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# Simultaneous stabilization of quadratic input nonlinear systems

Abderrahman Iggidr, Mohamed Oumoun

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Abderrahman Iggidr, Mohamed Oumoun. Simultaneous stabilization of quadratic input nonlinear systems. 2022. hal-04038270

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# Simultaneous stabilization of quadratic input nonlinear systems

Abderrahman Iggidr, Mohamed Oumoun

► To cite this version:

Abderrahman Iggidr, Mohamed Oumoun. Simultaneous stabilization of quadratic input nonlinear systems. International Journal of Control, 2025, pp.1-13. 10.1080/00207179.2025.2472263. hal-04038270v2

HAL Id: hal-04038270

https://inria.hal.science/hal-04038270v2

Submitted on 7 Oct 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
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# Simultaneous stabilization of quadratic input nonlinear systems

A. Iggidr

Université de Lorraine, CNRS, Inria, IECL
F57000 Metz, France.

Email adress: Abderrahman.Iggidr@inria.fr

M. Oumoun

Cadi Ayyad University, National School of Applied Sciences,
Av. Abdelkrim Khattabi, B.P. 575, Marrakech, Morocco.

Email adress: m.oumoun@uca.ac.ma

## Abstract

In this paper simultaneous stabilization of a collection of quadratic input nonlinear systems is considered. With the help of the technique of control Lyapunov functions (CLFs), necessary and sufficient conditions for the existence of simultaneously stabilizing state feedback controller are derived. Moreover, we present a constructive method for designing continuous simultaneously stabilizing controller when the provided sufficient conditions are satisfied. To verify the obtained results, an illustrative example is presented.

**Keywords.** Quadratic nonlinear systems, simultaneous stabilization, control Lyapunov function, state feedback.

**2000 Mathematics Subject Classification:** 93D15.

## 1 Introduction

The simultaneous stabilization problems arise due to several reasons: system's uncertainty, failure modes or systems with various modes of operation. The aim of this problem is to determine a single controller which simultaneously stabilizes a finite collection of systems. Current applications of simultaneous stabilization include many real systems, such as Hamiltonian systems [2, 18, 19], power converters [3], chemical reaction systems [25, 26], and descriptor systems [4] to name a few. Therefore, the investigation of simultaneous stabilization problems is quite relevant from a practical point of view, and many important results have been obtained for linear systems: see [8, 9, 13, 14, 16, 22] and references therein. For nonlinear deterministic and stochastic systems, the problem is more involved, and there are fewer results in the literature, see, for examples, [10, 20, 21, 24] for nonlinear deterministic systems case, and [11, 12] for nonlinear stochastic systems case. Recently, the problem of simultaneous stabilization has been addressed for nonlinear systems with delays in [5, 6, 7].

Till now, to our knowledge, no results have been reported about simultaneous stabilization of quadratic input nonlinear systems. Here, Motivated by the approach in [10, 12], in the context
# Simultaneous stabilization of quadratic input nonlinear systems

A. Iggidr<sup>a</sup> and M. Oumoun<sup>b</sup>

<sup>a</sup> Université de Lorraine, CNRS, Inria, IECL, F57000 Metz, France. Email: abderrahman.iggidr@inria.fr

<sup>b</sup> Université Cadi Ayyad, École Nationale des Sciences Appliquées, Av. Abdelkrim Khattabi, B.P. 575, Marrakech, Morocco. Email: m.oumoun@uca.ac.ma

## ABSTRACT

In this paper simultaneous stabilization of a collection of quadratic input nonlinear systems is considered. First, with the help of the technique of control Lyapunov functions (CLFs), a necessary condition, then sufficient conditions for the existence of simultaneously stabilizing state feedback controller are derived. Moreover, we present a constructive method for designing continuous simultaneously stabilizing controller when the provided sufficient conditions are satisfied. Second, by using the sliding mode technique, this constructive method has been extended to establish a discontinuous control that simultaneously stabilizes two levitation systems.

**Keywords.** Quadratic nonlinear systems, simultaneous stabilization, control Lyapunov function, state feedback.

**2000 Mathematics Subject Classification:** 93D15.

## 1. Introduction

The simultaneous stabilization problem arises frequently in practice, due to plant uncertainty, plant variation, failure modes or plants with various modes of operation. This problem is concerned with determining a single controller which simultaneously stabilizes a finite collection of systems (Wu, 2005). Current applications of simultaneous stabilization include many real systems, such as Hamiltonian systems (Cai, 2015; Wang et al., 2007; Wei & Wang, 2014), power converters (Chaudhuri et al., 2006), chemical reaction systems (Zhu & Yang, 2016, 2017), and descriptor systems (Chen et al., 2012) to name a few.

Therefore, the investigation of simultaneous stabilization problems is quite relevant from a practical point of view, and many important results have been obtained for linear systems: see (Galindo, 2012; Guan et al., 2011; Kohan-Sedgh et al., 2016; Li et al., 2016; Miller & Chen, 2002; Yu & Chi, 2014) and references therein. For nonlinear systems, we note that it has been pointed out in (Wu, 2010) that the simultaneous stabilization is more difficult to solve. Hence, there are fewer results in the literature related to this problem, see, for examples, (Iggidr & Oumoun, 2020; Wu, 2010, 2005; Zhang et al., 2011) for nonlinear deterministic systems case, and (Iggidr & Oumoun, 2019; Iggidr et al., 2020) for nonlinear stochastic systems case. Recently, the problem of simultaneous stabilization has been addressed for nonlinear systems with delays in (Dastaviz & Binazadeh, 2019, 2020; Dastaviz et al., 2021).
of quadratic input nonlinear systems, we study the problem of designing constructively a simultaneously asymptotically stabilizing control for these systems. Based on the control Lyapunov functions approach (CLF), we first give a necessary condition for the existence of simultaneously asymptotically stabilizing feedback. After that, we present sufficient conditions for the existence of simultaneously stabilizing feedback. Furthermore, we give an explicit formula for constructing the simultaneously stabilizing controller. It is shown that, under the small control property (SCP) assumption, the obtained feedback is continuous everywhere in the state space.

# 2 Problem formulation and preliminaries

## 2.1 Problem formulation

Consider a collection of $m$ quadratic input nonlinear systems described as follows:

$$S_i: \dot{x} = f_i(x) + ug_i(x) + u^2h_i(x), \quad i \in \{1, \dots, m\} = I, \qquad (2.1)$$

where $x \in \mathbb{R}^n$ is the state, $u \in \mathbb{R}$ is the control, $f_i(\cdot), g_i(\cdot)$ and $h_i(\cdot)$ are continuous functions with $f_i(0) = 0$.

The objective of this paper is to find a continuous function $k(\cdot)$ such that the state feedback controller $u = k(x)$ globally asymptotically stabilizes the collection of systems in (2.1) simultaneously, that is, the origin is a globally asymptotically stable equilibrium for the resulting closed-loop systems

$$S_i: \dot{x} = f_i(x) + k(x)g_i(x) + k(x)^2h_i(x), \quad i \in I. \qquad (2.2)$$

## 2.2 Preliminaries

We first give some notations and basic definitions concerning control Lyapunov functions, then we recall some results on the stabilization of quadratic input nonlinear systems.

A function $V: \mathbb{R}^n \to \mathbb{R}$ is said to be positive definite if $V(x) \ge 0$ and $V(x) = 0 \iff x = 0$. $V$ is said to be proper if for any $\rho \ge 0$, the set $\{x \in \mathbb{R}^n: V(x) \le \rho\}$ is a compact set of $\mathbb{R}^n$.

For each $i = 1, 2, \dots, m$, let $V_i$ be a $C^1$ positive definite and proper function defined on $\mathbb{R}^n$. Its derivative, denoted by $\dot{V}_i$, along the trajectories of system $S_i$ can be written as

$$\dot{V}_i(x) = a_i(x)u^2 + b_i(x)u + c_i(x), \qquad (2.3)$$

with, $a_i = \langle h_i, \nabla V_i \rangle$, $b_i = \langle g_i, \nabla V_i \rangle$, $c_i = \langle f_i, \nabla V_i \rangle$.

In the sequel, we will denote by $\Delta_i(x), i \in I$, the discriminant of the right-hand side of (2.3), viewed as a polynomial in $u$, at every $x \in \mathbb{R}^n$, i.e., $\Delta_i(x) = b_i^2(x) - 4a_i(x)c_i(x)$. Moreover, when $\Delta_i(x) \ge 0$, we denote by $\lambda_{1i}(x)$ and $\lambda_{2i}(x)$ the two roots of the equation $a_i(x)\lambda^2+b_i(x)\lambda+c_i(x) = 0$ given by

$$\lambda_{1i}(x) = \frac{-b_i(x) - \sqrt{\Delta_i(x)}}{2a_i(x)},$$

$$\lambda_{2i}(x) = \frac{-b_i(x) + \sqrt{\Delta_i(x)}}{2a_i(x)}.$$

We start by recalling the following definitions from [1, 17, 23].
Nowadays, quadratic input nonlinear systems represents an important class of non-linear systems which can model many practical applications, such as in magnetic problems (Moulay & Perruquetti, 2005) and oscillation problems of the electromagnetic oscillator (Sane & Bernstein, 2002). Therefore, it is interesting to investigate such systems.

To the best of our knowledge, there are even fewer results on simultaneous stabilization of quadratic nonlinear systems because of the difficulties involved in synthesis of these systems. Following the stream of research on the simultaneous stabilization of nonlinear systems and motivated by the approach in (Iggidr & Oumoun, 2020; Wu, 2010, 2005; Zhang et al., 2011), in the context of quadratic input nonlinear systems, we study the problem of designing constructively a simultaneously asymptotically stabilizing control for these systems. Based on the control Lyapunov functions approach (CLF), we first give a necessary condition for the existence of simultaneously asymptotically stabilizing feedback. After that, we present sufficient conditions for the existence of simultaneously stabilizing feedback. Furthermore, we give an explicit formula for constructing the simultaneously stabilizing controller. It is shown that, under the small control property (SCP) assumption, the obtained feedback is continuous everywhere in the state space.

In the end, based on the sliding mode technique, the design procedure developed in this paper is also used to construct a discontinuous control that simultaneously stabilizes two Levitation systems. Note that magnetic levitation systems have practical importance in many engineering systems such as in high-speed Japanese “Maglev” and German “Transrapid” passenger trains, frictionless bearings ... to name a few.

## 2. Problem formulation and preliminaries

### 2.1. Problem formulation

Consider a collection of $m$ quadratic input nonlinear systems described as follows:

$$S_i: \dot{x} = f_i(x) + ug_i(x) + u^2h_i(x), \quad i \in \{1, \dots, m\} = I, \qquad (2.1)$$

where $x \in \mathbb{R}^n$ is the state, $u \in \mathbb{R}$ is the control, $f_i(\cdot)$, $g_i(\cdot)$ and $h_i(\cdot)$ are continuous functions with $f_i(0) = 0$.

The objective of this paper is to find a continuous function $k(\cdot)$ such that the state feedback controller $u = k(x)$ globally asymptotically stabilizes the collection of systems in (2.1) simultaneously, that is, the origin is a globally asymptotically stable equilibrium for the resulting closed-loop systems

$$S_i: \dot{x} = f_i(x) + k(x)g_i(x) + k(x)^2h_i(x), \quad i \in I. \qquad (2.2)$$

### 2.2. Preliminaries

We first give some notations and basic definitions concerning control Lyapunov functions, then we recall some results on the stabilization of quadratic input nonlinear systems.

For each $i = 1, 2, \dots, m$, let $V_i$ be a $C^1$ positive definite and proper function defined on $\mathbb{R}^n$. Its derivative, denoted by $\dot{V}_i$, along the trajectories of system $S_i$ can be written as
**Definition 2.1.** ([1, 17, 23]) For $i \in I$. A $C^1$ positive definite and proper function $V_i$, defined on $\mathbb{R}^n$ is a control Lyapunov function (CLF) for system $S_i$ if for any $x \in \mathbb{R}^n \setminus \{0\}$

$$ \inf_{u \in \mathbb{R}} \dot{V}_i(x) = \inf_{u \in \mathbb{R}} \left( a_i(x)u^2 + b_i(x)u + c_i(x) \right) < 0. $$

A CLF $V_i$ for System $S_i$ is said to satisfy the small control property (SCP), if for each $\varepsilon > 0$ there is $\delta > 0$ such that, if $x \neq 0$ satisfies $\|x\| < \delta$, then there is some $u$ with $\|u\| < \varepsilon$ such that $\dot{V}_i(x) = a_i(x)u^2 + b_i(x)u + c_i(x) < 0$.

As far as possible, our objective is to explicitly design a common state feedback law $u$, which is continuous on $\mathbb{R}^n \setminus \{0\}$, and is such that the resulting closed-loop systems (2.2) are globally asymptotically stable. To be more precise, if $V_i$ is a known collection of CLFs for the systems $S_i$, $i \in I$, we shall construct a feedback law $u$ such that for each $i \in I$, $V_i$ is a Lyapunov function for the closed-loop system $\dot{x} = f_i(x) + u(x)g_i(x) + u^2(x)h_i(x)$. This means to construct a continuous feedback such that, for $x \neq 0$, $i \in I$,

$$ \dot{V}_i(x) = a_i(x)u^2(x) + b_i(x)u(x) + c_i(x) < 0. \quad (2.4) $$

**Remark 2.1. I.** Since $V_i$ is a CLF for system $S_i$, $i \in I$, it is easy to see that for $x \neq 0$, if $a_i(x) > 0$, then $\Delta_i(x) > 0$.

**II.** Inequality (2.4) shows that in the setting of simultaneous quadratic input nonlinear systems, a continuous feedback $u$ simultaneously stabilizes the collection of systems in (2.1) has to verify for each $i \in I$,

$$ 1. u(x) \in [\lambda_{1i}(x), \lambda_{2i}(x)], \quad \text{if } a_i(x) > 0, $$

$$ 2. u(x) < -c_i(x)/b_i(x), \quad \text{if } a_i(x) = 0 \text{ and } b_i(x) > 0, $$

$$ 3. u(x) > -c_i(x)/b_i(x), \quad \text{if } a_i(x) = 0 \text{ and } b_i(x) < 0, $$

$$ 4. u(x) \notin [\lambda_{2i}(x), \lambda_{1i}(x)], \quad \text{if } a_i(x) < 0 \text{ and } \Delta_i(x) \geq 0. $$

In what follows, we recall some results (from [15]) that will be used in this paper.

**Lemma 2.1.** ([15]) Assume that $V_i$ is a CLF for system $S_i$, $i \in I$. Then the following assertions hold:

(1) the inequality $\Delta_i(x) < 0$ implies $a_i(x) < 0$;

(2) $\Delta_i(x) = 0$, $a_i(x) \neq 0$ implies $a_i(x) < 0$;

(3) $\Delta_i(x) = 0$, $a_i(x) = 0$ implies $b_i(x) = 0$ and $c_i(x) < 0$.

**Lemma 2.2.** ([15]) Assume that $V_i$ is a CLF for System $S_i$, $i \in I$. Then for any $x_0 \in \mathbb{R}^n \setminus \{0\}$ such that $a_i(x_0) = 0$, one has:

$$ \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{1i}(x) = -\frac{c_i(x_0)}{b_i(x_0)}, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{2i}(x) = +\infty \quad \text{if } b_i(x_0) < 0. \quad (2.5) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) < 0}} \lambda_{1i}(x) = -\frac{c_i(x_0)}{b_i(x_0)}, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) < 0}} \lambda_{2i}(x) = -\infty \quad \text{if } b_i(x_0) < 0. \quad (2.6) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{1i}(x) = -\infty, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{2i}(x) = +\infty \quad \text{if } b_i(x_0) = 0. \quad (2.7) $$
$$ \dot{V}_i(x) = a_i(x)u^2 + b_i(x)u + c_i(x), \qquad (2.3) $$

with, $a_i = \langle h_i, \nabla V_i \rangle$, $b_i = \langle g_i, \nabla V_i \rangle$, $c_i = \langle f_i, \nabla V_i \rangle$.

In the sequel, we will denote by $\Delta_i(x), i \in I$, the discriminant of the right-hand side of (2.3), viewed as a polynomial in $u$, at every $x \in \mathbb{R}^n$, i.e., $\Delta_i(x) = b_i^2(x) - 4a_i(x)c_i(x)$. Moreover, when $\Delta_i(x) \ge 0$, we denote by $\lambda_{1i}(x)$ and $\lambda_{2i}(x)$ the two roots of the equation $a_i(x)\lambda^2 + b_i(x)\lambda + c_i(x) = 0$ given by

$$ \lambda_{1i}(x) = \frac{-b_i(x) - \sqrt{\Delta_i(x)}}{2a_i(x)}, $$

$$ \lambda_{2i}(x) = \frac{-b_i(x) + \sqrt{\Delta_i(x)}}{2a_i(x)}. $$

We start by recalling the following definitions.

**Definition 2.1.** (Maniar et al., 2017; Moulay & Perruquetti, 2005) For $i \in I$. The system $S_i$ is almost stabilizable, if there exists a feedback control law $u: \mathcal{V} \to \mathbb{R}$ continuous on $\mathcal{V} \setminus \{0\}$, such that

* $u(0)=0$;

* the origin is an asymptotically stable equilibrium of the closed-loop system deduced from $S_i$ and $u = u(x)$.

Moreover, if $u$ is continuous on $\mathcal{V}$, we say that the system $S_i$ is stabilizable. The system $S_i$ is globally stabilizable if $\mathcal{V} = \mathbb{R}^n$.

