# On homogeneity with respect to a part of variables and accelerated regulation

Denis Efimov, Ilya Kolmanovsky

*Abstract*—The paper addresses the problem of transforming a locally asymptotically stabilizing time-varying control law to a global one with accelerated finite/fixed-time convergence rates. The proposed approach relies on an extension of the theory of homogeneous systems to homogeneity only with respect to a part of the state variables and on the associated partial stability properties. The proposed control design builds upon the kind of approaches first studied in [1] and uses the implicit Lyapunov function framework. A sampled implementation scheme of the control law is also presented and its properties are characterized. The proposed approach is illustrated by finite-time and nearly fixed-time stabilization of a nonholonomic integrator.

## I. INTRODUCTION

In many real world applications the rates of convergence of estimation or regulation errors to zero represent principal performance characteristics. Beyond the simplest asymptotic convergence, rated exponential convergence, or more advanced finite-time or fixed-time convergence (in the latter cases the transients are finished after a finite time interval), are often desired. Different approaches can be used to guarantee these properties [2]–[7]. Another important issue in nonlinear control and estimation deals with the domain of convergence for the errors, where the local characteristics are much easier to obtain (using, for example, linear approximation for control law design), but the global ones are more desirable.

An appealing class of nonlinear systems, whose local behavior is the same as global one (e.g., local attractiveness implies global asymptotic stability), is composed of systems with homogeneous dynamics, see [7]–[9] and references therein. The theory of homogeneous systems has a rich history with contributions by J. M. Coron, W. P. Dayawansa, L. Grüne, H. Hermes, M. Kawski, W. Lin, L. Rosier, V. Zubov and others (here we mention only the most relevant titles to the present work). Homogenization of the closed-loop systems by a proper dilation of the control laws, which allows (local) asymptotic stabilizing controls to be transformed to ones with a rated convergence, was studied in [1] for the nonlinear time-varying systems (where asymptotic non-exponential stabilizers were converted to exponential

stabilizers) and in [10]–[14] for linear autonomous models (where finite/fixed-time convergence has been obtained). In [1], [11]–[14] the implicitly defined Lyapunov functions (see also [15], [16]) have been used for stability analysis.

In this work, the method of [1] (see also [17], [18]) for exponential stabilization of drift-free nonlinear systems via a time-varying feedback is extended to finite-time or fixed-time stabilization, thereby achieving accelerated convergence. Large classes of nonholonomic systems emerging in applications can be represented by such models; they are not smoothly or even continuously stabilizable due to the violation of Brockett's necessary condition. To this end, the theory of homogeneous systems is generalized to the case of partial symmetry, when only a part of the state is dilated, and the variables representing generator dynamics (time dependence) are not scaled (see also the extension of the theory of homogeneous systems to time-varying models in [19], [20]). In such a case stability and convergence only with respect to a part of the state variables are considered in the sense of uniform output stability from [21], [22] or [23], [24].

The paper is organized as follows. The brief preliminaries are given in Section II. The control problem statement is introduced in Section III. The properties of partially homogeneous and stable systems are investigated in Section IV. The control design is considered in Section V. The approach for digital implementation of the proposed control algorithms is described and investigated in Section VI. The results of numeric experiments are shown in Section VII.

Preliminary results have been included into our conference work [25]. As compared to [25], this paper contains additional details and proofs. In particular, the analysis of partial homogeneity is extended with theorems 3 and 4, while the case of semi-definite derivative of Lyapunov function is investigated in Theorem 6. Furthermore, the developments in Section VI related to discrete-time implementation and the theoretical analysis of the corresponding closed-loop properties are not covered in [25], but addressed in the present paper; these developments are especially important as discrete-time implementation is required for practical applications, brings about significant challenges, and yet often not solved in finite-time stabilization literature. Accordingly, the simulation example in Section VII has been also enhanced.

Denis Efimov is with Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France.

Ilya Kolmanovsky is with Department of Aerospace Engineering, the University of Michigan, US.

The second author would like to acknowledge the support of the National Science Foundation under award number ECCS-1931738.
### Notation

* $\mathbb{R}_+ = \{x \in \mathbb{R} : x \ge 0\}$, where $\mathbb{R}$ is the set of real numbers.

* $|\cdot|$ denotes the absolute value in $\mathbb{R}$, $\|\cdot\|$ is used for the Euclidean norm on $\mathbb{R}^n$.

* For a (Lebesgue) measurable function $d: \mathbb{R}_+ \to \mathbb{R}^s$ and $[t_0, t_1) \subset \mathbb{R}_+$ define the norm $\|d\|_{[t_0, t_1)} = \sup_{t \in [t_0, t_1)} \|d(t)\|$, then $\|d\|_{\infty} = \|d\|_{[0, +\infty)}$ and the set of $d$ with the property $\|d\|_{\infty} < +\infty$ we further denote as $\mathcal{L}_{\infty}^s$ (the set of essentially bounded measurable functions).

* A continuous function $\alpha: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}$ if $\alpha(0) = 0$ and it is strictly increasing. The function $\alpha: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}_{\infty}$ if $\alpha \in \mathcal{K}$ and it is increasing to infinity. A continuous function $\beta: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{KL}$ if $\beta(\cdot, t) \in \mathcal{K}$ for each fixed $t \in \mathbb{R}_+$ and $\beta(s, \cdot)$ is decreasing to zero for each fixed $s \in \mathbb{R}_+$.

