsto \sigma(t)^{r_1}$ and $t \mapsto \sigma(t)^{r_2}\zeta(t)$ are integrable on $\mathbb{R}_+$.

Then the system (S) is iISS.

In this case we consider a general real sector (the Hurwitz property of $A+BKC$ is necessary for absolute stability). The property 1) implies O-GAS of (S) (which is also
necessary for absolute stability [11]) for any $\phi$, and the only requirement here is linearity of the upper bound on $|y(t)|$ in $|x_0|$. Condition 2) relaxes the previously imposed global Lipschitz continuity requirement to a kind of local one (note that $\zeta$ may be growing with time). The third condition is technical. *Proof:* Let $r = \max(r_1, r_2 + 1)$, then $t \mapsto \sigma(t)^r$ and $t \mapsto \sigma(t)^{r-1}\zeta(t)$ are integrable on $\mathbb{R}_+$. $A+BKC$ is Hurwitz, therefore, from the sector condition, and the conclusion of 1st Part in the proof of Theorem 3 we obtain a candidate Lyapunov function $U_r: \mathbb{R}^n \to \mathbb{R}_+$ satisfying $\exists \gamma_1, \omega, \sigma > 0$ such that $\forall x, d \in \mathbb{R}^n$:

$$ \nabla U_r(x)f(x, d) \leq \omega |Cx|^r + \sigma |d|^r - \gamma_1 |x|^r. \quad (13) $$

Let $\gamma > \omega$, and define $W(x_0) := \gamma \int_0^\infty |Cx(t, x_0)|^r dt$. $W$ satisfies $\forall x_0 \in \mathbb{R}^n, t \ge 0$:

$$ 0 \le W(x_0) \le \gamma \kappa^r \int_0^\infty \sigma(t)^r dt |x_0|^r, \quad (14) $$

$$ DW(x)f(x, 0) \le -\gamma |Cx|^r, $$

and $\forall x_1, x_2 \in \mathbb{R}^n$:

$$ \begin{aligned} |W(x_1) - W(x_2)| &= \gamma \left| \int_0^\infty (|Cx(s, x_1)|^r - |Cx(s, x_2)|^r) ds \right| \\ &\le \gamma \int_0^\infty |C(x(s, x_1) - x(s, x_2))| r \kappa^{r-1} (\max(|x_1|, |x_2|))^{r-1} \\ &\qquad \sigma(s)^{r-1} ds \end{aligned} $$

$$ \le \chi(\max(|x_1|, |x_2|))^{r-1} (L + |x_1| + |x_2|) |x_1 - x_2|, $$

where $\chi := \gamma r \kappa^{r-1} \int_0^\infty \sigma(s)^{r-1} \zeta(s) ds$.

As a result, $W$ is locally Lipschitz. In the first step we used the same inequality (11) as before and the O-GAS hypothesis, and the last step is a direct consequence of the second output condition. Through the same calculations as in the last proof we obtain $\forall x, d \in \mathbb{R}^n$:

$$ DW(x)f(x, d) \le -\gamma |Cx|^r + \chi |x|^{r-1} (L + 2|x|) |d|. \quad (15) $$

Denoting $V_1 = U_r + W$, from (5), (14) we have $\exists \alpha_1 > 0, \alpha_2 > 0$ such that $\forall x \in \mathbb{R}^n$:

$$ \alpha_1 |x|^r \le V_1(x) \le \alpha_2 |x|^r, $$

and from (13), (15) we have:

$$ DV_1(x)f(x, d) \le -\gamma_1 |x|^r + \sigma |d|^r + \chi |x|^{r-1} (L + 2|x|) |d|. $$

Consider the candidate Lyapunov function $V_2(x) := \ln(1 + V_1(x))$. We have $\forall x, d \in \mathbb{R}^n$:

$$ \ln(1 + \alpha_1 |x|^r) \le V_2(x) \le \ln(1 + \alpha_2 |x|^r), $$

and form the lower bound of $V_1$ we obtain:

$$ \begin{aligned} DV_2(x)f(x, d) \le & -\gamma_1 \frac{|x|^r}{1 + \alpha_2 |x|^r} + \sigma |d|^r \\ & + \frac{\chi |x|^{r-1} (L + 2|x|)}{1 + \alpha_1 |x|^r} |d|, \end{aligned} $$

the map $x \in \mathbb{R}^n \mapsto \frac{|x|^{r-1}(L+2|x|)}{1+\alpha_1|x|^r}$ is bounded, and the map $x \in \mathbb{R}^n \mapsto \gamma_1 \frac{|x|^r}{1+\alpha_2|x|^r}$ is positive-definite, therefore, $V_2$ is

an iISS-Lyapunov function for the system (S), thus, (S) is iISS. ■

If we restrict the sector instead, we can relax the Lipschitz property even more and obtain ISS:

**Theorem 5:** Given some $K \in \mathbb{R}^{m \times q}$ and $\tilde{r}_1 > 0$, with $A+BKC$ being Hurwitz, for any $\phi \in S_{\mathbb{R}}(K, \tilde{r}_1)$ satisfying $\exists 0 \le \alpha < 1$ such that $\forall y \in \mathbb{R}^q: |\phi(y) - Ky| \le \tilde{r}_2|y|^\alpha$, let the associated system (S) with $d=0$ satisfy:

1) $\exists \sigma \in \mathcal{L}, k > 0$ such that $\forall x_0 \in \mathbb{R}^n, t \ge 0$:

$$ |Cx(t, x_0)| \le \kappa |x_0| \sigma(t), $$

2) $\exists \zeta: \mathbb{R}_+ \to \mathbb{R}_+$, $L \ge 1$ and $r_0 \ge 0$ such that $\forall x_1, x_2 \in \mathbb{R}^n, t \ge 0$:

$$ |Cx(t, x_1) - Cx(t, x_2)| \le |x_1 - x_2|(L + |x_1| + |x_2|)^{r_0} \zeta(t), $$

3) $\exists r_1, r_2 > 0$ such that $t \mapsto \sigma(t)^{r_1}$ and $t \mapsto \sigma(t)^{r_2} \zeta(t)$ are integrable on $\mathbb{R}_+$.

Then the system (S) is ISS.

**Proof:** Suppose with no loss of generality that $r_0$ is an integer (we can always replace it with $\lfloor r_0 \rfloor + 1$).

Let $\tilde{r} = \max(\tilde{r}_1, \tilde{r}_2)$ and $r > \max(r_1, r_2 + 1, \frac{\alpha r_0}{1-\alpha}, 2)$ be an integer, then $t \mapsto \sigma(t)^r$ and $t \mapsto \sigma(t)^{r-1}\zeta(t)$ are integrable on $\mathbb{R}_+$. $A+BKC$ is Hurwitz, therefore, we get from the conclusion of 1st Part in the proof of Theorem 3 a candidate Lyapunov function $U_{r+r_0}$ satisfying $\exists \gamma_1, \omega, \sigma > 0$ such that $\forall x, d \in \mathbb{R}^n$:

$$ \nabla U_{r+r_0}(x)f(x, d) \le \omega |\phi_k(Cx)|^{r+r_0} + \sigma |d|^{r+r_0} - \gamma_1 |x|^{r+r_0}. $$

We have $\forall x \in \mathbb{R}^n: |\phi_k(Cx)| \le \tilde{r} \min(|Cx|, |Cx|^\alpha)$, hence, $\forall x \in \mathbb{R}^n$:

$$ |\phi_k(Cx)|^{r+r_0} \le \tilde{r}^{r+r_0} \min(|Cx|^{r+r_0}, |Cx|^{\alpha(r+r_0)}). $$

For $x \in \mathbb{R}^n$, if $|Cx| \le 1$, then $|Cx|^{r+r_0} \le |Cx|^r$, and if $|Cx| > 1$, since $r \ge \frac{\alpha r_0}{1-\alpha}$ we obtain $|Cx|^{\alpha(r+r_0)} \le |Cx|^r$, therefore, $\forall x \in \mathbb{R}^n: |\phi_k(Cx)|^{r+r_0} \le \tilde{r}^{r+r_0} |Cx|^r$, thus, $\forall x, d \in \mathbb{R}^n$:

$$ \nabla U_{r+r_0}(x)f(x, d) \le \tilde{r}^{r+r_0} \omega |Cx|^r + \sigma |d|^{r+r_0} - \gamma_1 |x|^{r+r_0}. $$