**Definition 2.2.** (Maniar et al., 2017; Moulay & Perruquetti, 2005) For $i \in I$. A $C^1$ positive definite and proper function $V_i$, defined on $\mathbb{R}^n$ is a control Lyapunov function (CLF) for system $S_i$ if for any $x \in \mathbb{R}^n \setminus \{0\}$

$$ \inf_{u \in \mathbb{R}} \dot{V}_i(x) = \inf_{u \in \mathbb{R}} (a_i(x)u^2 + b_i(x)u + c_i(x)) < 0. $$

A CLF $V_i$ for System $S_i$ is said to satisfy the small control property (SCP), if for each $\varepsilon > 0$ there is $\delta > 0$ such that, if $x \ne 0$ satisfies $\|x\| < \delta$, then there is some $u$ with $\|u\| < \varepsilon$ such that $\dot{V}_i(x) = a_i(x)u^2 + b_i(x)u + c_i(x) < 0$.

As far as possible, our objective is to explicitly design a common state feedback law $u$, which is continuous on $\mathbb{R}^n \setminus \{0\}$, and is such that the resulting closed-loop systems (2.2) are globally asymptotically stable. To be more precise, if $V_i$ is a known collection of CLFs for the systems $S_i, i \in I$, we shall construct a feedback law $u$ such that for each $i \in I$, $V_i$ is a Lyapunov function for the closed-loop system $\dot{x} = f_i(x) + u(x)g_i(x) + u^2(x)h_i(x)$. This means to construct a continuous feedback such that, for $x \ne 0, i \in I$,

$$ \dot{V}_i(x) = a_i(x)u^2(x) + b_i(x)u(x) + c_i(x) < 0. \qquad (2.4) $$
$$ \lim_{\substack{x \to x_0 \\ a_i(x) < 0 \\ b_i(x) < 0}} \lambda_{1i}(x) = -\infty, \text{ and } \lim_{\substack{x \to x_0 \\ a_i(x) < 0 \\ b_i(x) < 0}} \lambda_{2i}(x) = -\infty \text{ if } b_i(x_0) = 0. \quad (2.8) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) < 0 \\ b_i(x) > 0}} \lambda_{1i}(x) = +\infty, \text{ and } \lim_{\substack{x \to x_0 \\ a_i(x) < 0 \\ b_i(x) > 0}} \lambda_{2i}(x) = +\infty \text{ if } b_i(x_0) = 0. \quad (2.9) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{1i}(x) = -\infty, \text{ and } \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{2i}(x) = -\frac{c_i(x_0)}{b_i(x_0)} \text{ if } b_i(x_0) > 0. \quad (2.10) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) < 0}} \lambda_{1i}(x) = +\infty, \text{ and } \lim_{\substack{x \to x_0 \\ a_i(x) < 0}} \lambda_{2i}(x) = -\frac{c_i(x_0)}{b_i(x_0)} \text{ if } b_i(x_0) > 0. \quad (2.11) $$

**Theorem 2.1.** ([15]) For $i \in I$, assume $V_i$ is a CLF for System $S_i$, and satisfies the following assumption:

**Assumption 2.1.** For every $x \in \mathbb{R}^n \setminus \{0\}$, $(a_i(x) < 0 \text{ and } \Delta_i(x) \ge 0) \implies b_i(x) \ne 0$.

Then, the feedback

$$ k_i(x) = \begin{cases} v_i(x), & \text{if } \Delta_i(x) > 0, \\ 0, & \text{if } \Delta_i(x) \le 0, \end{cases} \quad (2.12) $$

where

$$ v_i(x) = \begin{cases} \min \left( \lambda_{2i}(x) - \frac{\sqrt{\Delta_i(x)}}{2(1+a_i^2(x))}, 0 \right) & \text{if } a_i(x) \neq 0 \text{ and } b_i(x) > 0, \\ \max \left( \lambda_{1i}(x) + \frac{\sqrt{\Delta_i(x)}}{2(1+a_i^2(x))}, 0 \right) & \text{if } a_i(x) \neq 0 \text{ and } b_i(x) < 0, \\ \min \left( -\frac{c_i(x)}{b_i(x)} - \frac{b_i(x)}{2}, 0 \right) & \text{if } a_i(x) = 0 \text{ and } b_i(x) > 0, \\ \max \left( -\frac{c_i(x)}{b_i(x)} - \frac{b_i(x)}{2}, 0 \right) & \text{if } a_i(x) = 0 \text{ and } b_i(x) < 0, \\ 0 & \text{if } a_i(x) > 0 \text{ and } b_i(x) = 0, \end{cases} $$

is continuous on $\mathbb{R}^n \setminus \{0\}$ and globally asymptotically stabilizes the system $S_i$. In addition, if $V_i$ satisfies the small control property (SCP), then $k_i$ is continuous all over $\mathbb{R}^n$.

Hereafter we give a result that shows how to verify the small control property (SCP).

**Lemma 2.3.** For $i \in I$, assume $V_i$ is a CLF for System $S_i$. Then $V_i$ satisfy the SCP if and only if for all $\varepsilon > 0$, there exists $\delta > 0$ such that for all $x \in B(0, \delta) \setminus \{0\}$ and $\Delta_i(x) \ge 0$, we have

$$ \begin{aligned} \lambda_{1i}(x) < \varepsilon \quad \text{and} \quad -\varepsilon < \lambda_{2i}(x) & \quad \text{if } a_i(x) > 0, \\ \lambda_{2i}(x) < \lambda_{1i}(x) < \varepsilon & \quad \text{if } a_i(x) < 0 \text{ and } b_i(x) < 0, \\ -\varepsilon < \lambda_{2i}(x) < \lambda_{1i}(x) & \quad \text{if } a_i(x) < 0 \text{ and } b_i(x) > 0, \\ -\lambda_{2i}(x) = \lambda_{1i}(x) < \varepsilon & \quad \text{if } a_i(x) < 0 \text{ and } b_i(x) = 0, \\ -\frac{c_i(x)}{b_i(x)} < \varepsilon & \quad \text{if } a_i(x) = 0 \text{ and } b_i(x) < 0, \\ -\varepsilon < -\frac{c_i(x)}{b_i(x)} & \quad \text{if } a_i(x) = 0 \text{ and } b_i(x) > 0. \end{aligned} $$
**Remark 2.1. (I)** Since $V_i$ is a CLF for system $S_i$, $i \in I$, it is easy to see that for $x \neq 0$, if $a_i(x) > 0$, then $\Delta_i(x) > 0$.

**(II) Inequality (2.4)** shows that in the setting of simultaneous quadratic input non-linear systems, a continuous feedback u simultaneously stabilizes the collection of systems in (2.1) has to verify for each $i \in I$,

(1) $u(x) \in [\lambda_{1i}(x), \lambda_{2i}(x)]$, if $a_i(x) > 0$,

(2) $u(x) < -c_i(x)/b_i(x)$, if $a_i(x) = 0$ and $b_i(x) > 0$,

(3) $u(x) > -c_i(x)/b_i(x)$, if $a_i(x) = 0$ and $b_i(x) < 0$,

(4) $u(x) \notin [\lambda_{2i}(x), \lambda_{1i}(x)]$, if $a_i(x) < 0$ and $\Delta_i(x) \ge 0$.

In what follows, we recall some results (from (Maniar et al. , 2017)) that will be used in this paper.

**Lemma 2.1. (Maniar et al. , 2017)** Assume that $V_i$ is a CLF for system $S_i$, $i \in I$. Then the following assertions hold:

(1) the inequality $\Delta_i(x) < 0$ implies $a_i(x) < 0$;

(2) $\Delta_i(x) = 0$, $a_i(x) \neq 0$ implies $a_i(x) < 0$;

(3) $\Delta_i(x) = 0$, $a_i(x) = 0$ implies $b_i(x) = 0$ and $c_i(x) < 0$.

**Lemma 2.2. (Maniar et al. , 2017)** Assume that $V_i$ is a CLF for System $S_i$, $i \in I$. Then for any $x_0 \in \mathbb{R}^n \setminus \{0\}$ such that $a_i(x_0) = 0$, one has:

$$ \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{1i}(x) = -\frac{c_i(x_0)}{b_i(x_0)}, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{2i}(x) = +\infty \quad \text{if } b_i(x_0) < 0. \quad (2.5) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) < 0}} \lambda_{1i}(x) = -\frac{c_i(x_0)}{b_i(x_0)}, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) < 0}} \lambda_{2i}(x) = -\infty \quad \text{if } b_i(x_0) < 0. \quad (2.6) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{1i}(x) = -\infty, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{2i}(x) = +\infty \quad \text{if } b_i(x_0) = 0. \quad (2.7) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) < 0 \\ b_i(x) < 0}} \lambda_{1i}(x) = -\infty, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) < 0 \\ b_i(x) < 0}} \lambda_{2i}(x) = -\infty \quad \text{if } b_i(x_0) = 0. \quad (2.8) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) < 0 \\ b_i(x) > 0}} \lambda_{1i}(x) = +\infty, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) < 0 \\ b_i(x) > 0}} \lambda_{2i}(x) = +\infty \quad \text{if } b_i(x_0) = 0. \quad (2.9) $$

$$ \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{1i}(x) = -\infty, \quad \text{and} \quad \lim_{\substack{x \to x_0 \\ a_i(x) > 0}} \lambda_{2i}(x) = -\frac{c_i(x_0)}{b_i(x_0)} \quad \text{if } b_i(x_0) > 0. \quad (2.10) $$
*Proof.* The proof is an immediate consequence of the definition of the SCP and the study of the sign of the polynomial in $u$: $a_i(x)u^2 + b_i(x)u + c_i(x)$, according to the sign of $a_i(x)$, $x$ in a neighborhood of the origin.

Note that for the second assertion, we take into account the fact that $\lambda_{2i}(x) < 0$ and $|\lambda_{1i}(x)| < |\lambda_{2i}(x)|$ if $a_i(x) < 0$ and $b_i(x) < 0$. Similarly, for the third one, we use the fact that $\lambda_{1i}(x) > 0$ and $|\lambda_{2i}(x)| < \lambda_{1i}(x)$ if $a_i(x) < 0$ and $b_i(x) > 0$. $\square$

**Remark 2.2.** Notice that if $V_i$, $i \in I$, is a CLF satisfying Assumption 2.1, then the set $\{x \in \mathbb{R}^n | a_i(x) < 0, \Delta_i(x) \ge 0, b_i(x) = 0\}$ is empty. So, in our case, to verify the SCP, it is sufficient to find functions $d_{ji}: \mathbb{R}^n \to \mathbb{R}$, $j = 1, ..., 5$, continuous in a neighborhood of the origin and $d_{ji}(0) = 0$, such that for $x \ne 0$ close enough to 0 and $\Delta_i(x) \ge 0$, one has:

$$ 
\begin{array}{ll}
\lambda_{1i}(x) < d_{1i}(x) < \lambda_{2i}(x) & \text{if } a_i(x) > 0, \\
\lambda_{1i}(x) < d_{2i}(x) & \text{if } a_i(x) < 0 \text{ and } b_i(x) < 0, \\
d_{3i}(x) < \lambda_{2i}(x) & \text{if } a_i(x) < 0 \text{ and } b_i(x) > 0, \\
-\frac{c_i(x)}{b_i(x)} < d_{4i}(x) & \text{if } a_i(x) = 0 \text{ and } b_i(x) < 0, \\
d_{5i}(x) < -\frac{c_i(x)}{b_i(x)} & \text{if } a_i(x) = 0 \text{ and } b_i(x) > 0.
\end{array}
$$

### 3 Main results: Design method of simultaneously stabilizing controller

As a first step towards the construction of a simultaneously stabilizing feedback law for the collection of quadratic systems (2.1), and for easily presenting the main results, we define the following sets and functions: For each $i \in I$, suppose that $V_i$ is a CLF for System $S_i$ and let for each $x \in \mathbb{R}^n$

$$ I_1(x) = \{i \in I | a_i(x) > 0\}, $$

$$ I_2(x) = \{i \in I | a_i(x) < 0, b_i(x) < 0, \Delta_i(x) \ge 0\}, $$

$$ I_3(x) = \{i \in I | a_i(x) < 0, b_i(x) > 0, \Delta_i(x) \ge 0\}, $$

$$ I_4(x) = \{i \in I | a_i(x) = 0, b_i(x) < 0\} $$
$$ \lim_{\substack{x \to x_0 \\ a_i(x) < 0}} \lambda_{1i}(x) = +\infty, \text{ and } \lim_{\substack{x \to x_0 \\ a_i(x) < 0}} \lambda_{2i}(x) = -\frac{c_i(x_0)}{b_i(x_0)} \text{ if } b_i(x_0) > 0. \quad (2.11) $$

**Assumption 2.1.** Let $i \in I$. For every $x \in \mathbb{R}^n \setminus \{0\}$, $(a_i(x) < 0 \text{ and } \Delta_i(x) \ge 0) \implies b_i(x) \ne 0$.

**Theorem 2.2.** (Maniar et al., 2017) For $i \in I$, assume $V_i$ is a CLF for System $S_i$, and Assumption 2.1 holds. Then, $S_i$ is almost stabilizable with the feedback

$$ k_i(x) = \begin{cases} v_i(x), & \text{if } \Delta_i(x) > 0, \\ 0, & \text{if } \Delta_i(x) \le 0, \end{cases} \quad (2.12) $$

where

$$ v_i(x) = \begin{cases} \min \left( \lambda_{2i}(x) - \frac{\sqrt{\Delta_i(x)}}{2(1+a_i^2(x))}, 0 \right) & \text{if } a_i(x) \neq 0 \text{ and } b_i(x) > 0, \\ \max \left( \lambda_{1i}(x) + \frac{\sqrt{\Delta_i(x)}}{2(1+a_i^2(x))}, 0 \right) & \text{if } a_i(x) \neq 0 \text{ and } b_i(x) < 0, \\ \min \left( -\frac{c_i(x)}{b_i(x)} - \frac{b_i(x)}{2}, 0 \right) & \text{if } a_i(x) = 0 \text{ and } b_i(x) > 0, \\ \max \left( -\frac{c_i(x)}{b_i(x)} - \frac{b_i(x)}{2}, 0 \right) & \text{if } a_i(x) = 0 \text{ and } b_i(x) < 0, \\ 0 & \text{if } a_i(x) > 0 \text{ and } b_i(x) = 0, \end{cases} $$

In addition, if $V_i$ satisfies the small control property, then $S_i$ is stabilizable.

Hereafter we give a result that shows how to verify the small control property (SCP).

**Lemma 2.3.** For $i \in I$, assume $V_i$ is a CLF for System $S_i$. Then $V_i$ satisfy the SCP if and only if for all $\varepsilon > 0$, there exists $\delta > 0$ such that for all $x \in B(0, \delta) \setminus \{0\}$ and $\Delta_i(x) \ge 0$, we have

$$ \begin{aligned} \lambda_{1i}(x) < \varepsilon \quad \text{and} \quad -\varepsilon < \lambda_{2i}(x) & \quad \text{if } a_i(x) > 0, \\ \lambda_{2i}(x) < \lambda_{1i}(x) < \varepsilon & \quad \text{if } a_i(x) < 0 \text{ and } b_i(x) < 0, \\ -\varepsilon < \lambda_{2i}(x) < \lambda_{1i}(x) & \quad \text{if } a_i(x) < 0 \text{ and } b_i(x) > 0, \\ -\lambda_{2i}(x) = \lambda_{1i}(x) < \varepsilon & \quad \text{if } a_i(x) < 0 \text{ and } b_i(x) = 0, \\ -\frac{c_i(x)}{b_i(x)} < \varepsilon & \quad \text{if } a_i(x) = 0 \text{ and } b_i(x) < 0, \\ -\varepsilon < -\frac{c_i(x)}{b_i(x)} & \quad \text{if } a_i(x) = 0 \text{ and } b_i(x) > 0. \end{aligned} $$

**Proof.** The proof is an immediate consequence of the definition of the SCP and the study of the sign of the polynomial in $u: a_i(x)u^2 + b_i(x)u + c_i(x)$, according to the sign of $a_i(x)$, $x$ in a neighborhood of the origin.

Note that for the second assertion, we take into account the fact that $\lambda_{2i}(x) < 0$ and $|\lambda_{1i}(x)| < |\lambda_{2i}(x)|$ if $a_i(x) < 0$ and $b_i(x) < 0$. Similarly, for the third one, we use
$I_5(x) = \{i \in I | a_i(x) = 0, b_i(x) > 0\}.$

$$ \varphi_{1i}(x) = \begin{cases} \lambda_{1i}(x) & \text{if } i \in I_1 \cup I_2, \\ -\frac{c_i(x)}{b_i(x)} & \text{if } i \in I_4, \\ -\infty & \text{otherwise,} \end{cases} $$

$$ \varphi_{2i}(x) = \begin{cases} \lambda_{2i}(x) & \text{if } i \in I_1 \cup I_3, \\ -\frac{c_i(x)}{b_i(x)} & \text{if } i \in I_5, \\ +\infty & \text{otherwise,} \end{cases} $$

$$ \psi_{1i}(x) = \begin{cases} k_i(x) & \text{if } b_i(x) < 0, \\ 0 & \text{otherwise,} \end{cases} $$

$$ \psi_{2i}(x) = \begin{cases} k_i(x) & \text{if } b_i(x) > 0, \\ 0 & \text{otherwise.} \end{cases} $$

$$ \varphi_1(x) = \max_{i \in I} \varphi_{1i}(x), \quad \varphi_2(x) = \min_{i \in I} \varphi_{2i}(x), $$

$$ \psi_1(x) = \max_{i \in I} \psi_{1i}(x), \quad \psi_2(x) = \min_{i \in I} \psi_{2i}(x). $$

Finally, define

$$ w_1(x) = \min(\psi_1(x), \varphi_2(x)) \quad \text{and} \quad w_2(x) = \max(\psi_2(x), \varphi_1(x)). $$

**Remark 3.1.** Thanks to definitions of $\psi_{1i}$ and $\psi_{2i}$, it is easy to see that $\psi_{1i}(x) \ge 0$ and $\psi_{2i}(x) \le 0$ for all $x \in \mathbb{R}^n$ and for all $i \in I$. Thus, $\psi_1(x) \ge 0$ and $\psi_2(x) \le 0$, $\forall x \in \mathbb{R}^n$.