* For a set $\mathcal{A} \subset \mathbb{R}^n$, denote its boundary and interior by $\partial \mathcal{A}$ and $\text{int}(\mathcal{A})$, respectively.

* A finite series of integers $1, 2, \dots, n$ is denoted by $\overline{1, n}$, and $\{\overline{1, n}\} = \{1, 2, \dots, n\}$.

### Abbreviations

* uOS: uniformly output stable

* SIuOS: state independent uniformly output stable

* UBIBS: uniformly bounded input bounded state stable

* uFTOS: uniformly finite-time output stable

* unFxTOS: uniformly nearly fixed-time output stable

* uFxTOS: uniformly fixed-time output stable

## II. PRELIMINARIES

The standard stability notions and their definitions can be found in [4], and in this section we briefly review only that is needed in our setting.

### A. Uniform output stability

Consider a non-autonomous differential equation:

$$ 
\begin{aligned}
\dot{x}(t) &= f(x(t), d(t)), \quad t \ge 0, \\
y(t) &= h(x(t)),
\end{aligned}
\quad (1) $$

where $x(t) \in \mathbb{R}^n$ is the state vector, $d(t) \in D \subseteq \mathbb{R}^s$ is the vector of external inputs and $d \in D \subseteq \mathcal{L}_{\infty}^s$ ($D$ is the set of admissible inputs), $y(t) \in \mathbb{R}^p$ is the output; $f: \mathbb{R}^n \times \mathbb{R}^s \to \mathbb{R}^n$ and $h: \mathbb{R}^n \to \mathbb{R}^p$ are locally Lipschitz continuous functions, $f(0, 0) = 0$ and $h(0) = 0$. The unique maximal solution of the system (1) for an initial condition $x_0 \in \mathbb{R}^n$ and $d \in D$ is denoted by $X(t, x_0, d)$ for those $t \ge 0$ for which the solution exists; then $Y(t, x_0, d) = h(X(t, x_0, d))$.

The system (1) is called forward complete if for any $x_0 \in \mathbb{R}^n$ and $d \in D$ the solution $X(t, x_0, d)$ exists for all $t \ge 0$.

In the sequel, we assume that the inputs $d(t)$ take values in a compact set $D \subset \mathbb{R}^s$ for almost all time instants $t \ge 0$, i.e., $\sup_{d \in D} \|d\|_{\infty} < +\infty$.

**Definition 1.** [21] A forward complete system (1) is uniformly output stable (uOS) with respect to the inputs in $D$ if there exists $\beta \in \mathcal{KL}$ such that the property

$$ \|Y(t, x_0, d)\| \le \beta(\|x_0\|, t), \quad \forall t \ge 0 $$

holds for all $x_0 \in \mathbb{R}^n$ and $d \in D$. It is state-independent uniformly output stable (SIuOS) with respect to the inputs in $D$ if this estimate can be strengthened to

$$ \|Y(t, x_0, d)\| \le \beta(\|h(x_0)\|, t), \quad \forall t \ge 0 $$

which must be satisfied for all $x_0 \in \mathbb{R}^n$ and $d \in D$.

**Definition 2.** [21] A system (1) is uniformly bounded input bounded state stable (UBIBS) if it is forward complete and there exists $\sigma \in \mathcal{K}$ such that

$$ \|X(t, x_0, d)\| \le \max\{\sigma(\|x_0\|), \sigma(\|d\|_{\infty})\}, \quad \forall t \ge 0 $$

for all $x_0 \in \mathbb{R}^n$ and $d \in D$.

**Theorem 1.** [21] A UBIBS system (1) is uOS if and only if it admits a smooth Lyapunov function $V: \mathbb{R}^n \to \mathbb{R}_+$ satisfying for some $\alpha_1, \alpha_2 \in \mathcal{K}_{\infty}$ and $\alpha_3 \in \mathcal{KL}$ the following properties:

$$ 
\begin{aligned}
\alpha_1(\|h(x)\|) &\le V(x) \le \alpha_2(\|x\|), \quad \forall x \in \mathbb{R}^n; \\
\frac{\partial V(x)}{\partial x} f(x, d) &\le -\alpha_3(V(x), \|x\|), \quad \forall x \in \mathbb{R}^n, \forall d \in D.
\end{aligned}
 $$

Moreover, a forward complete system (1) is SIuOS if and only if there exists a smooth Lyapunov function $V$ such that

$$ 
\begin{aligned}
\alpha_1(\|h(x)\|) &\le V(x) \le \alpha_2(\|h(x)\|), \quad \forall x \in \mathbb{R}^n; \\
\frac{\partial V(x)}{\partial x} f(x, d) &\le -\alpha_4(V(x)), \quad \forall x \in \mathbb{R}^n, \forall d \in D
\end{aligned}
 $$

for some $\alpha_1, \alpha_2 \in \mathcal{K}_{\infty}$ and $\alpha_4 \in \mathcal{K}$.