Let $\gamma > \tilde{r}^{r+r_0}\omega$, and define $W(x_0) := \gamma \int_0^\infty |Cx(t, x_0)|^r dt$. By repeating the same derivations as before, it is possible to show that $\forall x_0 \in \mathbb{R}^n, t \ge 0$:

$$ 0 \le W(x_0) \le \gamma \kappa^r \int_0^\infty \sigma(t)^r dt |x_0|^r, $$

$$ DW(x)f(x, 0) \le -\gamma |Cx|^r, $$

For any $x_1, x_2 \in \mathbb{R}^n$:

$$ \begin{aligned} |W(x_1) - W(x_2)| &= \gamma \left| \int_0^\infty (|Cx(s, x_1)|^r - |Cx(s, x_2)|^r) ds \right| \\ &\le \gamma \int_0^\infty |C(x(s, x_1) - x(s, x_2))| r \kappa^{r-1} (\max(|x_1|, |x_2|))^{r-1} \\ &\qquad \sigma(s)^{r-1} ds \end{aligned} $$

$$ \le \chi(\max(|x_1|, |x_2|))^{r-1} (L + |x_1| + |x_2|)^{r_0} |x_1 - x_2|, $$

where $\chi := \gamma r \kappa^{r-1} \int_0^\infty \sigma(s)^{r-1} \zeta(s) ds$.
Through the same calculations as in the proof of Theorem 4 we obtain $\forall x, d \in \mathbb{R}^n$:

$$DW(x)f(x, d) \leq -\gamma |Cx|^r + \chi |x|^{r-1} (L + 2|x|)^{r_0} |d|,$$

if $|x| \le 1$, then $\exists \theta_1 > 0$ such that $\chi|x|^{r-1}(L+2|x|)^{r_0} \le \theta_1$, and if $|x| > 1$, then $|x|^{r-1} \le |x|^{r+r_0-1}$, hence, $\exists \theta_2 > 0$ such that $\chi|x|^{r-1}(L+2|x|)^{r_0}|d| \le \theta_2|x|^{r+r_0-1}|d|$, therefore, $\forall x, d \in \mathbb{R}^n$:

$$DW(x)f(x, d) \leq -\gamma |Cx|^r + (\theta_1 + \theta_2|x|^{r+r_0-1})|d|.$$

Young's Inequality implies that $\forall \alpha_1 > 0$, $\exists \alpha_2 > 0$ such that $\forall x, d \in \mathbb{R}^n$:

$$\theta_2|x|^{r+r_0-1}|d| \leq \frac{r+r_0-1}{r+r_0} \alpha_1^{\frac{r+r_0}{r+r_0-1}} |x|^{r+r_0} + \alpha_2|d|^{r+r_0}$$

Choosing $\alpha_1 > 0$ such that $\frac{r+r_0-1}{r+r_0} \alpha_1^{\frac{r+r_0}{r+r_0-1}} < \gamma_1$, and defining $V(x) := U_{r+r_0}(x) + W(x)$, $x \in \mathbb{R}^n$, we get $\exists \nu_1 > 0, \nu_2 > 0, \nu_3 > 0, \mu_1 > 0, \mu_2 > 0$ such that $\forall x, d \in \mathbb{R}^n$:

$$\nu_1|x|^{r+r_0} \leq V_1(x) \leq \nu_2|x|^{r+r_0} + \nu_3|x|^r \\ DV(x)f(x, d) \leq -\mu_1|x|^{r+r_0} + \mu_2(|d| + |d|^{r+r_0})$$

As a result, the system (S) is ISS.

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

The obtained in theorems 4 and 5 conditions of iISS and ISS, respectively, have a clear physical meaning (e.g., 1) defines a kind of output stability, while 2) fixes local Lipschitz continuity of the solutions), but their verification requires a separate analysis of (S), which may be based on design and investigation of other Lyapunov functions (for example, to establish the needed output asymptotic stability). On the other hand, the restrictions imposed on a real sector for $\phi$ are mild. The kind of regularity of the nonlinearity can be a less constraining hypothesis in applications than absolute stability in a complex sector.

Here is a simple example showing how one may use these results:

Example 1: Let:

$$A = \begin{pmatrix} -a_1 & b & c \\ 0 & -a_2 & 0 \\ 0 & 1 & -a_3 \end{pmatrix}, B = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, C = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix},$$

with $b, c \in \mathbb{R}$, $a_1, a_2, a_3 > 0$, therefore, A is Hurwitz.

Consider the system:

$$\dot{x} = Ax + B\phi(Cx) + d, \quad (16)$$

where $\phi$ is globally Lipschitz continuous.