In preparation for the construction of a continuous simultaneously stabilizing control $u$, we need the following lemma.

**Lemma 3.1.** Consider the collection of systems in (2.1). Suppose that there is a CLF $V_i$, satisfying Assumption 2.1, for every $S_i$, $i \in I$. Then the functions $\psi_1$ and $\psi_2$ are continuous on $\mathbb{R}^n \setminus \{0\}$. Moreover, if the functions $V_i$, $i \in I$, satisfy the SCP, then $\psi_1$ and $\psi_2$ are continuous on $\mathbb{R}^n$.

*Proof.* By Assumption 2.1 and the definition of $k_i$ (2.12), $i \in I$, we have $k_i(x) = 0$ whenever $b_i(x) = 0$. From this and the continuity of $k_i$, it then follows that $\psi_{1i}$ and $\psi_{2i}$, $i \in I$, are continuous on $\mathbb{R}^n \setminus \{0\}$, then so are $\psi_1$ and $\psi_2$. If in addition, the functions $V_i$, $i \in I$, satisfy the SCP, then $k_i$, $i \in I$, are continuous on $\mathbb{R}^n$. So, as above, $\psi_{1i}$ and $\psi_{2i}$, $i \in I$, are continuous on $\mathbb{R}^n$ since $k_i(0) = b_i(0) = 0$, $i \in I$. Consequently, $\psi_1$ and $\psi_2$ are also continuous on $\mathbb{R}^n$. This completes the proof of Lemma 3.1. $\square$
the fact that $\lambda_{1i}(x) > 0$ and $|\lambda_{2i}(x)| < \lambda_{1i}(x)$ if $a_i(x) < 0$ and $b_i(x) > 0$.

**Remark 2.2.** Notice that if $V_i$, $i \in I$, is a CLF satisfying Assumption 2.1, then the set $\{x \in \mathbb{R}^n | a_i(x) < 0, \Delta_i(x) \ge 0, b_i(x) = 0\}$ is empty. So, in our case, to verify the SCP, it is sufficient to find functions $d_{ji}: \mathbb{R}^n \to \mathbb{R}$, $j = 1,...,5$, continuous in a neighborhood of the origin and $d_{ji}(0) = 0$, such that for $x \ne 0$ close enough to 0 and $\Delta_i(x) \ge 0$, one has:

$$ 
\begin{array}{ll}
\lambda_{1i}(x) < d_{1i}(x) < \lambda_{2i}(x) & \text{if } a_i(x) > 0, \\
\lambda_{1i}(x) < d_{2i}(x) & \text{if } a_i(x) < 0 \text{ and } b_i(x) < 0, \\
d_{3i}(x) < \lambda_{2i}(x) & \text{if } a_i(x) < 0 \text{ and } b_i(x) > 0, \\
-\frac{c_i(x)}{b_i(x)} < d_{4i}(x) & \text{if } a_i(x) = 0 \text{ and } b_i(x) < 0, \\
d_{5i}(x) < -\frac{c_i(x)}{b_i(x)} & \text{if } a_i(x) = 0 \text{ and } b_i(x) > 0.
\end{array}
$$

### 3. Main results: Design method of simultaneously stabilizing controller

As a first step towards the construction of a simultaneously stabilizing feedback law for the collection of quadratic systems (2.1), and for easily presenting the main results, we define the following sets and functions: For each $i \in I$, suppose that $V_i$ is a CLF for System $S_i$ and let for each $x \in \mathbb{R}^n$,

$$ I_1(x) = \{i \in I | a_i(x) > 0\}, $$

$$ I_2(x) = \{i \in I | a_i(x) < 0, b_i(x) < 0, \Delta_i(x) \ge 0\}, $$

$$ I_3(x) = \{i \in I | a_i(x) < 0, b_i(x) > 0, \Delta_i(x) \ge 0\}, $$

$$ I_4(x) = \{i \in I | a_i(x) = 0, b_i(x) < 0\} $$

$$ I_5(x) = \{i \in I | a_i(x) = 0, b_i(x) > 0\}. $$

$$ \varphi_{1i}(x) = \begin{cases} \lambda_{1i}(x) & \text{if } i \in I_1 \cup I_2, \\ -\frac{c_i(x)}{b_i(x)} & \text{if } i \in I_4, \\ -\infty & \text{otherwise,} \end{cases} $$

$$ \varphi_{2i}(x) = \begin{cases} \lambda_{2i}(x) & \text{if } i \in I_1 \cup I_3, \\ -\frac{c_i(x)}{b_i(x)} & \text{if } i \in I_5, \\ +\infty & \text{otherwise,} \end{cases} $$

$$ \psi_{1i}(x) = \begin{cases} k_i(x) & \text{if } b_i(x) < 0, \\ 0 & \text{otherwise,} \end{cases} $$

$$ \psi_{2i}(x) = \begin{cases} k_i(x) & \text{if } b_i(x) > 0, \\ 0 & \text{otherwise.} \end{cases} $$
To easily present the following results, define the sets

$$A_1(x) = \{i \in I \mid (a_i(x) < 0, b_i(x) > 0, \Delta_i(x) > 0) \text{ or } (a_i(x) > 0) \text{ or } (a_i(x) = 0, b_i(x) > 0)\},$$

$$A_2(x) = \{i \in I \mid (a_i(x) \le 0, b_i(x) \le 0) \text{ or } (a_i(x) < 0, b_i(x) > 0, \Delta_i(x) < 0)\},$$

$$A_3(x) = \{i \in I \mid a_i(x) < 0, b_i(x) > 0, \Delta_i(x) = 0\},$$

$$B_1(x) = \{i \in I \mid (a_i(x) < 0, b_i(x) < 0, \Delta_i(x) > 0) \text{ or } (a_i(x) > 0) \text{ or } (a_i(x) = 0, b_i(x) < 0)\},$$

$$B_2(x) = \{i \in I \mid (a_i(x) \le 0, b_i(x) \ge 0) \text{ or } (a_i(x) < 0, b_i(x) < 0, \Delta_i(x) < 0)\},$$

$$B_3(x) = \{i \in I \mid a_i(x) < 0, b_i(x) < 0, \Delta_i(x) = 0\}.$$

Clearly, $I = A_1(x) \cup A_2(x) \cup A_3(x) = B_1(x) \cup B_2(x) \cup B_3(x)$.

For convenience, define

$$A_{31}(x) = \{i \in A_3(x) \mid \forall \eta > 0, \exists y \in B(x, \eta) \setminus \{x\} \text{ such that, } \Delta_i(y) \ge 0\},$$

and $A_{32}(x) = A_3(x) \setminus A_{31}(x)$.

We can, similarly, define $B_{31}(x)$ and $B_{32}(x)$.

For guaranteeing the continuity of the simultaneous feedback laws (which will be derived later) for the collection of quadratic input nonlinear systems in (2.1), here we assume

**Assumption 3.1.** If $A_3(x) \neq \emptyset$, 
$$\min(\psi_1(x), \min_{i \in A_1(x)} \varphi_{2i}(x)) \le \min_{j \in A_3(x)} -\frac{b_j(x)}{2a_j(x)}.$$

**Assumption 3.2.** If $B_3(x) \neq \emptyset$, 
$$\max_{j \in B_3(x)} -\frac{b_j(x)}{2a_j(x)} \le \max(\psi_2(x), \max_{i \in B_1(x)} \varphi_{1i}(x)).$$

**Lemma 3.2.** Consider the collection of systems in (2.1). Suppose that there is a CLF $V_i$, satisfying Assumption 2.1 for every $S_i$, $i \in I$ and Assumptions 3.1, 3.2 hold, Then the functions $w_1$ and $w_2$ are continuous on $\mathbb{R}^n \setminus \{0\}$. If in addition the functions $V_i$, $i \in I$, satisfy the small control property, then $w_1$ and $w_2$ are continuous on $\mathbb{R}^n$.

*Proof.* To establish the continuity of $w_1$ at $x_0 \in \mathbb{R}^n \setminus \{0\}$, we will verify that the functions $\varphi_{2i}$, $i \in A_1(x_0)$, are continuous on a neighborhood of $x_0$ and:

* If $A_{31}(x_0) = \emptyset$, then $w_1(x) = \min(\psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x))$ on a neighborhood of $x_0$. The conclusion follows since $\psi_1$ and $\varphi_{2i}$, $i \in A_1(x_0)$, are continuous on a neighborhood of $x_0$.

* If $A_{31}(x_0) \neq \emptyset$, then $w_1(x) = \min_{k \in A_{31}(x_0)} w_1^k(x)$, with $w_1^k$, $k \in A_{31}(x_0)$ are continuous functions on a neighborhood of $x_0$.
$$ \varphi_1(x) = \max_{i \in I} \varphi_{1i}(x), \quad \varphi_2(x) = \min_{i \in I} \varphi_{2i}(x), $$

$$ \psi_1(x) = \max_{i \in I} \psi_{1i}(x), \quad \psi_2(x) = \min_{i \in I} \psi_{2i}(x). $$

Finally, define

$$ w_1(x) = \min(\psi_1(x), \varphi_2(x)) \quad \text{and} \quad w_2(x) = \max(\psi_2(x), \varphi_1(x)). $$

**Remark 3.1.** Due to definitions of $\psi_{1i}$ and $\psi_{2i}$, it is easy to see that $\psi_{1i}(x) \ge 0$ and $\psi_{2i}(x) \le 0$ for all $x \in \mathbb{R}^n$ and for all $i \in I$. Thus, $\psi_1(x) \ge 0$ and $\psi_2(x) \le 0$, $\forall x \in \mathbb{R}^n$.

In preparation for the construction of a continuous simultaneously stabilizing control $u$, we need the following lemma.

**Lemma 3.1.** Consider the collection of systems in (2.1). Suppose that there is a CLF $V_i$, satisfying Assumption 2.1, for every $S_i$, $i \in I$. Then the functions $\psi_1$ and $\psi_2$ are continuous on $\mathbb{R}^n \setminus \{0\}$. Moreover, if the functions $V_i$, $i \in I$, satisfy the SCP, then $\psi_1$ and $\psi_2$ are continuous on $\mathbb{R}^n$.

**Proof.** By Assumption 2.1 and the definition of $k_i$ (2.12), $i \in I$, we have $k_i(x) = 0$ whenever $b_i(x) = 0$. From this and the continuity of $k_i$, it then follows that $\psi_{1i}$ and $\psi_{2i}$, $i \in I$, are continuous on $\mathbb{R}^n \setminus \{0\}$, then so are $\psi_1$ and $\psi_2$. If in addition, the functions $V_i$, $i \in I$, satisfy the SCP, then $k_i$, $i \in I$, are continuous on $\mathbb{R}^n$. So, as above, $\psi_{1i}$ and $\psi_{2i}$, $i \in I$, are continuous on $\mathbb{R}^n$ since $k_i(0) = b_i(0) = 0$, $i \in I$. Consequently, $\psi_1$ and $\psi_2$ are also continuous on $\mathbb{R}^n$. This completes the proof of Lemma 3.1. $\square$

To easily present the following results, define the sets

$$ A_1(x) = \{i \in I \mid (a_i(x) < 0, b_i(x) > 0, \Delta_i(x) > 0) \text{ or } (a_i(x) > 0) \text{ or } (a_i(x) = 0, b_i(x) > 0)\}, $$

$$ A_2(x) = \{i \in I \mid (a_i(x) \le 0, b_i(x) \le 0) \text{ or } (a_i(x) < 0, b_i(x) > 0, \Delta_i(x) < 0)\}, $$

$$ A_3(x) = \{i \in I \mid a_i(x) < 0, b_i(x) > 0, \Delta_i(x) = 0\}, $$

$$ B_1(x) = \{i \in I \mid (a_i(x) < 0, b_i(x) < 0, \Delta_i(x) > 0) \text{ or } (a_i(x) > 0) \text{ or } (a_i(x) = 0, b_i(x) < 0)\}, $$

$$ B_2(x) = \{i \in I \mid (a_i(x) \le 0, b_i(x) \ge 0) \text{ or } (a_i(x) < 0, b_i(x) < 0, \Delta_i(x) < 0)\}, $$

$$ B_3(x) = \{i \in I \mid a_i(x) < 0, b_i(x) < 0, \Delta_i(x) = 0\}. $$

$$ \text{Clearly, } I = A_1(x) \cup A_2(x) \cup A_3(x) = B_1(x) \cup B_2(x) \cup B_3(x). $$
To do this, we first study the behavior of the functions $\varphi_{2i}$, $i \in I$, in a neighborhood of $x_0$ according to the sign of $a_i(x_0)$, $i \in I$.

**Case 1:** If $a_i(x_0) > 0$, by continuity of the function $a_i$, $a_i(x) > 0$ in a neighborhood $V_{x_0}$ of $x_0$. Then $\varphi_{2i}(x) = \lambda_{2i}(x)$ on $V_{x_0}$, so $\varphi_{2i}$ is continuous at $x_0$ since $\lambda_{2i}$ is continuous on $V_{x_0}$.

**Case 2:** If $a_i(x_0) = 0$ and $b_i(x_0) > 0$, then $\Delta_i(x_0) > 0$. By continuity, $b_i(x) > 0$ and $\Delta_i(x) > 0$ in a neighborhood $V_{x_0}$ of $x_0$, together with (2.10), (2.11) and definition of $\varphi_{2i}$, we can easily verify that $\varphi_{2i}$ is continuous at $x_0$.

**Case 3:** If $a_i(x_0) = 0$ and $b_i(x_0) = 0$. Let $x^j$ be a sequence of states that converges to $x_0$ and let us review the different possibilities according to the definition of $\varphi_{2i}$.

* If the sequence $x^j$ satisfies $a_i(x^j) > 0$, it then follows from (2.7) and definition of $\varphi_{2i}$

$$ \lim_{x^j \to x_0} \varphi_{2i}(x^j) = \lim_{x^j \to x_0} \lambda_{2i}(x^j) = +\infty. $$

* If the sequence $x^j$ satisfies $a_i(x^j) < 0$, $b_i(x^j) > 0$ and $\Delta_i(x^j) \ge 0$, (2.9) yields

$$ \lim_{x^j \to x_0} \varphi_{2i}(x^j) = +\infty. $$

* If the sequence $x^j$ satisfies $a_i(x^j) = 0$ and $b_i(x^j) > 0$, recall that since $V_i$ is a CLF of System $S_i$, we have $c_i(x_0) < 0$, hence by continuity, $c_i(x^j) < 0$ whenever $x^j$ is close enough to $x_0$. So

$$ \lim_{x^j \to x_0} \varphi_{2i}(x^j) = \lim_{x^j \to x_0} -\frac{c_i(x^j)}{b_i(x^j)} = +\infty. $$

* If the sequence $x^j$ does not satisfy any of the above situations, then we have $\varphi_{2i}(x) = +\infty$.

Therefore, in all cases, i.e., if $a_i(x_0) = 0$ and $b_i(x_0) = 0$, the following holds

$$ \forall L > 0, \exists \lambda_L^i > 0 \text{ such that } \varphi_{2i}(x) > L, \forall x \in B(x_0, \lambda_L^i). $$

**Case 4:** If $a_i(x_0) = 0$ and $b_i(x_0) < 0$, by continuity, $b_i(x) < 0$ in a neighborhood $V_{x_0}$ of $x_0$, so, $\varphi_{2i}(x) = +\infty$ on $V_{x_0}$.

**Case 5:** If $a_i(x_0) < 0$, $b_i(x_0) > 0$ and $\Delta_i(x_0) > 0$, this case is similar to the Case 1.

**Case 6:** If $a_i(x_0) < 0$, $b_i(x_0) > 0$ and $\Delta_i(x_0) = 0$, that is $i \in A_3(x_0)$. If $i \in A_{32}(x_0)$, then there exists $\delta_i > 0$ such that $\Delta_i(x) < 0$ for all $x \in B(x_0, \delta_i) \setminus \{x_0\}$. So, $\varphi_{2i}(x) = +\infty$ on $B(x_0, \delta_i) \setminus \{x_0\}$. If $i \in A_{31}(x_0)$, then for $x$ close enough to $x_0$ we have, $\varphi_{2i}(x) = +\infty$ if $\Delta_i(x) < 0$ and $\varphi_{2i}(x) = \lambda_{2i}(x)$ if $\Delta_i(x) \ge 0$.

**Case 7:** If $a_i(x_0) < 0$, $b_i(x_0) > 0$ and $\Delta_i(x_0) < 0$, by continuity, $\Delta_i(x) < 0$ in a neighborhood $V_{x_0}$ of $x_0$, so, $\varphi_{2i}(x) = +\infty$ on $V_{x_0}$.

**Case 8:** If $a_i(x_0) < 0$, $b_i(x_0) = 0$, this case is similar to Case 7 since from Assumption 2.1, we have $\Delta_i(x_0) < 0$.

**Case 9:** If $a_i(x_0) < 0$, $b_i(x_0) < 0$, this case is similar to Case 4.

Summarizing the above, we have:
For convenience, define

$$A_{31}(x) = \{i \in A_3(x) \mid \forall \eta > 0, \exists y \in B(x, \eta) \setminus \{x\} \text{ such that, } \Delta_i(y) \ge 0\},$$

and $A_{32}(x) = A_3(x) \setminus A_{31}(x)$. We can, similarly, define $B_{31}(x)$ and $B_{32}(x)$.

For guaranteeing the continuity of the simultaneous feedback laws (which will be derived later) for the collection of quadratic input nonlinear systems in (2.1), here we assume

**Assumption 3.1.** If $A_3(x) \neq \emptyset$,

$$\min (\psi_1(x), \min_{i \in A_1(x)} \varphi_{2i}(x)) \le \min_{j \in A_3(x)} -\frac{b_j(x)}{2a_j(x)}.$$

**Assumption 3.2.** If $B_3(x) \neq \emptyset$,

$$\max_{j \in B_3(x)} -\frac{b_j(x)}{2a_j(x)} \le \max (\psi_2(x), \max_{i \in B_1(x)} \varphi_{1i}(x)).$$

**Lemma 3.2.** Consider the collection of systems in (2.1). Suppose that there is a CLF $V_i$, satisfying Assumption 2.1 for every $S_i$, $i \in I$ and Assumptions 3.1, 3.2 hold, Then the functions $w_1$ and $w_2$ are continuous on $\mathbb{R}^n \setminus \{0\}$. If in addition the functions $V_i$, $i \in I$, satisfy the small control property, then $w_1$ and $w_2$ are continuous on $\mathbb{R}^n$.