The given Lyapunov conditions have been developed in [22] for the case of non-UBIBS systems. In the papers [26], [27] these characterizations of SIuOS property have been extended to the case when $f$ is locally Lipschitz continuous on $\mathbb{R}^n \setminus \{x \in \mathbb{R}^n : h(x) = 0\} \times \mathbb{R}^s$ and continuous everywhere, $h: \mathbb{R}^n \to \mathbb{R}^p$ is continuously differentiable (in that work the case $D = \{0\}$ was considered, but the proof stays valid under mild modifications for any compact set $D$).

Following [27], the accelerated output convergence rates can be defined:

**Definition 3.** A forward complete system (1) is uniformly finite-time output stable (uFTOS) if it is uOS and for any $x_0 \in \mathbb{R}^n$ there exists $T_{x_0} \in [0, +\infty)$ such that $Y(t, x_0, d) = 0$ for all $t \ge T_{x_0}$ and $d \in D$. The function $T_0(x_0) = \inf\{T_{x_0} \ge 0 : Y(t, x_0, d) = 0 \forall t \ge T_{x_0}, \forall d \in D\}$ is called the settling-time function.

A forward complete system (1) is called uniformly nearly fixed-time output stable (unFxTOS) if (1) is uOS and for any $\rho > 0$ there exists $T_{\rho} \in [0, +\infty)$ such that $\|Y(t, x_0, d)\| \le \rho$ for all $t \ge T_{\rho}$, all $x_0 \in \mathbb{R}^n$ and $d \in D$.
The system (1) is uniformly fixed-time output stable (uFxTOS) if it is uFTOS and unFxTOS simultaneously, i.e., $\sup_{x_0 \in \mathbb{R}^n} T_0(x_0) < +\infty$.

Note that for unFxTOS systems, it could happen that $\sup_{\rho>0} T_\rho = +\infty$. If $h(x) = x$, then these properties reduce to the conventional (uniform) finite/fixed-time stability [28], [29]. The respective Lyapunov characterizations of the output stability properties with accelerated convergence can be found in [26], [27], [30]. These stability concepts can also be defined on a finite interval of time $[0, T(x_0))$.

### B. Homogeneity

For any $r_i > 0$, $i = 1, \dots, n$ and $\lambda > 0$, define the vector of weights $\mathbf{r} = [r_1, \dots, r_n]$ and the dilation matrix $\Lambda_r(\lambda) = \text{diag}\{\lambda^{r_i}\}_{i=1}^n$; $r_{\min} = \min_{i=1, n} r_i$ and $r_{\max} = \max_{i=1, n} r_i$.

**Definition 4.** [7], [31] A function $h: \mathbb{R}^n \to \mathbb{R}$ is called $\mathbf{r}$-homogeneous, if for some $\nu \in \mathbb{R}$ the relation

$$h(\Lambda_r(\lambda)x) = \lambda^{\nu}h(x)$$

holds for any $x \in \mathbb{R}^n$ and all $\lambda > 0$.

A vector field $f: \mathbb{R}^n \to \mathbb{R}^n$ is called $\mathbf{r}$-homogeneous, if for some $\nu \ge -r_{\min}$ the relation

$$f(\Lambda_r(\lambda)x) = \lambda^{\nu}\Lambda_r(\lambda)f(x)$$

holds for any $x \in \mathbb{R}^n$ and all $\lambda > 0$.

In both cases, the constant $\nu$ is called the degree of homogeneity.

A dynamical system

$$\dot{x}(t) = f(x(t)), \ t \ge 0, \ x(t) \in \mathbb{R}^n$$

is called $\mathbf{r}$-homogeneous of degree $\nu$ if this property is satisfied for $f$ in the sense of Definition 4.

For any $x \in \mathbb{R}^n$ and $\varpi \ge r_{\min}$, a homogeneous norm can be defined as follows

$$\|x\|_r = \left( \sum_{i=1}^{n} |x_i|^{\varpi/r_i} \right)^{1/\varpi}.$$

For all $x \in \mathbb{R}^n$, its Euclidean norm $\|x\|$ is related with the homogeneous one:

$$\underline{\sigma}_r(\|x\|_r) \le \|x\| \le \bar{\sigma}_r(\|x\|_r)$$

for some $\underline{\sigma}_r, \bar{\sigma}_r \in \mathcal{K}_\infty$ [32]. In the following, due to this “equivalence”, stability analysis with respect to the norm $\| \cdot \|$ can be substituted with analysis for the norm $\| \cdot \|_r$. The homogeneous norm has an important property: it is $\mathbf{r}$-homogeneous of degree 1, that is $\|\Lambda_r(\lambda)x\|_r = \lambda\|x\|_r$ for all $x \in \mathbb{R}^n$ and $\lambda > 0$.