Consider the candidate Lyapunov function: $V(x_2, x_3) = \frac{1}{2}(x_2^2+kx_3^2)$, with $k > 0$. The derivative along the trajectories of the system (16) satisfies:

$$\dot{V} = -a_3kx_3^2 - a_2x_2^2 + kx_2x_3 \leq k(-a_3 + \frac{\rho}{2})x_3^2 + (-a_2 + \frac{k}{2\rho})x_2^2,$$

for any positive real $\rho$.

Selecting $k < 4a_2a_3$, always there is a value of $\rho$ such that $\dot{V} \leq -\kappa V$ for some $\kappa > 0$, then the output is globally exponentially stable, and Theorem 3 implies that the system is ISS.

V. CONCLUSION

In this work a series of conditions for (i)ISS of forced Lur'e systems were obtained by playing with its regularity, assuming that it is in a real sector only, and that the associated system satisfies some type of output asymptotic stability. As a trivial consequence, when all nonlinearities in a sector satisfy these conditions, we obtain (i)ISS for all the nonlinearities in that sector. So an extension of the proposed results to discontinuous nonlinearities as in [5] can be considered as a direction for future research.

REFERENCES

[1] Mark Aronovich Aizerman, Feliks Ruvimovich Gantmakher, and Elijah Polak. Absolute stability of regulator systems. 1964.

[2] David Angeli, Eduardo D Sontag, and Yuan Wang. A characterization of integral input-to-state stability. *IEEE Transactions on Automatic Control*, 45(6):1082–1097, 2000.

[3] Francis H Clarke, Yuri S Ledyaev, Ronald J Stern, and Peter R Wolenski. *Nonsmooth analysis and control theory*, volume 178. Springer Science & Business Media, 2008.

[4] JCA De Bruin, Apostolos Doris, Nathan van de Wouw, WPMH Heemels, and Henk Nijmeijer. Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities. *Automatica*, 45(2):405–415, 2009.

[5] Matteo Della Rossa, Aneel Tanwani, and Luca Zaccarian. Non-pathological ISS-Lyapunov functions for interconnected differential inclusions. *IEEE Transactions on Automatic Control*, 67(8):3774–3789, 2021.

[6] Chris Guiver and Hartmut Logemann. A circle criterion for strong integral input-to-state stability. *Automatica*, 111:108641, 2020.

[7] Chris Guiver, Hartmut Logemann, and Mark R Opmeer. Infinite-dimensional Lur'e systems: Input-to-state stability and convergence properties. *SIAM Journal on Control and Optimization*, 57(1):334–365, 2019.

[8] Wassim M Haddad and VijaySekhar Chellaboina. *Nonlinear dynamical systems and control: a Lyapunov-based approach*. Princeton university press, 2008.

[9] Diederich Hinrichsen and Anthony J Pritchard. *Mathematical systems theory I: modelling, state space analysis, stability and robustness*, volume 48. Springer, 2005.

[10] Hassan K Khalil. *Control of nonlinear systems*. Prentice Hall, New York, NY, 2002.

[11] Xiaoxin Liao and Pei Yu. *Absolute stability of nonlinear control systems*, volume 25. Springer Science & Business Media, 2008.

[12] Mark R Liberson. Lur'e problem of absolute stability-a historical essay. *IFAC Proceedings Volumes*, 34(6):25–28, 2001.

[13] Simone Mariano, Romain Postoyan, and Luca Zaccarian. Finite-time stability properties of Lur'e systems with piecewise continuous nonlinearities. *IEEE Transactions on Automatic Control*, 2024.

[14] Elvijs Sarkans and Hartmut Logemann. Input-to-state stability of Lur'e systems. *Mathematics of Control, Signals, and Systems*, 27:439–465, 2015.

[15] Eduardo D Sontag et al. Smooth stabilization implies coprime factorization. *IEEE transactions on automatic control*, 34(4):435–443, 1989.

[16] Alberto Tesi and Antonio Vicino. Robust absolute stability of Lur'e control systems in parameter space. In *Automatic Control 1990*, pages 225–230. Elsevier, 1991.

[17] Vladimir A Yakubovich. Popov's method and its subsequent development. *European Journal of Control*, 8(3):200–208, 2002.

[18] Vladimir Andreevich Yakubovich, Gennadii Alekseevich Leonov, and Arkadii Khaimovich Gelig. *Stability of stationary sets in control systems with discontinuous nonlinearities*, volume 14. World Scientific Singapore, 2004.

[19] Alexandr A Zevin and Mark A Pinsky. A new approach to the Lur'e problem in the theory of absolute stability. *SIAM journal on control and optimization*, 42(5):1895–1904, 2003.