**Proof.** To establish the continuity of $w_1$ at $x_0 \in \mathbb{R}^n \setminus \{0\}$, we will verify that the functions $\varphi_{2i}$, $i \in A_1(x_0)$, are continuous on a neighborhood of $x_0$ and:

* If $A_{31}(x_0) = \emptyset$, then $w_1(x) = \min(\psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x))$ on a neighborhood of $x_0$. The conclusion follows since $\psi_1$ and $\varphi_{2i}$, $i \in A_1(x_0)$, are continuous on a neighborhood of $x_0$.

* If $A_{31}(x_0) \neq \emptyset$, then $w_1(x) = \min_{k \in A_{31}(x_0)} w_1^k(x)$, with $w_1^k$, $k \in A_{31}(x_0)$ are continuous functions on a neighborhood of $x_0$.

To do this, we first study the behavior of the functions $\varphi_{2i}$, $i \in I$, in a neighborhood of $x_0$ according to the sign of $a_i(x_0)$, $i \in I$.

**Case 1:** If $a_i(x_0) > 0$, by continuity of the function $a_i$, $a_i(x) > 0$ in a neighborhood $V_{x_0}$ of $x_0$. Then $\varphi_{2i}(x) = \lambda_{2i}(x)$ on $V_{x_0}$, so $\varphi_{2i}$ is continuous at $x_0$ since $\lambda_{2i}$ is continuous on $V_{x_0}$.

**Case 2:** If $a_i(x_0) = 0$ and $b_i(x_0) > 0$, then $\Delta_i(x_0) > 0$. By continuity, $b_i(x) > 0$ and $\Delta_i(x) > 0$ in a neighborhood $V_{x_0}$ of $x_0$, together with (2.10), (2.11) and definition of $\varphi_{2i}$, we can easily verify that $\varphi_{2i}$ is continuous at $x_0$.

**Case 3:** If $a_i(x_0) = 0$ and $b_i(x_0) = 0$. Let $x^j$ be a sequence of states that converges to $x_0$ and let us review the different possibilities according to the definition of $\varphi_{2i}$.

* If the sequence $x^j$ satisfies $a_i(x^j) > 0$, it then follows from (2.7) and definition of $\varphi_{2i}$

$$\lim_{x^j \to x_0} \varphi_{2i}(x^j) = \lim_{x^j \to x_0} \lambda_{2i}(x^j) = +\infty.$$
* For $i \in A_1(x_0)$, the discussions of Cases 1, 2, 5 indicate that there exists $\varepsilon_1^i > 0$ such that, the function $\varphi_{2i}$ is continuous on $B(x_0, \varepsilon_1^i)$. Take $\varepsilon_1 = \min_{i \in A_1(x_0)} \varepsilon_1^i$, it follows that the functions $\varphi_{2i}, i \in A_1(x_0)$, are continuous on $B(x_0, \varepsilon_1)$.

* For $i \in A_2(x_0)$, Cases 3, 4, 7, 8, 9 imply that the following holds

$$ \forall L > 0, \exists \lambda_L^i > 0, \text{ s.t. } \varphi_{2i}(x) > L, \forall x \in B(x_0, \lambda_L^i). $$

Choosing $\lambda_L = \min_{i \in A_2(x_0)} \lambda_L^i$, it follows that

$$ \forall L > 0, \exists \lambda_L > 0, \text{ such that } \varphi_{2i}(x) > L, \forall x \in B(x_0, \lambda_L), \forall i \in A_2(x_0). \quad (3.1) $$

* For $i \in A_{32}(x_0)$, Case 6 indicates that there exists $\delta_i > 0$ such that $\varphi_{2i}(x) = +\infty$ on $B(x_0, \delta_i) \setminus \{x_0\}$. Take $\delta = \min_{i \in A_{32}(x)} \delta_i$, it follows that

$$ \varphi_{2i}(x) = +\infty, \forall x \in B(x_0, \delta) \setminus \{x_0\}, \forall i \in A_{32}(x_0). $$

From this and Assumption 3.1, we obtain, by taking $\mu = \min(\varepsilon_1, \delta)$, for all $x \in B(x_0, \mu)$

$$ \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x), \min_{i \in A_{32}(x_0)} \varphi_{2i}(x) \right) = \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x) \right). \quad (3.2) $$

Let $\varepsilon > 0$ and take $L = |\psi_1(x_0)| + \varepsilon$. The continuity of the function $\psi_1$ and inequality (3.1) ensure the existence of $\lambda_L^\varepsilon > 0$ such that for all $x \in B(x_0, \lambda_L^\varepsilon)$

$$ \psi_1(x) < |\psi_1(x_0)| + \varepsilon = L < \varphi_{2i}(x), \forall i \in A_2(x_0). \quad (3.3) $$

In terms of (3.2) and (3.3), choosing $\eta = \min(\mu, \lambda_L^\varepsilon)$, we have for all $x \in B(x_0, \eta)$

$$ \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x), \min_{i \in A_2(x_0)} \varphi_{2i}(x), \min_{i \in A_{32}(x_0)} \varphi_{2i}(x) \right) = \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x) \right). \quad (3.4) $$

Now, if $A_{31}(x_0) = \emptyset$. From (3.4), we get

$$ w_1(x) = \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x) \right), \forall x \in B(x_0, \eta). $$

Thus, $w_1$ is continuous at $x_0$ since $\psi_1$ and $\varphi_{2i}, i \in A_1(x_0)$, are continuous on $B(x_0, \eta)$.

If $A_{31}(x_0) \neq \emptyset$. For each $i \in A_{31}(x_0)$, we have $a_i(x_0) < 0$ and $b_i(x_0) > 0$. For reasons of continuity, $a_i(x) < 0$ and $b_i(x) > 0$ in a neighborhood $W_{x_0}^i$ of $x_0$. Recall that $\varphi_{2i}, i \in A_1(x_0)$, are continuous on $B(x_0, \varepsilon_1)$ and let

$$ W_{x_0} = \bigcap_{i \in A_3(x_0)} W_{x_0}^i \cap B(x_0, \varepsilon_1). $$

Now, let $k \in A_{31}(x_0)$ and for convenience, define the function $w_1^k$ on $W_{x_0}$ by:

$$ w_1^k(x) = \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x), \varphi_{2k}(x) \right), \forall x \in W_{x_0}. $$

Note that $\varphi_{2k}(x_0) = -\frac{b_k(x_0)}{2a_k(x_0)}$, so, Assumption 3.1 yields

$$ w_1^k(x_0) = \min(\psi_1(x_0), \min_{i \in A_1(x_0)} \varphi_{2i}(x_0)). $$
* If the sequence $x^j$ satisfies $a_i(x^j) < 0$, $b_i(x^j) > 0$ and $\Delta_i(x^j) \ge 0$, (2.9) yields

$$ \lim_{x^j \to x_0} \varphi_{2i}(x^j) = +\infty. $$

* If the sequence $x^j$ satisfies $a_i(x^j) = 0$ and $b_i(x^j) > 0$, recall that since $V_i$ is a CLF of System $S_i$, we have $c_i(x_0) < 0$, hence by continuity, $c_i(x^j) < 0$ whenever $x^j$ is close enough to $x_0$. So

$$ \lim_{x^j \to x_0} \varphi_{2i}(x^j) = \lim_{x^j \to x_0} -\frac{c_i(x^j)}{b_i(x^j)} = +\infty. $$

* If the sequence $x^j$ does not satisfy any of the above situations, then we have $\varphi_{2i}(x) = +\infty$.

Therefore, in all cases, i.e., if $a_i(x_0) = 0$ and $b_i(x_0) = 0$, the following holds

$$ \forall L > 0, \ \exists \lambda_L^i > 0 \text{ such that } \varphi_{2i}(x) > L, \ \forall x \in B(x_0, \lambda_L^i). $$

**Case 4:** If $a_i(x_0) = 0$ and $b_i(x_0) < 0$, by continuity, $b_i(x) < 0$ in a neighborhood $V_{x_0}$ of $x_0$, so, $\varphi_{2i}(x) = +\infty$ on $V_{x_0}$.

**Case 5:** If $a_i(x_0) < 0$, $b_i(x_0) > 0$ and $\Delta_i(x_0) > 0$, this case is similar to the Case 1.

**Case 6:** If $a_i(x_0) < 0$, $b_i(x_0) > 0$ and $\Delta_i(x_0) = 0$, that is $i \in A_3(x_0)$. If $i \in A_{32}(x_0)$, then there exists $\delta_i > 0$ such that $\Delta_i(x) < 0$ for all $x \in B(x_0, \delta_i) \setminus \{x_0\}$. So, $\varphi_{2i}(x) = +\infty$ on $B(x_0, \delta_i) \setminus \{x_0\}$.

If $i \in A_{31}(x_0)$, then for $x$ close enough to $x_0$ we have, $\varphi_{2i}(x) = +\infty$ if $\Delta_i(x) < 0$ and $\varphi_{2i}(x) = \lambda_{2i}(x)$ if $\Delta_i(x) \ge 0$.

**Case 7:** If $a_i(x_0) < 0$, $b_i(x_0) > 0$ and $\Delta_i(x_0) < 0$, by continuity, $\Delta_i(x) < 0$ in a neighborhood $V_{x_0}$ of $x_0$, so, $\varphi_{2i}(x) = +\infty$ on $V_{x_0}$.

**Case 8:** If $a_i(x_0) < 0$, $b_i(x_0) = 0$, this case is similar to Case 7 since from Assumption 2.1, we have $\Delta_i(x_0) < 0$.

**Case 9:** If $a_i(x_0) < 0$, $b_i(x_0) < 0$, this case is similar to Case 4.

Summarizing the above, we have:

* For $i \in A_1(x_0)$, the discussions of Cases 1, 2, 5 indicate that there exists $\varepsilon_1^i > 0$ such that, the function $\varphi_{2i}$ is continuous on $B(x_0, \varepsilon_1^i)$. Take $\varepsilon_1 = \min_{i \in A_1(x_0)} \varepsilon_1^i$, it follows that the functions $\varphi_{2i}$, $i \in A_1(x_0)$, are continuous on $B(x_0, \varepsilon_1)$.

* For $i \in A_2(x_0)$, Cases 3, 4, 7, 8, 9 imply that the following holds

$$ \forall L > 0, \ \exists \lambda_L^i > 0, \text{ s.t. } \varphi_{2i}(x) > L, \ \forall x \in B(x_0, \lambda_L^i). $$
Let $x^j \in W_{x_0}$ and $y^j \in W_{x_0}$ be two sequences of states that converges to $x_0$ satisfying $\Delta_k(x^j) \ge 0$ and $\Delta_k(y^j) < 0$. By definition of $\varphi_{2k}$, we have

$$ \lim_{x^j \to x_0} \varphi_{2k}(x^j) = \lim_{x^j \to x_0} \lambda_{2k}(x^j) = -\frac{b_k(x_0)}{2a_k(x_0)} = \varphi_{2k}(x_0). \quad (3.5) $$

and

$$ \varphi_{2k}(y^j) = +\infty. \quad (3.6) $$

On the one hand, Assumption 3.1, equality (3.5) and the continuity of the functions $\psi_1$ and $\varphi_{2i}$, $i \in A_1(x_0)$, on $W_{x_0}$, along with the definition of $w_1^k$, we have

$$ \begin{aligned} \lim_{x^j \to x_0} w_1^k(x^j) &= \min \left( \psi_1(x_0), \min_{i \in A_1(x_0)} \varphi_{2i}(x_0), -\frac{b_k(x_0)}{2a_k(x_0)} \right) \\ &= \min \left( \psi_1(x_0), \min_{i \in A_1(x_0)} \varphi_{2i}(x_0) \right) = w_1^k(x_0). \end{aligned} \quad (3.7) $$

On the other hand, equality (3.6) yields $w_1^k(y^j) = \min \left( \psi_1(y^j), \min_{i \in A_1(x_0)} \varphi_{2i}(y^j) \right)$, which implies

$$ \lim_{y^j \to x_0} w_1^k(y^j) = \min \left( \psi_1(x_0), \min_{i \in A_1(x_0)} \varphi_{2i}(x_0) \right) = w_1^k(x_0). \quad (3.8) $$

In terms of (3.7) and (3.8), the functions $w_1^k$, $k \in A_{31}(x_0)$, are continuous at $x_0$.

Finally, using (3.4) and choosing $W'_{x_0} = W_{x_0} \cap B(x_0, \eta)$, along with the definitions of $w_1$ and $w_1^k$, $k \in A_{31}(x_0)$, we obtain

$$ \begin{aligned} w_1(x) &= \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x), \min_{k \in A_{31}(x_0)} \varphi_{2k}(x) \right) \\ &= \min_{k \in A_{31}(x_0)} w_1^k(x), \quad \forall x \in W'_{x_0}. \end{aligned} $$

Thus, $w_1$ is continuous at $x_0$ since the functions $w_1^k$, $k \in A_{31}(x_0)$, are continuous at $x_0$.

Now, we establish the continuity of $w_1$ at the origin in the case of the small control property of $V_i$, $i \in I$.

Far each $i \in I$, take $\varepsilon > 0$, due to the small control property, there exists $\alpha_i > 0$ such that, for each $x \ne 0$ and $\|x\| < \alpha_i$, there exists $v \in [-\varepsilon, \varepsilon]$ that satisfies the inequality

$$ \dot{V}_i(x) = a_i(x)v^2 + b_i(x)v + c_i(x) < 0. \quad (3.9) $$

With this inequality in mind, we review the different items in the definition of $\varphi_{2i}$.

* If $a_i(x) > 0$, then (3.9) leads to $v \in [-\lambda_{1i}(x), \lambda_{2i}(x)]$, that is, $-\varepsilon < v < \lambda_{2i}(x) = \varphi_{2i}(x)$.

* If $a_i(x) = 0$ and $b_i(x) > 0$, then (3.9) leads to $-\varepsilon < v < -\frac{c_i(x)}{b_i(x)} = \varphi_{2i}(x)$.

* If $a_i(x) < 0$, $b_i(x) > 0$ and $\Delta_i(x) \ge 0$, then (3.9) leads to $v < \lambda_{2i}(x) \le \lambda_{1i}(x)$ or $\lambda_{2i}(x) \le \lambda_{1i}(x) < v$. But in this case, we have $\lambda_{1i}(x) > 0$ and $|\lambda_{2i}(x)| \le \lambda_{1i}(x)$. So, we always have $-\varepsilon < -|v| < \lambda_{2i}(x) = \varphi_{2i}(x)$.

* Otherwise, $\varphi_{2i}(x) = +\infty$.
Choosing $\lambda_L = \min_{i \in A_2(x_0)} \lambda_L^i$, it follows that

$$ \forall L > 0, \exists \lambda_L > 0, \text{ such that } \varphi_{2i}(x) > L, \forall x \in B(x_0, \lambda_L), \forall i \in A_2(x_0). \quad (3.1) $$

* For $i \in A_{32}(x_0)$, Case 6 indicates that there exists $\delta_i > 0$ such that $\varphi_{2i}(x) = +\infty$ on $B(x_0, \delta_i) \setminus \{x_0\}$. Take $\delta = \min_{i \in A_{32}(x)} \delta_i$, it follows that

$$ \varphi_{2i}(x) = +\infty, \forall x \in B(x_0, \delta) \setminus \{x_0\}, \forall i \in A_{32}(x_0). $$

From this and Assumption 3.1, we obtain, by taking $\mu = \min(\varepsilon_1, \delta)$, for all $x \in B(x_0, \mu)$

$$ \min(\psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x), \min_{i \in A_{32}(x_0)} \varphi_{2i}(x)) = \min(\psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x)). \quad (3.2) $$

Let $\varepsilon > 0$ and take $L = |\psi_1(x_0)| + \varepsilon$. The continuity of the function $\psi_1$ and inequality (3.1) ensure the existence of $\lambda_L^\varepsilon > 0$ such that for all $x \in B(x_0, \lambda_L^\varepsilon)$

$$ \psi_1(x) < |\psi_1(x_0)| + \varepsilon = L < \varphi_{2i}(x), \forall i \in A_2(x_0). \quad (3.3) $$

In terms of (3.2) and (3.3), choosing $\eta = \min(\mu, \lambda_L^\varepsilon)$, we have for all $x \in B(x_0, \eta)$

$$ \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x), \min_{i \in A_2(x_0)} \varphi_{2i}(x), \min_{i \in A_{32}(x_0)} \varphi_{2i}(x) \right) = \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x) \right). \quad (3.4) $$

Now, if $A_{31}(x_0) = \emptyset$. From (3.4), we get

$$ w_1(x) = \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x) \right), \forall x \in B(x_0, \eta). $$

Thus, $w_1$ is continuous at $x_0$ since $\psi_1$ and $\varphi_{2i}$, $i \in A_1(x_0)$, are continuous on $B(x_0, \eta)$.