### III. PROBLEM STATEMENT

Consider the problem of accelerated stabilization (in the sense of convergence rates given in Definition 3) of a controllable system of the form:

$$\dot{y}(t) = X_0(y(t)) + \sum_{i=1}^{m} X_i(y(t))u_i(t), \ t \ge 0, \quad (2)$$

where $y(t) \in \mathbb{R}^p$ is the state of (2) and $u(t) \in \mathbb{R}^m$ is the control to be designed; $X_j: \mathbb{R}^p \to \mathbb{R}^p$ are continuous vector fields for $j = 0, m$. If $X_0(y) = 0$, then (2) becomes driftless, such models frequently appear in the study of mechanical systems with nonholonomic constraints [1]. Asymptotic stabilization of such systems by a continuous time-invariant feedback $u(t) = k(y(t))$ (with a continuous function $k: \mathbb{R}^p \to \mathbb{R}^m$) is impossible [33], hence, frequently discontinuous [34] or time-varying controls $u(t) = k(t, y(t))$ (with continuous $k: \mathbb{R} \times \mathbb{R}^p \to \mathbb{R}^m$) are used [1], [35], [36]. In the latter case, the time dependence can be modeled by introducing auxiliary state variables $z(t) \in \mathbb{R}^s$ in the system dynamics, for which there is no requirement for convergence, and the problem of accelerated output $y(t)$ stabilization can be analyzed for an autonomous extended model with the combined state $x(t) = (y(t), z(t))$. Other situations, when additional dynamics can appear, correspond to the trajectory tracking or disturbance compensation scenarios, then $z(t)$ (or a function of it) becomes the reference or the internal model state; the adaptive control, where $z(t)$ represents the parameter estimates, is another example. Gauss Variational Equations that model the evolution of the orbital elements of a spacecraft orbit provide yet another practically relevant example [37].

In [1] the theory of homogeneous systems was applied to design a feedback law guaranteeing exponential stability of the closed-loop driftless dynamics. The method is based on scaling of existing controls, which provide just asymptotic stabilization of the systems. A homogeneous norm was applied for scaling, similarly as it has been suggested for linear systems in [15], [10], or using implicit Lyapunov functions as exemplified in later works [11]–[14]. In this paper, utilizing similar ideas and extending the theory of homogeneous systems to the case of partial symmetry and stability, finite/fixed-time stabilizing feedbacks are synthesized.

### IV. PARTIAL HOMOGENEITY

Consider an explicit decomposition of (1) for $d \equiv 0$ into output dynamics and the remaining variables (assume such a representation exists):

$$\dot{y}(t) = F(y(t), z(t)), \quad (3)$$

$$\dot{z}(t) = G(y(t), z(t)),$$

where $y(t) \in \mathbb{R}^p$ and $z(t) \in \mathbb{R}^s$ are the components of the state vector $x(t) = [y(t)^\top \ z(t)^\top]^\top \in \mathbb{R}^n$ with $n = p + s$, $F: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}^p$ and $G: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}^s$ are
locally Lipschitz continuous functions on $\mathbb{R}^p \setminus \{0\} \times \mathbb{R}^s$ and continuous everywhere, $F(0, z) = 0$ for all $z \in \mathbb{R}^s$. For specified initial conditions $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$, we denote the respective solutions of (3) by $Y(t, y_0, z_0)$ and $Z(t, y_0, z_0)$ for those $t \ge 0$ for which these solutions exist. For brevity of exposition, assume in the sequel that these solutions are unique and defined for all $t \ge 0$, i.e., (3) is forward complete.

**Assumption 1.** There exists a vector of weights $\mathbf{r} = [r_1, ..., r_p]$ with $r_i > 0$, $i = \overline{1, p}$ such that the following properties,

$$F(\Lambda_r(\lambda)y, z) = \lambda^\nu \Lambda_r(\lambda) F(y, z), \quad G(\Lambda_r(\lambda)y, z) = \lambda^\nu G(y, z)$$

hold for all $\lambda > 0$, $y \in \mathbb{R}^p$, $z \in \mathbb{R}^s$ and some $\nu \ge -r_{\min}$.

Hence, according to Definition 4, $F$ is an $\mathbf{r}$-homogeneous vector field and $G$ is an $\mathbf{r}$-homogeneous function of degree $\nu$ considering $z$ as a constant, i.e., they are partially $\mathbf{r}$-homogeneous with respect to $y$.

**Proposition 1.** Under Assumption 1, if $Y(t, y_0, z_0)$ and $Z(t, y_0, z_0)$ are solutions of (3) for some $y_0 \in \mathbb{R}^p \setminus \{0\}$ and $z_0 \in \mathbb{R}^s$, then $\Lambda_r(\lambda)Y(\lambda^\nu t, y_0, z_0)$ and $Z(\lambda^\nu t, y_0, z_0)$ are solutions to (3) for initial conditions $\Lambda_r(\lambda)y_0$ and $z_0$ for any $\lambda > 0$.

*Proof.* We need to show that

$$Y(t, \Lambda_r(\lambda)y_0, z_0) = \Lambda_r(\lambda)Y(\lambda^\nu t, y_0, z_0), \quad (4)$$

$$Z(t, \Lambda_r(\lambda)y_0, z_0) = Z(\lambda^\nu t, y_0, z_0), \quad t \ge 0.$$

Let $\bar{Y}(t)$, $\bar{Z}(t)$ denote functions on the right-hand side of (4). Differentiate $\bar{Y}(t)$ and $\bar{Z}(t)$ to demonstrate that they are solutions of (3):

$$\frac{\partial \bar{Y}(t)}{\partial t} = \Lambda_r(\lambda) \frac{\partial Y(\lambda^\nu t, y_0, z_0)}{\partial t} = \lambda^\nu \Lambda_r(\lambda) F(Y(\lambda^\nu t, y_0, z_0), Z(\lambda^\nu t, y_0, z_0)) = F(\bar{Y}(t), \bar{Z}(t)),$$

$$\frac{\partial \bar{Z}(t)}{\partial t} = \frac{\partial Z(\lambda^\nu t, y_0, z_0)}{\partial t} = \lambda^\nu G(Y(\lambda^\nu t, y_0, z_0), Z(\lambda^\nu t, y_0, z_0)) = G(\bar{Y}(t), \bar{Z}(t)).$$

The proof is completed by noting that $\bar{Y}(0) = \Lambda_r(\lambda)y_0$ and $\bar{Z}(0) = z_0$, and that the solution to (3) is unique. $\square$

**Remark 1.** If the requirement on uniqueness of solutions is not satisfied, then repeating the same arguments one can show that each solution issued from the pair of initial conditions $y_0$ and $z_0$ admits the symmetry relations (4). More detailed analysis of this case is left for further research.

Due to the symmetry of solutions (4), as in the case of the conventional $\mathbf{r}$-homogeneity [38], under Assumption 1, the local output stability of (3) implies global one:

**Lemma 1.** Let there exist $\rho > 0$ and $\beta \in \mathcal{KL}$ such that $\|Y(t, y_0, z_0)\| \le \beta(\|y_0\|, t)$ for all $t \ge 0$, $z_0 \in \mathbb{R}^s$ and all $y_0 \in \mathbb{R}^p$ with $\|y_0\| \le \rho$. Then, under Assumption 1, the system (3) is SIuOS.

*Proof.* Recalling the relations between a homogeneous norm and the Euclidean one, the following property is verified:

$$\|Y(t, y_0, z_0)\|_r \le \tilde{\beta}(\|y_0\|_r, t), \quad \forall t \ge 0,$$

for all $z_0 \in \mathbb{R}^s$ and $\|y_0\|_r \le \tilde{\rho}$, where $\tilde{\rho} = \underline{\sigma}_r^{-1}(\rho)$ and $\tilde{\beta}(q, t) = \underline{\sigma}_r^{-1} \circ \beta(\tilde{\sigma}_r(q), t)$ is a function from class $\mathcal{KL}$. Dilating an initial condition $y_0 \in \mathbb{R}^p$ with $\|y_0\|_r > \tilde{\rho}$ by $\lambda = \frac{\tilde{\rho}}{\|y_0\|_r} \in (0, 1)$ we get for $\tilde{y}_0 = \Lambda_r(\lambda)y_0$ that $\|\tilde{y}_0\|_r = \tilde{\rho}$. Then using (4), we obtain:

$$\lambda \|Y(\lambda^\nu t, y_0, z_0)\|_r = \|Y(t, \Lambda_r(\lambda)y_0, z_0)\|_r \\ = \|Y(t, \tilde{y}_0, z_0)\|_r \le \tilde{\beta}(\|\tilde{y}_0\|_r, t) = \tilde{\beta}(\tilde{\rho}, t), \quad \forall t \ge 0,$$

which implies that the following estimate is satisfied:

$$\|Y(t, y_0, z_0)\|_r \le \frac{\|y_0\|_r}{\tilde{\rho}} \tilde{\beta} \left( \tilde{\rho}, \left( \frac{\|y_0\|_r}{\tilde{\rho}} \right)^\nu t \right) \\ \le \tilde{\beta}(\|y_0\|_r, t), \quad \forall t \ge 0,$$

for all $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$, and some $\tilde{\beta} \in \mathcal{KL}$ (for example, $\tilde{\beta}(w, t) = \frac{w}{\tilde{\rho}} \tilde{\beta}(\tilde{\rho}, (\frac{\max\{w, \tilde{\rho}\}}{\tilde{\rho}})^\nu t)$). Hence, the SIuOS estimate is verified for $\max\{\tilde{\beta}(q, t), \tilde{\beta}(q, t)\} \in \mathcal{KL}$. $\square$

Following [38], a compact set $A \subset \mathbb{R}^p$ is called partially positively invariant for the system (3) if $y_0 \in A$ and $z_0 \in \mathbb{R}^s$ implies that $Y(t, y_0, z_0) \in A$ for all $t \ge 0$. Moreover, $A$ is called strictly partially positively invariant if $Y(t, y_0, z_0) \in \text{int}(A)$ for all $t > 0$.

**Lemma 2.** Under Assumption 1, if the system (3) has a compact partially positively invariant set $A \subset \mathbb{R}^p$ with $0 \in \text{int}(A)$, then $\|Y(t, y_0, z_0)\|_r \le \kappa \|y_0\|_r$ for all $t \ge 0$, all $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$ with some $\kappa > 0$. In addition, if $A$ is strictly partially positively invariant for (3), and $\|Z(t, y_0, z_0)\| \le \sigma(\|z_0\|) + \sigma_0$ for all $t \ge 0$, all $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$, for some $\sigma \in K$ and $\sigma_0 \ge 0$, then it is uOS.