If $A_{31}(x_0) \neq \emptyset$. For each $i \in A_{31}(x_0)$, we have $a_i(x_0) < 0$ and $b_i(x_0) > 0$. For reasons of continuity, $a_i(x) < 0$ and $b_i(x) > 0$ in a neighborhood $W_{x_0}^i$ of $x_0$. Recall that $\varphi_{2i}$, $i \in A_1(x_0)$, are continuous on $B(x_0, \varepsilon_1)$ and let

$$ W_{x_0} = \bigcap_{i \in A_3(x_0)} W_{x_0}^i \bigcap B(x_0, \varepsilon_1). $$

Now, let $k \in A_{31}(x_0)$ and for convenience, define the function $w_1^k$ on $W_{x_0}$ by:

$$ w_1^k(x) = \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x), \varphi_{2k}(x) \right), \forall x \in W_{x_0}. $$

Note that $\varphi_{2k}(x_0) = -\frac{b_k(x_0)}{2a_k(x_0)}$, so, Assumption 3.1 yields

$$ w_1^k(x_0) = \min(\psi_1(x_0), \min_{i \in A_1(x_0)} \varphi_{2i}(x_0)). $$
From the above discussion, we deduce that

$$ \forall \varepsilon > 0, \exists \alpha_i > 0 \text{ such that, } -\varepsilon < \varphi_{2i}(x), \forall x \in B(0, \alpha_i). $$

Take $\alpha = \min_{i \in I} \alpha_i$, it follows

$$ \forall \varepsilon > 0, \exists \alpha > 0 \text{ s.t., } -\varepsilon < \varphi_2(x), \forall x \in B(0, \alpha). \quad (3.10) $$

Continuity of $\psi_1$ at the origin leads to

$$ \forall \varepsilon > 0, \exists \beta > 0 \text{ s.t., } -\varepsilon < \psi_1(x) < \varepsilon, \forall x \in B(0, \beta). \quad (3.11) $$

Take $\gamma = \min(\alpha, \beta)$, it follows from (3.10) and (3.11)

$$ \forall \varepsilon > 0, \exists \gamma > 0 \text{ such that, } -\varepsilon < w_1(x) = \min(\psi_1(x), \varphi_2(x)) < \varepsilon, \forall x \in B(0, \gamma). $$

That is, $w_1$ is continuous at the origin and so on $\mathbb{R}^n$.

Using Assumptions 2.1, 3.2 and Lemma 2.2, the continuity of $w_2$ can be treated similarly and is omitted here. The proof of Lemma 3.2 is now completed. $\square$

**Remark 3.2.** Let $V_i$, $i \in I$ be a collection of CLFs for system $S_i$, respectively. It is easy to see that Assumptions 2.1, 3.1, 3.2 are trivial if $a_i(x) \ge 0$ for all $x \in \mathbb{R}^n$ and $i \in I$. Indeed, in this case we have $A_3(x) = B_3(x) = \emptyset$ for all $x \in \mathbb{R}^n$ and, for each $i \in I$, the set $\{x \in \mathbb{R}^n, | a_i(x) < 0\}$ is empty. So, by Lemma 3.2, if $a_i(x) \ge 0$ for all $x \in \mathbb{R}^n$ and $i \in I$, then the functions $w_1$ and $w_2$ are continuous on $\mathbb{R}^n \setminus \{0\}$ and continuous on $\mathbb{R}^n$ if the functions $V_i$ satisfy the SCP.

The following theorem presents a necessary condition for the existence of simultaneously asymptotically stabilizing state feedback for the collection of systems in (2.1).

**Theorem 3.1.** Consider the collection of systems in (2.1). If there exists a feedback $k: \mathbb{R}^n \to \mathbb{R}$ such that for each $i \in I$, the closed-loop system $\dot{x} = f_i(x)+k(x)g_i(x)+k^2(x)h_i(x)$ is globally asymptotically stable, then there exists a collection of CLFs $V_i$ for Systems $S_i$, respectively, satisfying for all $x \in \mathbb{R}^n$ such that $I_1(x) \cup I_4(x) \neq \emptyset$ and $I_1(x) \cup I_5(x) \neq \emptyset$

$$ \max_{i \in I_1(x) \cup I_4(x)} \varphi_{1i}(x) < \min_{i \in I_1(x) \cup I_5(x)} \varphi_{2i}(x). \quad (3.12) $$

*Proof.* Suppose that there exists a feedback $k$ such that the closed-loop systems $\dot{x} = f_i(x) + k(x)g_i(x) + k^2(x)h_i(x)$, $i \in I$, are globally asymptotically stable, then by converse Lyapunov theorem, there exist a collection of smooth, positive definite and radially unbounded functions $V_i$, $i \in I$, such that for each $i \in I$, $\dot{V}_i(x) = a_i(x)k^2(x) + b_i(x)k(x) + c_i(x) < 0$, for all $x \neq 0$. It then follows that for each $x \in \mathbb{R}^n \setminus \{0\}$, $\lambda_{1i}(x) < k(x) < \lambda_{2i}(x)$ if $a_i(x) > 0$,

$$ k(x) > -\frac{c_i(x)}{b_i(x)} \text{ if } a_i(x) = 0 \text{ and } b_i(x) < 0, $$

$$ k(x) < -\frac{c_i(x)}{b_i(x)} \text{ if } a_i(x) = 0 \text{ and } b_i(x) > 0. $$

That is, $\varphi_{1i}(x) < k(x)$ if $i \in I_1(x) \cup I_4(x)$ and, $k(x) < \varphi_{2i}(x)$ if $i \in I_1(x) \cup I_5(x)$, which is equivalent, for all $x \in \mathbb{R}^n$ such that $I_1(x) \cup I_4(x) \neq \emptyset$ and $I_1(x) \cup I_5(x) \neq \emptyset$, to

$$ \max_{i \in I_1(x) \cup I_4(x)} \varphi_{1i}(x) < k(x) < \min_{i \in I_1(x) \cup I_5(x)} \varphi_{2i}(x), \quad (3.13) $$

and this completes the proof of Theorem 3.1. $\square$
Let $x^j \in W_{x_0}$ and $y^j \in W_{x_0}$ be two sequences of states that converges to $x_0$ satisfying $\Delta_k(x^j) \ge 0$ and $\Delta_k(y^j) < 0$. By definition of $\varphi_{2k}$, we have

$$ \lim_{x^j \to x_0} \varphi_{2k}(x^j) = \lim_{x^j \to x_0} \lambda_{2k}(x^j) = -\frac{b_k(x_0)}{2a_k(x_0)} = \varphi_{2k}(x_0). \quad (3.5) $$

and

$$ \varphi_{2k}(y^j) = +\infty. \quad (3.6) $$

On the one hand, Assumption 3.1, equality (3.5) and the continuity of the functions $\psi_1$ and $\varphi_{2i}$, $i \in A_1(x_0)$, on $W_{x_0}$, along with the definition of $w_1^k$, we have

$$ \begin{aligned} \lim_{x^j \to x_0} w_1^k(x^j) &= \min \left( \psi_1(x_0), \min_{i \in A_1(x_0)} \varphi_{2i}(x_0), -\frac{b_k(x_0)}{2a_k(x_0)} \right) \\ &= \min \left( \psi_1(x_0), \min_{i \in A_1(x_0)} \varphi_{2i}(x_0) \right) = w_1^k(x_0). \end{aligned} \quad (3.7) $$

On the other hand, equality (3.6) yields $w_1^k(y^j) = \min \left( \psi_1(y^j), \min_{i \in A_1(x_0)} \varphi_{2i}(y^j) \right)$, which implies

$$ \lim_{y^j \to x_0} w_1^k(y^j) = \min \left( \psi_1(x_0), \min_{i \in A_1(x_0)} \varphi_{2i}(x_0) \right) = w_1^k(x_0). \quad (3.8) $$

In terms of (3.7) and (3.8), the functions $w_1^k$, $k \in A_{31}(x_0)$, are continuous at $x_0$.

Finally, using (3.4) and choosing $W'_{x_0} = W_{x_0} \cap B(x_0, \eta)$, along with the definitions of $w_1$ and $w_1^k$, $k \in A_{31}(x_0)$, we obtain

$$ \begin{aligned} w_1(x) &= \min \left( \psi_1(x), \min_{i \in A_1(x_0)} \varphi_{2i}(x), \min_{k \in A_{31}(x_0)} \varphi_{2k}(x) \right) \\ &= \min_{k \in A_{31}(x_0)} w_1^k(x), \quad \forall x \in W'_{x_0}. \end{aligned} $$

Thus, $w_1$ is continuous at $x_0$ since the functions $w_1^k$, $k \in A_{31}(x_0)$, are continuous at $x_0$.

Now, we establish the continuity of $w_1$ at the origin in the case of the small control property of $V_i$, $i \in I$.

Far each $i \in I$, take $\varepsilon > 0$, due to the small control property, there exists $\alpha_i > 0$ such that, for each $x \ne 0$ and $\|x\| < \alpha_i$, there exists $v \in [-\varepsilon, \varepsilon]$ that satisfies the inequality

$$ \dot{V}_i(x) = a_i(x)v^2 + b_i(x)v + c_i(x) < 0. \quad (3.9) $$

With this inequality in mind, we review the different items in the definition of $\varphi_{2i}$.

* If $a_i(x) > 0$, then (3.9) leads to $v \in [-\lambda_{1i}(x), \lambda_{2i}(x)]$, that is, $-\varepsilon < v < \lambda_{2i}(x) = \varphi_{2i}(x)$.

* If $a_i(x) = 0$ and $b_i(x) > 0$, then (3.9) leads to $-\varepsilon < v < -\frac{c_i(x)}{b_i(x)} = \varphi_{2i}(x)$.
The necessary condition 3.12 shows that a simultaneously stabilizing feedback $u$ for the collection of systems in (2.1) verify items 1, 2 and 3 in Remark 2.1. The feedback $u$ has also to verify item 4 in Remark 2.1, that is $u(x) < \lambda_{2i}(x) \le \lambda_{1i}(x)$ or $\lambda_{2i}(x) \le \lambda_{1i}(x) < u(x)$ for each $i \in I$ such that $a_i(x) < 0$ and $\Delta_i(x) \ge 0$.

But the following situation may occur: There exists an open set $V \subset \mathbb{R}^n$ and $i \in I$ such that:

(1) $a_i(x) < 0, b_i(x) > 0$ and $\Delta_i(x) \ge 0$, for every $x \in V$;

(2) there exists $x_0 \in \partial V$ (the border of $V$) such that $a_i(x_0) = 0$ and $b_i(x_0) > 0$.

Let $x_1 \in V$, if we choose $\lambda_{2i}(x_1) \le \lambda_{1i}(x_1) < u(x_1)$, for reasons of continuity we have $\lambda_{2i}(x) \le \lambda_{1i}(x) < u(x)$ for all $x \in V$. From this and using (2.11) we get,

$$ \lim_{x \to x_0} u(x) \ge \lim_{x \to x_0} \lambda_{1i}(x) = +\infty $$

which contradict the fact that $u$ is continuous at $x_0$. To eliminate this situation, we have to choose $u(x) < \lambda_{2i}(x) \le \lambda_{1i}(x)$ if $a_i(x) < 0, b_i(x) > 0$ and $\Delta_i(x) \ge 0$. Similarly, by using (2.6), we have to choose $\lambda_{2i}(x) \le \lambda_{1i}(x) < u(x)$ if $a_i(x) < 0, b_i(x) < 0$ and $\Delta_i(x) \ge 0$. That is, for each $x \in \mathbb{R}^n \setminus \{0\}$

$$ \lambda_{1i}(x) < u(x), \forall i \in I_2(x) \text{ and } u(x) < \lambda_{2i}(x), \forall i \in I_3(x). \quad (3.14) $$

(3.14) and definitions of $\varphi_{1i}$ and $\varphi_{2i}$ lead to, for each $x \in \mathbb{R}^n$

$$ \max_{i \in I_2(x)} \varphi_{1i}(x) < u(x) < \min_{i \in I_3(x)} \varphi_{2i}(x). \quad (3.15) $$

The discussion above and Theorem 3.1 (condition (3.12)) show that in order to construct a continuous feedback $u$ that simultaneously stabilizes the collection of systems in (2.1), the feedback $u$ should verify (3.13) and (3.15), which means for all $x \in \mathbb{R}^n$

$$ \max_{i \in I_N(x)} \varphi_{1i}(x) < u(x) < \min_{i \in I_P(x)} \varphi_{2i}(x), \quad (3.16) $$

with, $I_N(x) = I_1(x) \cup I_2(x) \cup I_4(x)$, $I_P(x) = I_1(x) \cup I_3(x) \cup I_5(x)$.

Therefore, to guarantee that the feedback that will be constructed below verify (3.16), we assume

$$ \max_{i \in I_N(x)} \varphi_{1i}(x) < \min_{i \in I_P(x)} \varphi_{2i}(x), \forall x \in \mathbb{R}^n. \quad (3.17) $$

**Remark 3.3.** Note that for each $x \in \mathbb{R}^n$, since $\varphi_{1i}(x) = -\infty$ for all $i \in I \setminus I_N(x)$ and $\varphi_{2i}(x) = +\infty$ for all $i \in I \setminus I_P(x)$, then inequality (3.17) is equivalent to the following assumption.

**Assumption 3.3.** For every $x \in \mathbb{R}^n$, $\varphi_1(x) < \varphi_2(x)$.

**Remark 3.4.** If the collection of CLFs in Theorem 3.1 verify $a_i(x) \ge 0$ for all $x \in \mathbb{R}^n$ and $i \in I$, then $I_2(x) = I_3(x) = \emptyset$ for all $x \in \mathbb{R}^n$. So, inequalities (3.12) and (3.17) are equivalent. That is, by Theorem 3.1 and Remark 3.3, Assumption 3.3 becomes necessary in this case.

Now, we can state our main result that gives the simultaneous stabilizing feedback law.

**Theorem 3.2.** Consider the collection of systems in (2.1). If there exists a collection of CLFs $V_i$, $i = 1, ..., m$, satisfying Assumption 2.1, such that Assumptions 3.1, 3.2 and 3.3 hold, then the feedback

$$ k(x) = \frac{w_1(x) + w_2(x)}{2} \quad (3.18) $$

is continuous on $\mathbb{R}^n \setminus \{0\}$ and globally asymptotically stabilizes the collection of systems in (2.1) simultaneously. Moreover, the feedback $k$ is continuous on $\mathbb{R}^n$ if the functions $V_i$, $i = 1, 2, ..., m$, satisfy the small control property.
* If $a_i(x) < 0$, $b_i(x) > 0$ and $\Delta_i(x) \ge 0$, then (3.9) leads to $v < \lambda_{2i}(x) \le \lambda_{1i}(x)$ or $\lambda_{2i}(x) \le \lambda_{1i}(x) < v$. But in this case, we have $\lambda_{1i}(x) > 0$ and $|\lambda_{2i}(x)| \le \lambda_{1i}(x)$. So, we always have $-\varepsilon < -|v| < \lambda_{2i}(x) = \varphi_{2i}(x)$.

* Otherwise, $\varphi_{2i}(x) = +\infty$.

From the above discussion, we deduce that

$$ \forall \varepsilon > 0, \exists \alpha_i > 0 \text{ such that, } -\varepsilon < \varphi_{2i}(x), \forall x \in B(0, \alpha_i). $$

Take $\alpha = \min_{i \in I} \alpha_i$, it follows

$$ \forall \varepsilon > 0, \exists \alpha > 0 \text{ s.t., } -\varepsilon < \varphi_{2}(x), \forall x \in B(0, \alpha). \quad (3.10) $$

Continuity of $\psi_1$ at the origin leads to

$$ \forall \varepsilon > 0, \exists \beta > 0 \text{ s.t., } -\varepsilon < \psi_{1}(x) < \varepsilon, \forall x \in B(0, \beta). \quad (3.11) $$

Take $\gamma = \min(\alpha, \beta)$, it follows from (3.10) and (3.11)

$$ \forall \varepsilon > 0, \exists \gamma > 0 \text{ such that, } -\varepsilon < w_{1}(x) = \min(\psi_{1}(x), \varphi_{2}(x)) < \varepsilon, \forall x \in B(0, \gamma). $$

That is, $w_1$ is continuous at the origin and so on $\mathbb{R}^n$.

Using Assumptions 2.1, 3.2 and Lemma 2.2, the continuity of $w_2$ can be treated similarly and is omitted here. The proof of Lemma 3.2 is now completed. □

**Remark 3.2.** Let $V_i$, $i \in I$ be a collection of CLFs for system $S_i$, respectively. It is easy to see that Assumptions 2.1, 3.1, 3.2 are trivial if $a_i(x) \ge 0$ for all $x \in \mathbb{R}^n$ and $i \in I$. Indeed, in this case we have $A_3(x) = B_3(x) = \emptyset$ for all $x \in \mathbb{R}^n$ and, for each $i \in I$, the set $\{x \in \mathbb{R}^n, | a_i(x) < 0 \}$ is empty. So, by Lemma 3.2, if $a_i(x) \ge 0$ for all $x \in \mathbb{R}^n$ and $i \in I$, then the functions $w_1$ and $w_2$ are continuous on $\mathbb{R}^n \setminus \{0\}$ and continuous on $\mathbb{R}^n$ if the functions $V_i$ satisfy the SCP.

The following theorem presents a necessary condition for the existence of simultaneously stabilizing state feedback for the collection of systems in (2.1).

**Theorem 3.3.** Consider the collection of systems in (2.1). If there exists a continuous feedback $k: \mathbb{R}^n \to \mathbb{R}$, with $k(0) = 0$, so that for each $i \in I$, the closed-loop system $\dot{x} = f_i(x)+k(x)g_i(x)+k^2(x)h_i(x)$ is globally stabilizable, then there exists a collection of CLFs $V_i$ for Systems $S_i$, respectively, satisfying for all $x \in \mathbb{R}^n$ such that $I_1(x) \cup I_4(x) \ne \emptyset$ and $I_1(x) \cup I_5(x) \ne \emptyset$

$$ \max_{i \in I_1(x) \cup I_4(x)} \varphi_{1i}(x) < \min_{i \in I_1(x) \cup I_5(x)} \varphi_{2i}(x). \quad (3.12) $$

*Proof.* Suppose that there exists a continuous feedback $k$ such that the closed-loop systems $\dot{x} = f_i(x) + k(x)g_i(x) + k^2(x)h_i(x)$, $i \in I$, (whose coefficients are continuous) are stabilizable, then by converse Lyapunov theorem (see (Massera, 1956, Theorem 14), (Kurzweil, 1956), one can also see a survey in (Kellett, 2015)), there exists a collection of smooth, positive definite and radially unbounded functions $V_i$, $i \in I$, such
*Proof.* According to Lemmas 3.1 and 3.2, the continuity of $k$ is immediate. For the asymptotic stability, we only have to verify that for each $i \in I$

$$ \dot{V}_i(x) = a_i(x)k^2(x) + b_i(x)k(x) + c_i(x) < 0, \quad \forall x \in \mathbb{R}^n \setminus \{0\}, $$

according to the sign of $\Delta_i(x)$.