*Proof.* Select $\rho > 0$ such that $\mathcal{R} = \{y \in \mathbb{R}^p : \|y\|_r = \rho\} \subset A$ (such a $\rho$ always exists due to the properties of the set $A$), then

$$\|Y(t, \tilde{y}_0, z_0)\|_r \le R, \quad \forall t \ge 0$$

for all $\tilde{y}_0 \in \mathcal{R}$ and $z_0 \in \mathbb{R}^s$, where $R = \sup_{y \in A} \|y\|_r$ by partial positive invariance of $A$. Again, dilating an initial condition $y_0 \in \mathbb{R}^p \setminus \{0\}$ by $\lambda = \frac{\rho}{\|y_0\|_r}$ we get $\Lambda_r(\lambda)y_0 = \tilde{y}_0$ for some $\tilde{y}_0 \in \mathcal{R}$, then using (4), we obtain:

$$\lambda \|Y(\lambda^\nu t, y_0, z_0)\|_r = \|Y(t, \Lambda_r(\lambda)y_0, z_0)\|_r \\ = \|Y(t, \tilde{y}_0, z_0)\|_r \le R, \quad \forall t \ge 0,$$

which implies that the following estimate is satisfied:

$$\|Y(t, y_0, z_0)\|_r \le \kappa \|y_0\|_r, \quad \forall t \ge 0,$$

for all $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$, where $\kappa = \frac{R}{\rho}$. Now, if $A$ is strictly partially positively invariant for (3) and $z(t)$ is bounded, then

$$\|Y(t, \tilde{y}_0, z_0)\|_a < \|\tilde{y}_0\|_a, \quad \forall t \ge 0$$
for all $\tilde{y}_0 \in \partial A$ and $z_0 \in \mathbb{R}^s$ for a suitably defined norm $\|\cdot\|_a$, and utilizing this contraction property and dilating those initial conditions by $\lambda \in (0, 1)$ we get that the same property is satisfied for all $\tilde{y}_0 \in A \setminus \{0\}$, which together with proven stability leads to uOS. $\square$

It is a well-known fact that any (locally) asymptotically stable homogeneous system is globally finite-time or nearly fixed-time stable provided that it possesses negative or positive homogeneity degree, respectively [7], [38], [39]. The same convergence characteristics of the output stability can be obtained for (3):

**Theorem 2.** Under Assumption 1, if $\|Z(t, y_0, z_0)\| \le \sigma(\|z_0\|) + \sigma_0$ for all $t \ge 0$, all $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$, for some $\sigma \in K$ and $\sigma_0 \ge 0$, then a uOS system (3) is uFTOS or unFxTOS provided that the homogeneity degree $\nu$ is negative or positive, respectively, while for $\nu = 0$ the convergence rate of $Y(t, y_0, z_0)$ is exponential.

*Proof.* For a uOS system (3), due to the properties of a function from class $KL$, there exists a (locally bounded) map $T: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$ such that

$$ \|Y(t, y_0, z_0)\|_r \le \frac{1}{2} \|y_0\|_r, \quad \forall t \ge T(y_0, z_0) $$

for any $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$ (assume that $T(y_0, z_0)$ is defined by the smallest time instant verifying this inequality). Using (4), we obtain a symmetry relation

$$ T(\Lambda_r(\lambda)y_0, z_0) = \lambda^{-\nu}T(y_0, z_0) $$

for any $\lambda > 0$, $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$. Indeed, for any $\lambda > 0$,

$$ \begin{aligned} \lambda \|Y(\lambda^\nu t, y_0, z_0)\|_r &= \|Y(t, \Lambda_r(\lambda)y_0, z_0)\|_r \\ &\le \frac{1}{2} \|\Lambda_r(\lambda)y_0\|_r = \frac{\lambda}{2} \|y_0\|_r, \quad \forall t \ge T(\Lambda_r(\lambda)y_0, z_0), \end{aligned} $$

or, alternatively, considering only the first and the last items:

$$ \|Y(\lambda^\nu t, y_0, z_0)\|_r \le \frac{1}{2} \|y_0\|_r, \quad \forall t \ge \lambda^{-\nu}T(y_0, z_0), $$

which by equivalence gives the required property. Finally, for any $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$ define a uniform upper bound $\bar{T}$ on $T$:

$$ \begin{aligned} T(y_0, z_0) &\le \bar{T}_{z_0}(\|y_0\|_r) \\ &= \|y_0\|_r^{-\nu} \sup_{\|y\|_r=1, \|z\| \le \sigma(\|z_0\|) + \sigma_0} T(y, z), \end{aligned} $$

and, obviously, $\bar{T}$ inherits the same symmetry relation from $T$:

$$ \bar{T}_{z_0}(\|\Lambda_r(\lambda)y_0\|_r) = \lambda^{-\nu}\bar{T}_{z_0}(\|y_0\|_r) $$

for any $\lambda > 0$, $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$. Consider the case with $\nu < 0$, then the time of convergence to zero of $Y(t, y_0, z_0)$ for any $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$ can be expressed as follows

(we consider the segments of trajectory passing from $\|y_0\|_r$ to $\frac{1}{2}\|y_0\|_r$, next from $\frac{1}{2}\|y_0\|_r$ to $\frac{1}{4}\|y_0\|_r$, etc.):

$$ \begin{aligned} & T(y_0, z_0) + T(y(T(y_0, z_0)), z(T(y_0, z_0))) \\ & \quad + T(y(T(y(T(y_0, z_0)), z(T(y_0, z_0)))), \\ & \quad z(T(y(T(y_0, z_0)), z(T(y_0, z_0))))) + \dots \\ & \le \bar{T}_{z_0}(\|y_0\|_r) + \bar{T}_{z_0}(\frac{1}{2}\|y_0\|_r) + \bar{T}_{z_0}(\frac{1}{4}\|y_0\|_r) + \dots \\ & = \sum_{i=0}^{+\infty} \bar{T}_{z_0}(\frac{1}{2^i} \|y_0\|_r) = \bar{T}_{z_0}(\|y_0\|_r) \sum_{i=0}^{+\infty} 2^{\nu i} \\ & = \frac{\bar{T}_{z_0}(\|y_0\|_r)}{1 - 2^{\nu}} < +\infty \end{aligned} $$

and it is finite, hence, the system is uFTOS. The result for $\nu > 0$ can be obtained by applying similar arguments to the time of convergence from arbitrary initial conditions $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$ to the set where $\|Y(t, y_0, z_0)\|_r \le \rho$ for any $\rho > 0$. For the case $\nu = 0$, the conclusion follows the observation that the decrease of $\|Y(t, y_0, z_0)\|_r$ by a factor of 2 is upper bounded by $\bar{T}_{z_0}(1)$ for any $y_0 \in \mathbb{R}^p$. $\square$

**Remark 2.** The requirement that the output stability properties are verified for all $z_0 \in \mathbb{R}^s$ can be replaced by $z_0 \in \mathcal{Z}$ for some compact set $\mathcal{Z} \subset \mathbb{R}^s$, then global results on output $y$ stability and convergence are preserved but locally in $z$, for the initial conditions in $\mathcal{Z}$ (there is no scaling of $z_0$ in (4)). Moreover, the only properties of $z(t)$ used in this section are existence of solutions and their boundedness, then requirements on continuity of $G$ can be relaxed (especially when this variable represents generator dynamics simulated in the controller).

The results above extend [38] to the case of partial homogeneity formulated in Assumption 1. It is also possible to extend the result of [31], [40] on the existence of a partially homogeneous Lyapunov function to the case of the output stability:

**Theorem 3.** Under Assumption 1, the system (3) is SIuOS if and only if there is a smooth on $\mathbb{R}^p \setminus \{0\} \times \mathbb{R}^s$ Lyapunov function $W: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$, $C^k$ on $\mathbb{R}^p \times \mathbb{R}^s$ for any integer $k \ge 1$, such that for some $c_1, c_2, c_3 > 0$ and all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$:

$$ \begin{aligned} c_1 \|y\|_r^\mu &\le W(y, z) \le c_2 \|y\|_r^\mu, \\ \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) &\le -c_3 \|y\|_r^{\nu+\mu}, \\ W(\Lambda_r(\lambda)y, z) &= \lambda^\mu W(y, z), \quad \forall \lambda > 0, \end{aligned} $$

where $\mu > kr_{\max}$.

*Proof.* The existence of such a Lyapunov function $W$ (with the first two properties) implies SIuOS (see Theorem 1).

Inversely, if the system (3) is SIuOS, then according to the results given in [21], [26], there is a smooth function
$V: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$ such that for some $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ and $\alpha_3 \in \mathcal{K}$,

$$ 
\begin{aligned}
\alpha_1(\|y\|_r) &\le V(y, z) \le \alpha_2(\|y\|_r), \\
\frac{\partial V(y, z)}{\partial y} F(y, z) + \frac{\partial V(y, z)}{\partial z} G(y, z) &\le -\alpha_3(V(y, z))
\end{aligned}
$$

hold for all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$ (the norms $\|\cdot\|$ and $\|\cdot\|_r$ are interchangeable). Following [40], define a smooth function $a: \mathbb{R} \to \mathbb{R}_+$ such that

$$ a(q) = \begin{cases} 0 & \text{if } q \le 1, \\ 1 & \text{if } q \ge 2, \\ a'(q) = \frac{da(q)}{dq} > 0 & \text{otherwise.} \end{cases} \quad (5) $$