**Case 1.** If $\Delta_i(x) < 0$, then $a_i(x) < 0$ according to Lemma 2.1. Thus, the sign of $\dot{V}_i(x)$ regarded as a quadratic polynomial in $k$ is that of $a_i(x)$. So, we have $\dot{V}_i(x) < 0$ in this case.

**Case 2.** If $\Delta_i(x) = 0$, from Lemma 2.1, we know that $a_i(x) = b_i(x) = 0$ and $c_i(x) < 0$ or $a_i(x) < 0$.

i. If $a_i(x) = b_i(x) = 0$ and $c_i(x) < 0$, then $\dot{V}_i(x) = c_i(x) < 0$.

ii. If $a_i(x) < 0$, from Assumption 2.1 we have $b_i(x) \neq 0$.

- If $b_i(x) > 0$. On the one hand, From the definitions of $w_1$, $\varphi_2$ and $\varphi_{2i}$, we obtain

$$ w_1(x) = \min(\psi_1(x), \varphi_2(x)) \le \varphi_2(x) \le \varphi_{2i}(x) = \lambda_{2i}(x) = -\frac{b_i(x)}{2a_i(x)}. \quad (3.19) $$

On the other hand, definitions of $\psi_2$, $\psi_{2i}$ and $k_i$ give, $\psi_2(x) \le \psi_{2i}(x) = k_i(x) < \varphi_{2i}(x) = \lambda_{2i}(x) = -\frac{b_i(x)}{2a_i(x)}$. From Assumption 3.3, we get $\varphi_1(x) < \varphi_2(x) \le -\frac{b_i(x)}{2a_i(x)}$. Hence,

$$ w_2(x) = \max(\psi_2(x), \varphi_1(x)) < \lambda_{2i}(x) = -\frac{b_i(x)}{2a_i(x)}. \quad (3.20) $$

(3.19) and (3.20) lead to

$$ k(x) = \frac{w_1(x) + w_2(x)}{2} < \lambda_{2i}(x) = \lambda_{1i}(x) = -\frac{b_i(x)}{2a_i(x)}. $$

Thus, $\dot{V}_i(x) < 0$ since $a_i(x) < 0$.

- If $b_i(x) < 0$, then $\varphi_{1i}(x) = \lambda_{1i}(x) < k_i(x) = \psi_{1i}(x) \le \psi_1(x)$. Assumption 3.3 yields $\lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) < \varphi_2(x)$. So

$$ \lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) \le \max(\psi_2(x), \varphi_1(x)) = w_2(x), $$

and

$$ \lambda_{1i}(x) < \min(\psi_1(x), \varphi_2(x)) = w_1(x). $$

That is,

$$ \lambda_{2i}(x) = \lambda_{1i}(x) < \frac{w_1(x) + w_2(x)}{2} = k(x). $$

Thus, $\dot{V}_i(x) < 0$ since $a_i(x) < 0$.

**Case 3.** If $\Delta_i(x) > 0$. In this case, we consider the three following cases according to the sign of $b_i(x)$.
that for each $i \in I$, $\dot{V}_i(x) = a_i(x)k^2(x) + b_i(x)k(x) + c_i(x) < 0$, for all $x \neq 0$. It then follows that for each $x \in \mathbb{R}^n \setminus \{0\}$,

$$ \lambda_{1i}(x) < k(x) < \lambda_{2i}(x) \text{ if } a_i(x) > 0, $$

$$ k(x) > -\frac{c_i(x)}{b_i(x)} \text{ if } a_i(x) = 0 \text{ and } b_i(x) < 0, $$

$$ k(x) < -\frac{c_i(x)}{b_i(x)} \text{ if } a_i(x) = 0 \text{ and } b_i(x) > 0. $$

That is, $\varphi_{1i}(x) < k(x)$ if $i \in I_1(x) \cup I_4(x)$ and, $k(x) < \varphi_{2i}(x)$ if $i \in I_1(x) \cup I_5(x)$, which is equivalent, for all $x \in \mathbb{R}^n$ such that $I_1(x) \cup I_4(x) \neq \emptyset$ and $I_1(x) \cup I_5(x) \neq \emptyset$, to

$$ \max_{i \in I_1(x) \cup I_4(x)} \varphi_{1i}(x) < k(x) < \min_{i \in I_1(x) \cup I_5(x)} \varphi_{2i}(x), \quad (3.13) $$

and this completes the proof of Theorem 3.3. $\square$

The necessary condition (3.12) shows that a simultaneously stabilizing feedback $u$ for the collection of systems in (2.1) verify items 1, 2 and 3 in Remark 2.1. The feedback $u$ has also to verify item 4 in Remark 2.1, that is $u(x) < \lambda_{2i}(x) \le \lambda_{1i}(x)$ or $\lambda_{2i}(x) \le \lambda_{1i}(x) < u(x)$ for each $i \in I$ such that $a_i(x) < 0$ and $\Delta_i(x) \ge 0$.

But the following situation may occur: There exists an open set $V \subset \mathbb{R}^n$ and $i \in I$ such that:

(1) $a_i(x) < 0$, $b_i(x) > 0$ and $\Delta_i(x) \ge 0$, for every $x \in V$;

(2) there exists $x_0 \in \partial V$ (the border of $V$) such that $a_i(x_0) = 0$ and $b_i(x_0) > 0$.

Let $x_1 \in V$, if we choose $\lambda_{2i}(x_1) \le \lambda_{1i}(x_1) < u(x_1)$, for reasons of continuity we have $\lambda_{2i}(x) \le \lambda_{1i}(x) < u(x)$ for all $x \in V$. From this and using (2.11) we get,

$$ \lim_{x \to x_0} u(x) \ge \lim_{x \to x_0} \lambda_{1i}(x) = +\infty $$

which contradict the fact that $u$ is continuous at $x_0$. To eliminate this situation, we have to choose $u(x) < \lambda_{2i}(x) \le \lambda_{1i}(x)$ if $a_i(x) < 0$, $b_i(x) > 0$ and $\Delta_i(x) \ge 0$. Similarly, by using (2.6), we have to choose $\lambda_{2i}(x) \le \lambda_{1i}(x) < u(x)$ if $a_i(x) < 0$, $b_i(x) < 0$ and $\Delta_i(x) \ge 0$. That is, for each $x \in \mathbb{R}^n \setminus \{0\}$

$$ \lambda_{1i}(x) < u(x), \forall i \in I_2(x) \text{ and } u(x) < \lambda_{2i}(x), \forall i \in I_3(x). \quad (3.14) $$

(3.14) and definitions of $\varphi_{1i}$ and $\varphi_{2i}$ lead to, for each $x \in \mathbb{R}^n$

$$ \max_{i \in I_2(x)} \varphi_{1i}(x) < u(x) < \min_{i \in I_3(x)} \varphi_{2i}(x). \quad (3.15) $$

The discussion above and Theorem 3.12 show that in order to construct a continuous feedback $u$ that simultaneously stabilizes the collection of systems in (2.1), the feedback $u$ should verify (3.13) and (3.15), which means for all $x \in \mathbb{R}^n$

$$ \max_{i \in I_N(x)} \varphi_{1i}(x) < u(x) < \min_{i \in I_P(x)} \varphi_{2i}(x), \quad (3.16) $$

with, $I_N(x) = I_1(x) \cup I_2(x) \cup I_4(x)$, $I_P(x) = I_1(x) \cup I_3(x) \cup I_5(x)$.
i. If $b_i(x) < 0$, we consider again three cases according to the sign of $a_i(x)$.

- If $a_i(x) > 0$. On the one hand, since $k_i$ stabilizes System $S_i$, we have $\lambda_{1i}(x) < k_i(x) = \psi_{1i}(x) \le \psi_1(x)$. By Assumption 3.3, we get $\lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) < \varphi_2(x)$. So $\lambda_{1i}(x) < \min(\psi_1(x), \varphi_2(x)) = w_1(x)$. Together with the fact that $w_1(x) \le \varphi_2(x) \le \varphi_{2i}(x) = \lambda_{2i}(x)$, it then follows that

$$ \lambda_{1i}(x) < w_1(x) \le \lambda_{2i}(x). \qquad (3.21) $$

On the other hand, note that in this case $0 < \lambda_{2i}(x)$, hence $\psi_2(x) \le \psi_{2i}(x) = 0 < \lambda_{2i}(x)$. As above, by Assumption 3.3, we get $\varphi_1(x) < \varphi_2(x) \le \varphi_{2i}(x) = \lambda_{2i}(x)$. So $w_2(x) = \max(\psi_2(x), \varphi_1(x)) < \lambda_{2i}(x)$, and since $\lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) \le w_2(x)$, it then follows that

$$ \lambda_{1i}(x) \le w_2(x) < \lambda_{2i}(x). \qquad (3.22) $$

(3.21) and (3.22) give

$$ \lambda_{1i}(x) < \frac{w_1(x) + w_2(x)}{2} = k(x) < \lambda_{2i}(x). $$

Thus, $\dot{V}_i(x) < 0$ since $a_i(x) > 0$.

- If $a_i(x) = 0$. Then $-\frac{c_i(x)}{b_i(x)} < k_i(x) = \psi_{1i}(x) \le \psi_1(x)$. Assumption 3.3, yields $-\frac{c_i(x)}{b_i(x)} = \varphi_{1i}(x) \le \varphi_1(x) < \varphi_2(x)$. So $-\frac{c_i(x)}{b_i(x)} < w_1(x)$ and $-\frac{c_i(x)}{b_i(x)} \le w_2(x)$ since $\varphi_1(x) \le w_2(x)$. It then follows that

$$ -\frac{c_i(x)}{b_i(x)} < \frac{w_1(x) + w_2(x)}{2} = k(x). $$

Thus, $\dot{V}_i(x) = b_i(x)k(x) + c_i(x) < 0$ since $b_i(x) < 0$.

- If $a_i(x) < 0$. As above, $\lambda_{1i}(x) < k_i(x) = \psi_{1i}(x) \le \psi_1(x)$. Assumption 3.3 yields $\lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) < \varphi_2(x)$. Hence $\lambda_{1i}(x) < w_1(x)$ and $\lambda_{1i}(x) \le \varphi_1(x) \le w_2(x)$. So

$$ \lambda_{2i}(x) < \lambda_{1i}(x) < \frac{w_1(x) + w_2(x)}{2} = k(x). $$

That is, $\dot{V}_i(x) < 0$ since $a_i(x) < 0$.

ii. If $b_i(x) > 0$, this case can be treated similarly to the previous one.

iii. If $b_i(x) = 0$, by Assumption 2.1 and since $\Delta_i(x) > 0$, we have $a_i(x) > 0$, so, this case is similar to the case $a_i(x) > 0$ in Case 3, item i.

Thus, the proof of Theorem 3.2 is completed. $\square$

**Remark 3.5.** If the collection of CLFs in Theorem 3.2 verify $a_i(x) \ge 0$ for all $x \in \mathbb{R}^n$ and $i \in I$, then, by Remark 3.2, the feedback (3.18) simultaneously stabilizes all the systems in (2.1) if only Assumption 3.3 holds.
To guarantee that the feedback that will be constructed below verify (3.16), we assume

$$ \max_{i \in I_N(x)} \varphi_{1i}(x) < \min_{i \in I_P(x)} \varphi_{2i}(x), \forall x \in \mathbb{R}^n. \quad (3.17) $$

**Remark 3.3.** Note that for each $x \in \mathbb{R}^n$, since $\varphi_{1i}(x) = -\infty$ for all $i \in I \setminus I_N(x)$ and $\varphi_{2i}(x) = +\infty$ for all $i \in I \setminus I_P(x)$, then inequality (3.17) is equivalent to the following assumption.

**Assumption 3.4.** For every $x \in \mathbb{R}^n$, $\varphi_1(x) < \varphi_2(x)$.

**Remark 3.4.** If the collection of CLFs in Theorem 3.3 verify $a_i(x) \ge 0$ for all $x \in \mathbb{R}^n$ and $i \in I$, then $I_2(x) = I_3(x) = \emptyset$ for all $x \in \mathbb{R}^n$. So, inequalities (3.12) and (3.17) are equivalent. That is, by Theorem 3.3 and Remark 3.3, Assumption 3.4 becomes necessary in this case.

Now, we state the main result of this paper.

**Theorem 3.5.** Consider the collection of systems in (2.1). If there exists a collection of CLFs $V_i$, $i = 1, ..., m$, satisfying Assumption 2.1, such that Assumptions 3.1, 3.2 and 3.4 hold, then the collection of systems in (2.1) is simultaneously almost stabilizable with the feedback

$$ k(x) = \frac{w_1(x) + w_2(x)}{2} \quad (3.18) $$

Moreover, the collection of systems in (2.1) is simultaneously stabilizable by the feedback $k$ if the functions $V_i$, $i = 1, 2, ..., m$, satisfy the small control property.

**Proof.** According to Lemmas 3.1 and 3.2, the continuity of $k$ is immediate.

For the asymptotic stability, we only have to verify that for each $i \in I$

$$ \dot{V}_i(x) = a_i(x)k^2(x) + b_i(x)k(x) + c_i(x) < 0, \forall x \in \mathbb{R}^n \setminus \{0\}, $$

according to the sign of $\Delta_i(x)$.

**Case 1.** If $\Delta_i(x) < 0$, then $a_i(x) < 0$ according to Lemma 2.1. Thus, the sign of $\dot{V}_i(x)$ regarded as a quadratic polynomial in $k$ is that of $a_i(x)$. So, we have $\dot{V}_i(x) < 0$ in this case.

**Case 2.** If $\Delta_i(x) = 0$, from Lemma 2.1, we know that $a_i(x) = b_i(x) = 0$ and $c_i(x) < 0$ or $a_i(x) < 0$.

i. If $a_i(x) = b_i(x) = 0$ and $c_i(x) < 0$, then $\dot{V}_i(x) = c_i(x) < 0$.

ii. If $a_i(x) < 0$, from Assumption 2.1 we have $b_i(x) \neq 0$.

◦ If $b_i(x) > 0$. On the one hand, From the definitions of $w_1$, $\varphi_2$ and $\varphi_{2i}$, we obtain

$$ w_1(x) = \min(\psi_1(x), \varphi_2(x)) \le \varphi_2(x) \le \varphi_{2i}(x) = \lambda_{2i}(x) = -\frac{b_i(x)}{2a_i(x)}. \quad (3.19) $$
# 4 Illustrative example

In this section, we present an example to illustrate the effectiveness of the results obtained in this paper to design a simultaneously stabilizing feedback for some nonlinear quadratic systems. Consider a system with the following two possible modes

$$ S_1: \begin{cases} \dot{x}_1 = -x_1 + \frac{x_1 x_2^4}{4} + \frac{x_1}{1+x_1^2} u^2 \\ \dot{x}_2 = -x_2 + x_1^2 x_2 u, \end{cases} \qquad (4.1) $$

$$ S_2: \begin{cases} \dot{x}_1 = -x_1 + x_1^3 x_2^4 + (4x_1 - 4x_2)u - 4x_1 u^2 \\ \dot{x}_2 = -x_2 + x_1 u. \end{cases} \qquad (4.2) $$

Let

$$ V_1(x) = \frac{x_1^2 + x_2^2}{2} \quad \text{and} \quad V_2(x) = \frac{x_1^2 + 4x_2^2}{8}. $$

The Lie derivatives of $V_1$ and $V_2$ along the trajectories of Systems $S_1$ and $S_2$, respectively, write:

$$ \dot{V}_1(x) = a_1(x)u^2 + b_1(x)u + c_1(x), $$

$$ \dot{V}_2(x) = a_2(x)u^2 + b_2(x)u + c_2(x), $$

with $a_1(x) = x_1^2/(1+x_1^2)$, $b_1(x) = x_1^2 x_2^2$, $c_1(x) = -x_1^2 - x_2^2 + (1/4)x_1^2 x_2^4$, $a_2(x) = -x_1^2$, $b_2(x) = x_1^2$, and $c_2(x) = (1/4)(-x_1^2 + x_1^4 x_2^4) - x_2^2$.

By the definitions in Section 2, we have

$$ \Delta_1(x) = \frac{x_1^6 x_2^4 + 4x_1^2 x_2^2 + 4x_1^4}{1 + x_1^2}, \quad \Delta_2(x) = x_1^2 x_2^2 (x_1^4 x_2^2 - 4), $$

$$ \lambda_{11}(x) = \frac{-x_1^2 x_2^2 - x_2^2}{2} - \frac{\sqrt{(1+x_1^2)(x_1^6 x_2^4 + 4x_1^2 x_2^2 + 4x_1^4)}}{2x_1^2}, $$

$$ \lambda_{21}(x) = \frac{-x_1^2 x_2^2 - x_2^2}{2} + \frac{\sqrt{(1+x_1^2)(x_1^6 x_2^4 + 4x_1^2 x_2^2 + 4x_1^4)}}{2x_1^2}, $$

$$ \lambda_{12}(x) = \frac{1}{2} + \frac{\sqrt{x_1^2 x_2^2 (x_1^4 x_2^2 - 4)}}{2x_1^2}, $$

$$ \lambda_{22}(x) = \frac{1}{2} - \frac{\sqrt{x_1^2 x_2^2 (x_1^4 x_2^2 - 4)}}{2x_1^2}. $$

We can prove that $V_1$ and $V_2$ are CLFs for Systems $S_1$ and $S_2$, respectively, and satisfy the small control property. In fact, $V_2$ satisfies the small control property since $\Delta_2(x) < 0$ whenever $x \neq 0$ is close enough to 0 and then $a_2(x) < 0$ if $x_1 \neq 0$ or $c_2(x) < 0$ if $a_2(x) = b_2(x) = 0$.
On the other hand, definitions of $\psi_2$, $\psi_{2i}$ and $k_i$ give, $\psi_2(x) \le \psi_{2i}(x) = k_i(x) < \varphi_{2i}(x) = \lambda_{2i}(x) = -\frac{b_i(x)}{2a_i(x)}$. From Assumption 3.4, we get $\varphi_1(x) < \varphi_2(x) \le -\frac{b_i(x)}{2a_i(x)}$. Hence,

$$w_2(x) = \max(\psi_2(x), \varphi_1(x)) < \lambda_{2i}(x) = -\frac{b_i(x)}{2a_i(x)}. \quad (3.20)$$

(3.19) and (3.20) lead to

$$k(x) = \frac{w_1(x) + w_2(x)}{2} < \lambda_{2i}(x) = \lambda_{1i}(x) = -\frac{b_i(x)}{2a_i(x)}.$$

Thus, $\dot{V}_i(x) < 0$ since $a_i(x) < 0$.

o If $b_i(x) < 0$, then $\varphi_{1i}(x) = \lambda_{1i}(x) < k_i(x) = \psi_{1i}(x) \le \psi_1(x)$. Assumption 3.4 yields $\lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) < \varphi_2(x)$. So

$$\lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) \le \max(\psi_2(x), \varphi_1(x)) = w_2(x),$$

and

$$\lambda_{1i}(x) < \min(\psi_1(x), \varphi_2(x)) = w_1(x).$$

That is,

$$\lambda_{2i}(x) = \lambda_{1i}(x) < \frac{w_1(x) + w_2(x)}{2} = k(x).$$

Thus, $\dot{V}_i(x) < 0$ since $a_i(x) < 0$.

**Case 3.** If $\Delta_i(x) > 0$. In this case, we consider the three following cases according to the sign of $b_i(x)$.

i. If $b_i(x) < 0$, we consider again three cases according to the sign of $a_i(x)$.

o If $a_i(x) > 0$. On the one hand, since $k_i$ stabilizes System $S_i$, we have $\lambda_{1i}(x) < k_i(x) = \psi_{1i}(x) \le \psi_1(x)$. By Assumption 3.4, we get $\lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) < \varphi_2(x)$. So $\lambda_{1i}(x) < \min(\psi_1(x), \varphi_2(x)) = w_1(x)$. Together with the fact that $w_1(x) \le \varphi_2(x) \le \varphi_{2i}(x) = \lambda_{2i}(x)$, it then follows that

$$\lambda_{1i}(x) < w_1(x) \le \lambda_{2i}(x). \quad (3.21)$$

On the other hand, note that in this case $0 < \lambda_{2i}(x)$, hence $\psi_2(x) \le \psi_{2i}(x) = 0 < \lambda_{2i}(x)$. As above, by Assumption 3.4, we get $\varphi_1(x) < \varphi_2(x) \le \varphi_{2i}(x) = \lambda_{2i}(x)$. So $w_2(x) = \max(\psi_2(x), \varphi_1(x)) < \lambda_{2i}(x)$, and since $\lambda_{1i}(x) = \varphi_{1i}(x) \le$
Using notations of the Section 3, we have

$$
\varphi_1(x) = \begin{cases} \lambda_{11}(x), & \text{if } x_1 \neq 0, \\ -\infty, & \text{if } x_1 = 0, \end{cases}
$$
$$
\varphi_2(x) = \begin{cases} \min(\lambda_{21}(x), \lambda_{22}(x)), & \text{if } x_1^4 x_2^2 - 4 \ge 0 \text{ or } x_1 \neq 0 \text{ and } x_2 = 0, \\ \lambda_{21}(x), & \text{if } x_1^4 x_2^2 - 4 < 0 \text{ and } x_1 \neq 0 \text{ and } x_2 \neq 0, \\ +\infty, & \text{if } x_1 = 0, \end{cases}
$$

$$
\psi_1(x) = 0, \quad \forall x \in \mathbb{R}^2,
$$

$$
\psi_2(x) = \begin{cases}
\min(\lambda_{21}(x) - \frac{\sqrt{\Delta_1(x)}}{2(1+a_1^2(x))}, \lambda_{22}(x) - \frac{\sqrt{\Delta_2(x)}}{2(1+a_2^2(x))}, 0), & \text{if } x_1 \neq 0, x_2 \neq 0, \\
\min(\lambda_{22}(x) - \frac{\sqrt{\Delta_2(x)}}{2(1+a_2^2(x))}, 0), & \text{if } x_1 \neq 0, x_2 = 0, \\
0, & \text{if } x_1 = 0.
\end{cases}
$$

It is not difficult to verify that Systems $S_1$ and $S_2$ satisfy Assumptions 2.1, 3.2.
To verify Assumption 3.1, define $S = \{x \in \mathbb{R}^2 | x_1 = 0 \text{ and } x_2 = 0, \text{ or } x_1^4 x_2^2 = 4\}$.

• For $x \notin S$, $A_3(x) = \emptyset$ and then Assumption 3.1 holds.

• For $x \in S$, we have $A_1(x) = \{1\}$, $A_3(x) = \{2\}$, $\psi_1(x) = 0$ and $-\frac{b_2(x)}{2a_2(x)} = \frac{1}{2}$. Hence,
$$
\min(\psi_1(x), \varphi_{21}(x)) \le \psi_1(x) < -\frac{b_2(x)}{2a_2(x)}, \text{ that is Assumption 3.1 holds.}
$$

It remains to verify Assumption 3.3. To this end define $D_1 = \{x \in \mathbb{R}^2 | x_1 = 0 \text{ or, } x_1^4 x_2^2 < 4 \text{ and } x_1 x_2 \neq 0\}$ and $D_2 = \{x \in \mathbb{R}^2 | x_1 \neq 0 \text{ and } x_2 = 0 \text{ or, } x_1^4 x_2^2 \geq 4\}$. Clearly, $\mathbb{R}^2 = D_1 \cup D_2$.

• For $x \in D_1$, from definitions of $\varphi_1$ and $\varphi_2$, we have $\varphi_1(x) < \varphi_2(x)$.

• For $x \in D_2$, we have $\varphi_1(x) = \lambda_{11}(x)$ and $\varphi_2(x) = \min(\lambda_{21}(x), \lambda_{22}(x))$. Notice that
$\lambda_{11}(x) < \lambda_{21}(x)$ for all $x \in D_2$. Notice also that since $-\sqrt{(1+x_1^2)(x_1^6x_2^4+4x_1^2x_2^2+4x_1^4)} \le$
$-\sqrt{x_1^2x_2^2(x_1^4x_2^2-4)}$ for all $x \in D_2$, it follows that $\lambda_{11}(x) < \lambda_{22}(x)$ for all $x \in D_2$. Thus
$\varphi_1(x) < \varphi_2(x)$ for all $x \in D_2$.

In both cases, $\varphi_1(x) < \varphi_2(x)$ and thus Assumption 3.3 holds.
Therefore, by Theorem 3.2, the feedback

$$
u(x) = \frac{w_1(x) + w_2(x)}{2} \tag{4.3}
$$

is continuous on $\mathbb{R}^2$ and globally asymptotically stabilizes the Systems $S_1$ and $S_2$ simultaneously, with $w_1(x) = \min(\psi_1(x), \varphi_2(x))$ and $w_2(x) = \max(\psi_2(x), \varphi_1(x))$.

The behaviors of the closed-loop systems (4.1) and (4.2) under the action of the simultaneously stabilizing feedback are drawn in Figures 1-2. The evolution of the stabilizing feedback along the closed-loop systems (4.1) and (4.2) is shown in Figures 3-4.
$\varphi_1(x) \le w_2(x)$, it then follows that

$$ \lambda_{1i}(x) \le w_2(x) < \lambda_{2i}(x). \qquad (3.22) $$

(3.21) and (3.22) give

$$ \lambda_{1i}(x) < \frac{w_1(x) + w_2(x)}{2} = k(x) < \lambda_{2i}(x). $$

Thus, $\dot{V}_i(x) < 0$ since $a_i(x) > 0$.

o If $a_i(x) = 0$. Then $-\frac{c_i(x)}{b_i(x)} < k_i(x) = \psi_{1i}(x) \le \psi_1(x)$. Assumption 3.4, yields $-\frac{c_i(x)}{b_i(x)} = \varphi_{1i}(x) \le \varphi_1(x) < \varphi_2(x)$. So $-\frac{c_i(x)}{b_i(x)} < w_1(x)$ and $-\frac{c_i(x)}{b_i(x)} \le w_2(x)$ since $\varphi_1(x) \le w_2(x)$. It then follows that

$$ -\frac{c_i(x)}{b_i(x)} < \frac{w_1(x) + w_2(x)}{2} = k(x). $$

Thus, $\dot{V}_i(x) = b_i(x)k(x) + c_i(x) < 0$ since $b_i(x) < 0$.

o If $a_i(x) < 0$. As above, $\lambda_{1i}(x) < k_i(x) = \psi_{1i}(x) \le \psi_1(x)$. Assumption 3.4 yields $\lambda_{1i}(x) = \varphi_{1i}(x) \le \varphi_1(x) < \varphi_2(x)$. Hence $\lambda_{1i}(x) < w_1(x)$ and $\lambda_{1i}(x) \le \varphi_1(x) \le w_2(x)$. So

$$ \lambda_{2i}(x) < \lambda_{1i}(x) < \frac{w_1(x) + w_2(x)}{2} = k(x). $$

That is, $\dot{V}_i(x) < 0$ since $a_i(x) < 0$.

ii. If $b_i(x) > 0$, this case can be treated similarly to the previous one.

iii. If $b_i(x) = 0$, by Assumption 2.1 and since $\Delta_i(x) > 0$, we have $a_i(x) > 0$, so, this case is similar to the case $a_i(x) > 0$ in Case 3, item i.

Thus, the proof of Theorem 3.5 is completed. □

**Remark 3.5.** 1. Notice that if the collection of CLFs in Theorem 3.5 verify $a_i(x) \ge 0$ for all $x \in \mathbb{R}^n$ and $i \in I$, then, by Remark 3.2, the feedback (3.18) simultaneously stabilizes all the systems in (2.1) if only Assumption 3.4 holds.

2. Notice also that if $V_i$ is a CLF for the system $S_i$ such that $a_i(x) < 0$ for all $x \in \mathbb{R}^n \setminus \{0\}$, then, according to Theorem 3.4 in Maniar et al. (2017), the system $S_i$ is almost stabilizable with the feedback, ($\varepsilon > 0$).

$$ u_i(x) = \begin{cases} \frac{-b_i(x) - \sqrt{|\Delta_i(x)|}}{2a_i(x)} + \varepsilon & \text{if } x \neq 0, \\ 0 & \text{if } x = 0. \end{cases} $$

So, if $V_i$ is a collection of CLFs $V_i$ for systems $S_i$ satisfying $a_i(x) < 0$ for all $x \in \mathbb{R}^n \setminus \{0\}$ and $i \in I$, then the collection of systems in (2.1) is simultaneously almost stabilizable with the feedback

$$ u(x) = \max_{i \in I} u_i(x). $$
<figure><img src="image_2.png" /><figcaption>Figure 1: The responses of the closed-loop system $S_1$ and (4.3) (initial state $x(0) = (3, -3)$).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: The responses of the closed-loop system $S_2$ and (4.3) (initial state $x(0) = (3, -3)$).</figcaption></figure>
## 4. Illustrative example

In this section, we present an example to illustrate the effectiveness of the results obtained in this paper to design a simultaneously stabilizing feedback for some nonlinear quadratic systems.

Consider a system with the following two possible modes

$$ S_1: \begin{cases} \dot{x}_1 = -x_1 + \frac{x_1 x_2^4}{4} + \frac{x_1}{1+x_1^2} u^2 \\ \dot{x}_2 = -x_2 + x_1^2 x_2 u, \end{cases} \qquad (4.1) $$

$$ S_2: \begin{cases} \dot{x}_1 = -x_1 + x_1^3 x_2^4 + (4x_1 - 4x_2)u - 4x_1 u^2 \\ \dot{x}_2 = -x_2 + x_1 u. \end{cases} \qquad (4.2) $$

Let

$$ V_1(x) = \frac{x_1^2 + x_2^2}{2} \quad \text{and} \quad V_2(x) = \frac{x_1^2 + 4x_2^2}{8}. $$

The Lie derivative of $V_1$ and $V_2$ along the trajectories of Systems $S_1$ and $S_2$, respectively, write:

$$ \dot{V}_1(x) = a_1(x)u^2 + b_1(x)u + c_1(x), $$

$$ \dot{V}_2(x) = a_2(x)u^2 + b_2(x)u + c_2(x), $$

with $a_1(x) = x_1^2/(1+x_1^2)$, $b_1(x) = x_1^2 x_2^2$, $c_1(x) = -x_1^2 - x_2^2 + (1/4)x_1^2 x_2^4$, $a_2(x) = -x_1^2$, $b_2(x) = x_1^2$, and $c_2(x) = (1/4)(-x_1^2 + x_1^4 x_2^4) - x_2^2$.

By the definitions in Section 2, we have

$$ \Delta_1(x) = \frac{x_1^6 x_2^4 + 4x_1^2 x_2^2 + 4x_1^4}{1 + x_1^2}, \quad \Delta_2(x) = x_1^2 x_2^2 (x_1^4 x_2^2 - 4), $$

$$ \lambda_{11}(x) = \frac{-x_1^2 x_2^2 - x_2^2}{2} - \frac{\sqrt{(1+x_1^2)(x_1^6 x_2^4 + 4x_1^2 x_2^2 + 4x_1^4)}}{2x_1^2}, $$

$$ \lambda_{21}(x) = \frac{-x_1^2 x_2^2 - x_2^2}{2} + \frac{\sqrt{(1+x_1^2)(x_1^6 x_2^4 + 4x_1^2 x_2^2 + 4x_1^4)}}{2x_1^2}, $$

$$ \lambda_{12}(x) = \frac{1}{2} + \frac{\sqrt{x_1^2 x_2^2 (x_1^4 x_2^2 - 4)}}{2x_1^2}, $$
<figure><img src="image_4.png" /><figcaption>Figure 3: The evolution of the stabilizing control along the solutions of the closed-loop system $S_1$ and (4.3) (initial state $x(0) = (3, -3)$).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: The evolution of the stabilizing control along the solutions of the closed-loop system $S_2$ and (4.3) (initial state $x(0) = (3, -3)$).</figcaption></figure>
$$ \lambda_{22}(x) = \frac{1}{2} - \frac{\sqrt{x_1^2 x_2^2 (x_1^4 x_2^2 - 4)}}{2x_1^2}. $$

We can prove that $V_1$ and $V_2$ are CLFs for Systems $S_1$ and $S_2$, respectively, and satisfy the small control property. In fact, $V_2$ satisfies the small control property since $\Delta_2(x) < 0$ whenever $x \neq 0$ is close enough to 0 and then $a_2(x) < 0$ if $x_1 \neq 0$ or $c_2(x) < 0$ if $a_2(x) = b_2(x) = 0$.

Using notations of the Section 3, we have

$$ \varphi_1(x) = \begin{cases} \lambda_{11}(x), & \text{if } x_1 \neq 0, \\ -\infty, & \text{if } x_1 = 0, \end{cases} $$ 
$$ \varphi_2(x) = \begin{cases} \min(\lambda_{21}(x), \lambda_{22}(x)), & \text{if } x_1^4 x_2^2 - 4 \ge 0 \text{ or } x_1 \neq 0 \text{ and } x_2 = 0, \\ \lambda_{21}(x), & \text{if } x_1^4 x_2^2 - 4 < 0 \text{ and } x_1 \neq 0 \text{ and } x_2 \neq 0, \\ +\infty, & \text{if } x_1 = 0, \end{cases} $$

$$ \psi_1(x) = 0, \quad \forall x \in \mathbb{R}^2, $$

$$ \psi_2(x) = \begin{cases} \min(\lambda_{21}(x) - \frac{\sqrt{\Delta_1(x)}}{2(1+a_1^2(x))}, \lambda_{22}(x) - \frac{\sqrt{\Delta_2(x)}}{2(1+a_2^2(x))}, 0), & \text{if } x_1 \neq 0, x_2 \neq 0, \\ \min(\lambda_{22}(x) - \frac{\sqrt{\Delta_2(x)}}{2(1+a_2^2(x))}, 0), & \text{if } x_1 \neq 0, x_2 = 0, \\ 0, & \text{if } x_1 = 0. \end{cases} $$

It is not difficult to verify that Systems $S_1$ and $S_2$ satisfy Assumptions 2.1, 3.2.

To verify Assumption 3.1, define $S = \{x \in \mathbb{R}^2 | x_1 = 0 \text{ and } x_2 = 0, \text{ or } x_1^4 x_2^2 = 4\}$.

* For $x \notin S$, $A_3(x) = \emptyset$ and then Assumption 3.1 holds.

* For $x \in S$, we have $A_1(x) = \{1\}$, $A_3(x) = \{2\}$, $\psi_1(x) = 0$ and $-\frac{b_2(x)}{2a_2(x)} = \frac{1}{2}$.

Hence, $\min(\psi_1(x), \varphi_{21}(x)) \le \psi_1(x) < -\frac{b_2(x)}{2a_2(x)}$, that is Assumption 3.1 holds.

It remains to verify Assumption 3.4. To this end define $D_1 = \{x \in \mathbb{R}^2 | x_1 = 0 \text{ or, } x_1^4 x_2^2 < 4 \text{ and } x_1 x_2 \neq 0\}$ and $D_2 = \{x \in \mathbb{R}^2 | x_1 \neq 0 \text{ and } x_2 = 0 \text{ or, } x_1^4 x_2^2 \ge 4\}$. Clearly, $\mathbb{R}^2 = D_1 \cup D_2$.

* For $x \in D_1$, from definitions of $\varphi_1$ and $\varphi_2$, we have $\varphi_1(x) < \varphi_2(x)$.

* For $x \in D_2$, we have $\varphi_1(x) = \lambda_{11}(x)$ and $\varphi_2(x) = \min(\lambda_{21}(x), \lambda_{22}(x))$. Notice that $\lambda_{11}(x) < \lambda_{21}(x)$ for all $x \in D_2$. Notice also that since $-\sqrt{(1+x_1^2)(x_1^6 x_2^4 + 4x_1^2 x_2^2 + 4x_1^4)} \le -\sqrt{x_1^2 x_2^2 (x_1^4 x_2^2 - 4)}$ for all $x \in D_2$, it follows that $\lambda_{11}(x) < \lambda_{22}(x)$ for all $x \in D_2$. Thus $\varphi_1(x) < \varphi_2(x)$ for all $x \in D_2$.

In both cases, $\varphi_1(x) < \varphi_2(x)$ and thus Assumption 3.4 holds.

Therefore, by Theorem 3.5, the feedback

$$ u(x) = \frac{w_1(x) + w_2(x)}{2} \qquad (4.3) $$
References

[1] Z. Artstein. Stabilization with relaxed controls. *Nonlinear Analysis: Theory, Methods & Applications*, 7(11):1163–1173, 1983.

[2] L. C. Cai, *Simultaneous stabilization of Port-Hamiltonian systems subject to actuation saturation and input delay*, Int. J. Autom. Comput., (2015). https://doi.org/10.1007/s11633-015-0928-4.

[3] B. Chaudhuri, P. Korba and B. C. Pal, *Simultaneous stabilization technique for damping controller design in power systems*, Intelligent Automation & Soft Computing, **12** (2006), 41–49.

[4] Y. Chen, T. Xu, C. Zeng, Z. Zhou and Q. Zhang, *Simultaneous stabilization for uncertain descriptor systems with input saturation*, International Journal of Robust and Nonlinear Control, **22** (2012), 1938–1951.

[5] A. Dastaviz and T. Binazadeh. Simultaneous stabilization of a collection of uncertain discrete-time systems with time-varying state-delay via discrete-time sliding mode control. *Journal of Vibration and Control*, 25(16):2261–2273, jun 2019.

[6] Dastaviz, A., and T. Binazadeh, *Simultaneous stabilization for a collection of uncertain time-delay systems using sliding-mode output feedback control*, International Journal of Control, **93** (2020), 2135–2144.

[7] A. Dastaviz, T. Binazadeh, and Y. Cui. A novel control lyapunov-krasovskii functional methodology for simultaneous stabilization of a set of multiple time-delays nonlinear systems. *Journal of the Franklin Institute*, 358(12):6101–6120, aug 2021.

[8] R. Galindo, *Simultaneous stabilization of full state information Linear Time-Invariant systems*, Asian Journal of Control **14** (2012), 1382–1386.

[9] Q. Guan, G. He, L. Wang and W. Yu, *Simultaneous stabilization of linear systems*, Control Theory and Applications, **28** (2011), 58–73.

[10] A. Iggidr and M. Oumoun, "Simultaneous Stabilization of Single-Input Nonlinear Systems With Bounded Controls," *International Journal of Dynamics and Control*, (2020). https://doi.org/10.1007/s40435-020-00658-3

[11] A. Iggidr and M. Oumoun, "Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs," *International Conference on Control, Automation and Diagnosis (ICCAD)*, Grenoble, France, (2019). https://doi.org/10.1109/ICCAD46983.2019.9037892

[12] A. Iggidr, M. Oumoun and M. Souza "Continuous simultaneous stabilization of single-input nonlinear stochastic systems," *Asian Journal of Control*, (2020). https://doi.org/10.1002/asjc.2401

[13] P. Kohan-Sedgh, A. Khayatian and M. H. Asemani, *Conservatism reduction in simultaneous output feedback stabilisation of linear systems*, IET Control Theory & Applications **10** (2016), 2243–2250.

[14] W. Li, L. Wang and W. Yu, *Some Open Problems on Simultaneous Stabilization of Linear Systems*, Journal of Systems Science and Complexity **29** (2016), 289–299.
is continuous on $\mathbb{R}^2$ and globally asymptotically stabilizes the Systems $S_1$ and $S_2$ simultaneously, with $w_1(x) = \min(\psi_1(x), \varphi_2(x))$ and $w_2(x) = \max(\psi_2(x), \varphi_1(x))$.

The behaviors of the closed-loop systems (4.1) and (4.2) under the action of the simultaneously stabilizing feedback are drawn in Figures 1-2. The evolution of the stabilizing feedback along the closed-loop systems (4.1) and (4.2) is shown in Figures 3-4.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** The responses of the closed-loop system $S_1$ and (4.3) (initial state $x(0) = (3, -3)$).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 2.** The responses of the closed-loop system $S_2$ and (4.3) (initial state $x(0) = (3, -3)$).</figcaption></figure>
[15] L. Maniar, M. Oumoun and J.C. Vivalda, “On the stabilization of quadratic nonlinear systems”, *European Journal of Control*, vol. 35, pp. 28–33, 2017.

[16] D. E. Miller and T. Chen, *Simultaneous stabilization with near-optimal $H^\infty$ performance*, IEEE Trans. Autom. Control, **47** (2002), 1986–1998.

[17] E. D. Sontag. A “universal” construction of Artstein’s theorem on nonlinear *Systems Control Lett.*, 13(2):117–123, 1989.

[18] Y. Z. Wang, G. Feng, and D. Z. Cheng, *Simultaneous stabilization of a set of nonlinear port-controlled Hamiltonian systems*, Automatica **43** (2007), 403–415.

[19] A. Wei and Y. Wang, *Adaptive parallel simultaneous stabilization of a set of uncertain port-controlled hamiltonian systems subject to actuator saturation*, International Journal of Adaptive Control and Signal Processing, **28** (2014), 1128–1144.

[20] J. L. Wu, “Simultaneous quadratic stabilization for discrete-time nonlinear systems”, *IEEE Trans. Autom. Control*, vol. 55, pp. 1443–1448, 2010.

[21] J. L. Wu, “Simultaneous stabilization for a collection of single-input nonlinear systems”, *IEEE Trans. Autom. Control*, vol. 50, pp. 328–337, 2005.

[22] T. Yu and W. Chi, (2014). *Sufficient conditions for simultaneous stabilization of three linear systems within the framework of nest algebras*, Journal of the Franklin Institute **351** (2014), 5310–5325.

[23] J. Zhong, D. Cheng, X. Hu, *Constructive stabilization for quadratic input nonlinear systems*, Automatica 44 (2008) 1996–2005.

[24] W. Zhang, Z. Z. Han and X. S. Cai, “Simultaneous stabilization of a collection of single-input nonlinear systems based on CLFs”, *Asian Journal of Control*, vol. 13, pp. 582–589, 2011.

[25] Y. Zhu and F. Yang, *Simultaneous $H_2/H_\infty$ stabilization for chemical reaction systems based on orthogonal complement space*, International Journal of Automation and Computing, **13** (2016), 19–30.

[26] Y. Zhu and F. Yang, *Network-based simultaneous $H_\infty$ stabilization for chemical reaction systems with multiple packet dropouts*, International Journal of Control, Automation and Systems, **15** (2017), 104–112.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** The evolution of the stabilizing control along the solutions of the closed-loop system $S_1$ and (4.3) (initial state $x(0) = (3, -3)$).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4.** The evolution of the stabilizing control along the solutions of the closed-loop system $S_2$ and (4.3) (initial state $x(0) = (3, -3)$).</figcaption></figure>
## 5. Simultaneous stabilization by discontinuous controller: The magnetic ball levitation systems

**Remark 5.1.** In the following, using the sliding mode (SM) technique, we will show that to achieve the simultaneous stabilization, by a discontinuous feedback, of two levitation systems is reduced to solving a quadratic inequalities of the form (2.4). So, the method and notations are similar to those in Section 3.

Also, note that since the feedback is discontinuous, we will focus on the negativity of quadratic inequalities, that is we only use Assumptions 2.1 and 3.4.

Consider two ferromagnetic balls suspended in a voltage-controlled magnetic field. Only the vertical motion is considered. The goal is to keep each ball in levitation at a desired position with respect to the ground.

The two levitation systems, $\Sigma_1$ and $\Sigma_2$ modeling, respectively, the motion of the ball 1 and ball 2, are described by the following equations (see (Bethoux et al., 2003; Moulay & Perruquetti, 2005) for more details):

$$ \Sigma_i: \left\{ \begin{array}{l} \dot{e}_1 = e_2 \\ \dot{e}_2 = g - \frac{k_i}{(l_i + e_1 + x_{1\text{refi}})^2} u^2. \end{array} \right. \qquad (5.1) $$

where, for each system $\Sigma_i$, $i = 1, 2$, $e = (e_1, e_2) \in \mathbb{R}^2$. $e_1$ denotes the distance between $x_1$ the vertical position of the ball $i$ with respect to the ground and $x_{1\text{refi}}$ (a constant desired position), $e_2$ its speed, $g = 9.8$ the acceleration of gravity, $u$ the coil current is the control and, $k_i$ and $l_i$ are some positive parameters depending on the electromagnet, the mass and the permeability of the free space (Bethoux et al., 2003).

The objective is to find a common control law $u$ which stabilizes each ball $i$ at a desired position $x_{1\text{refi}}$. Toward this end, and in view of the sliding mode (SM) technique, we first design two appropriate sliding surfaces $\sigma_1$ and $\sigma_2$ on which, respectively, systems $\Sigma_1$ and $\Sigma_2$ are asymptotically stable. Second, we deal with the design of a single feedback law by explicit formulas in such a way that the so-called $\eta$-reachability condition $\dot{\sigma}_i\sigma_i < -\eta|\sigma_i|$ is satisfied, with $\eta > 0$.

Notice that, for each system $\Sigma_i$, $i = 1, 2$, the $\eta$-reachability condition guarantees that all trajectories are directed toward the switching surface $\sigma_i = 0$ in a finite time, and are maintained on this surface once it has been reached. For more details on this subject, we refer the reader to (Khalil, 2002; Perruquetti & Barbot, 2002).

Now, let us define the sliding surfaces as: $\sigma_1 = \sigma_2 = \beta e_1 + e_2$, where $\beta$ is a positive scalar.

Note that the choice of the switching surfaces guarantees that $e = (e_1, e_2)$ converges to 0 as $t \to \infty$ when we have sliding (i.e., $\sigma_i = 0$), since on this manifold, the motion is governed by $\dot{e}_1 = -\beta e_1$. Note also that $\sigma_1$ and $\sigma_2$ have the same sign.

Differentiating $\sigma_1$ and $\sigma_2$ with respect to time along the trajectories, respectively,
of $\Sigma_1$ and $\Sigma_2$ leads to, for each $i = 1, 2$

$$ \dot{\sigma}_i = \beta \dot{e}_1 + \dot{e}_2 = - \frac{k_i}{(l_i + e_1 + x_{1\text{refi}})^2} u^2 + \beta e_2 + g. \quad (5.2) $$

Take $\eta > 0$, from (5.2) we have

$$ \dot{\sigma}_i < - \frac{k_i}{(l_i + e_1 + x_{1\text{refi}})^2} u^2 + \beta e_2 + g + \eta. \quad (5.3) $$

and

$$ -\dot{\sigma}_i < \frac{k_i}{(l_i + e_1 + x_{1\text{refi}})^2} u^2 - \beta e_2 - g + \eta. \quad (5.4) $$

Now, to construct a control $u$ that constrain both systems $\Sigma_1$ and $\Sigma_2$ to verify the $\eta$-reachability $\dot{\sigma}_i \sigma_i < -\eta |\sigma_i|$, one sees that it is sufficient that the right-hand side of (5.3) and (5.4), viewed as a polynomials in $u$, have negative values. For this, consider two cases for each system $\Sigma_i$, $i = 1, 2$, according to the sign of $\sigma_i$.

* If $\sigma_i > 0$, $i = 1, 2$. Consider the right-hand side of (5.3). These polynomials are of the form $a_i(e)u^2 + c_i(e)$ with

$$ a_i(e) = - \frac{k_i}{(l_i + e_1 + x_{1\text{refi}})^2}, \quad c_i(e) = \beta e_2 + g + \eta, $$

and $\Delta_{1i}(e) = -4a_i(e)c_i(e)$, $i = 1, 2$, their discriminants which are always positive in a neighborhood of the origin.

Since $a_i(e) < 0$, according to second item in Remark 3.5, the control

$$ u_p(e) = \max_{i=1,2} - \frac{\sqrt{\Delta_{1i}(e)}}{2a_i(e)} + \varepsilon, \quad (\varepsilon > 0), $$

renders both polynomials, $i = 1, 2$, in (5.3) negative, that is $\dot{\sigma}_i < -\eta$ and so, $\dot{\sigma}_i \sigma_i < -\eta |\sigma_i|$ as desired.

* If $\sigma_i < 0$, $i = 1, 2$. As above, consider polynomials in the right-hand side of (5.4) which are of the form $a'_i(e)u^2 + c'_i(e)$ with

$$ a'_i(e) = \frac{k_i}{(l_i + e_1 + x_{1\text{refi}})^2} \quad \text{and} \quad c'_i(e) = -\beta e_2 - g + \eta. $$

These polynomials have negative values if their discriminants $\Delta_{2i}(e) = -4a'_i(e)c'_i(e)$ are positive, that is, $-\beta e_2 < g - \eta$, which occurs in a neighborhood of the origin if $g - \eta > 0$. It then follows that, in this situation, one can process as in Section 3 to construct $u_n$ such that $a'_i(e)u_n^2(e) + c'_i(e) < 0$.

In this case, the two roots of each polynomial $a'_i(e)u^2 + c'_i(e)$, $i = 1, 2$ are

$$ \lambda_{1i}(e) = - \frac{\sqrt{\Delta_{2i}(e)}}{2a'_i(e)} \quad \text{and} \quad \lambda_{2i}(e) = \frac{\sqrt{\Delta_{2i}(e)}}{2a'_i(e)}. $$
It is easy to verify that ( notations below are those in Section 3)

$$ \varphi_{1i}(e) = \lambda_{1i}(e), \quad \varphi_{2i}(e) = \lambda_{2i}(e) \quad \text{and} \quad \psi_{1i}(e) = \psi_{2i}(e) = 0. $$

Thus

$$ \varphi_1(e) = \max(\lambda_{11}(e), \lambda_{12}(e)), \quad \varphi_2(e) = \min(\lambda_{21}(e), \lambda_{22}(e)) \quad \text{and} \quad \psi_1(e) = \psi_2(e) = 0. $$

One sees that $\varphi_1(e) < 0 < \varphi_2(e)$ that is, Assumption 3.4 holds. It is obvious that Assumption 2.1 holds too since $a'_i(e) > 0$. One sees also that

$$ w_1(x) = \min(\psi_1(x), \varphi_2(x)) = 0 \quad \text{and} \quad w_2(x) = \max(\psi_2(x), \varphi_1(x)) = 0. $$

Consequently, the control

$$ u_n(e) = \frac{w_1(e) + w_2(e)}{2} = 0 $$

represents both polynomials, $i = 1, 2$, in (5.4) negative, that is $-\dot{\sigma}_i < -\eta$ which leads to $\dot{\sigma}_i\sigma_i < -\eta|\sigma_i|$.

Finally, the discontinuous control

$$ u(e) = \begin{cases} 0 & \text{if } \sigma_i < 0 \\ \max_{i=1,2} -\frac{\sqrt{\Delta_{1i}(e)}}{2a_i(e)} + \varepsilon & \text{if } \sigma_i \ge 0, \end{cases} \quad (5.5) $$

guarantees the $\eta$-reachability condition and, from the discussion above, simultaneously stabilizes $\Sigma_1$ and $\Sigma_2$.

## Simulations:

$\Sigma_1$: $l_1 = 0.01, k_1 = 1, x_{ref1} = 5, \beta = 1, \eta = 0.01, \varepsilon = 0.0001$.

$\Sigma_2$: $l_2 = 0.03, k_2 = 1.5, x_{ref2} = 4, \beta = 1, \eta = 0.01, \varepsilon = 0.0001$.

## References

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Cai, L. C. (2015). Simultaneous stabilization of Port-Hamiltonian systems subject to actuation saturation and input delay, *Int. J. Autom. Comput.*, https://doi.org/10.1007/s11633-015-0928-4.

Chaudhuri, B., Korba, P., & Pal, B. C. (2006). Simultaneous stabilization technique for damping controller design in power systems, *Intelligent Automation & Soft Computing*, 12, 41-49.

Chen, Y., Xu, T., Zeng, C., Zhou, Z., & Zhang, Q. (2012). Simultaneous stabilization for uncertain descriptor systems with input saturation, *International Journal of Robust and Nonlinear Control*, 22, 1938-1951.

Dastaviz, A. & Binazadeh, T. (2019). Simultaneous stabilization of a collection of uncertain discrete-time systems with time-varying state-delay via discrete-time sliding mode control. *Journal of Vibration and Control*, 25.
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Evolution of the state $e_1(t)$ of the system $\Sigma_1$ and (5.5) (initial state $e(0) = (0.1, 0.05)$).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Evolution of the state $e_2(t)$ of the system $\Sigma_1$ and (5.5) (initial state $e(0) = (0.1, 0.05)$).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 7.** Evolution of the state $e_1(t)$ of the system $\Sigma_2$ and (5.5) (initial state $e(0) = (0.1, 0.05)$).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 8.** Evolution of the state $e_2(t)$ of the system $\Sigma_2$ and (5.5) (initial state $e(0) = (0.1, 0.05)$).</figcaption></figure>
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