Take any $\mu > \max\{0, -\nu\}$ and consider a candidate Lyapunov function:

$$ W(y, z) = \begin{cases} \int_{0}^{+\infty} \frac{a(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+1}} d\lambda & \text{if } y \neq 0, \\ 0 & \text{if } y = 0, \end{cases} $$

which is smooth on $\mathbb{R}^p \setminus \{0\} \times \mathbb{R}^s$ by construction. Due to the growth properties of $V$ and the restrictions on $a$, there exist $0 < \underline{l}_y < \bar{l}_y < +\infty$ such that

$$ W(y, z) = \frac{1}{\mu \bar{l}_y^\mu} + \int_{\underline{l}_y}^{\bar{l}_y} \frac{a(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+1}} d\lambda $$

for any $y \in \mathbb{R}^p \setminus \{0\}$ and $z \in \mathbb{R}^s$. Since $\lim_{\|y\| \to 0} \bar{l}_y = +\infty$ and $\lim_{\|y\| \to +\infty} \underline{l}_y = 0$ (the behavior of $V(y, z)$ is governed by the first argument, and $W(y, z) = 0$ whenever $V(y, z) = 0$, i.e., for $y = 0$), there are $\tilde{\alpha}_1, \tilde{\alpha}_2 \in \mathcal{K}_\infty$ such that

$$ \tilde{\alpha}_1(\|y\|_r) \le W(y, z) \le \tilde{\alpha}_2(\|y\|_r) $$

for all $y \in \mathbb{R}^p$, $z \in \mathbb{R}^s$. By a simple change of integral variables it is easy to demonstrate that

$$ W(\Lambda_r(\lambda)y, z) = \lambda^\mu W(y, z) $$

for any $\lambda > 0$ and all $y \in \mathbb{R}^p$, $z \in \mathbb{R}^s$. Note that for any $y \in \mathbb{R}^p$ there is $\tilde{y} \in \mathbb{R}^p$ with $\|\tilde{y}\|_r = 1$ and $y = \Lambda_r(\|y\|_r)\tilde{y}$, then

$$ c_1 \|y\|_r^\mu \le W(y, z) = \|y\|_r^\mu W(\tilde{y}, z) \le c_2 \|y\|_r^\mu, $$

where $c_1 = \tilde{\alpha}_1(1)$ and $c_2 = \tilde{\alpha}_2(1)$. Finally, for all $y \in \mathbb{R}^p \setminus \{0\}$ and $z \in \mathbb{R}^s$,

$$ 
\begin{aligned}
& \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) \\
&= \int_{0}^{+\infty} \frac{a'(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+\nu+1}} \left[ \frac{\partial V(\Lambda_r(\lambda)y, z)}{\partial y} \lambda^\nu \Lambda_r(\lambda) F(y, z) + \frac{\partial V(\Lambda_r(\lambda)y, z)}{\partial z} \lambda^\nu G(y, z) \right] d\lambda
\end{aligned}
$$

$$ 
\begin{aligned}
&= \int_{0}^{+\infty} \frac{a'(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+\nu+1}} \left[ \frac{\partial V(\Lambda_r(\lambda)y, z)}{\partial y} F(\Lambda_r(\lambda)y, z) + \frac{\partial V(\Lambda_r(\lambda)y, z)}{\partial z} G(\Lambda_r(\lambda)y, z) \right] d\lambda \\
&\le - \int_{0}^{+\infty} \frac{a'(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+\nu+1}} \alpha_3(V(\Lambda_r(\lambda)y, z)) d\lambda \\
&\le - \int_{\underline{l}_y}^{\bar{l}_y} \frac{a'(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+\nu+1}} \alpha_3 \circ \alpha_1(\lambda \|y\|_r) d\lambda \le -\alpha_4(\|y\|_r),
\end{aligned}
$$

where $\alpha_4: \mathbb{R}_+ \to \mathbb{R}_+$ is a positive definite continuous function. Recall the definition of $\tilde{y}$ used above to calculate $c_1$ and $c_2$, then

$$ 
\begin{aligned}
& \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) \\
&= \frac{\partial W(\Lambda_r(\|y\|_r)\tilde{y}, z)}{\partial \Lambda_r(\|y\|_r)\tilde{y}} F(\Lambda_r(\|y\|_r)\tilde{y}, z) \\
&\quad + \frac{\partial W(\Lambda_r(\|y\|_r)\tilde{y}, z)}{\partial z} G(\Lambda_r(\|y\|_r)\tilde{y}, z) \\
&= \|y\|_r^{\mu+\nu} \left( \frac{\partial W(\tilde{y}, z)}{\partial \tilde{y}} F(\tilde{y}, z) + \frac{\partial W(\tilde{y}, z)}{\partial z} G(\tilde{y}, z) \right) \\
&\le -c_3 \|y\|_r^{\mu+\nu}, \quad c_3 = \alpha_4(1).
\end{aligned}
$$

The property of $C^k$-continuity under the constraint $\mu > kr_{\max}$ can be proven using the same arguments as in [40]. Thus, $W$ is a required SIuOS Lyapunov function. $\square$

A more advanced version of this theorem for zero homogeneity degree, where also certain periodicity in $z$ is taken into account, has been obtained in [35]. A similar to Theorem 3 but a less constructive result can be formulated for uOS case:

**Theorem 4.** Under Assumption 1, if $F, G$ are locally Lipschitz continuous vector functions, the UBIBS system (3) is uOS if and only if there is a Lyapunov function $W: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$, $C^k$ on $\mathbb{R}^p \times \mathbb{R}^s$ for any integer $k \ge 1$, such that for some $\tilde{\alpha}_1, \tilde{\alpha}_2 \in \mathcal{K}_\infty$ and all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$:

$$ 
\begin{aligned}
& \tilde{\alpha}_1(\|y\|) \le W(y, z) \le \tilde{\alpha}_2(\|x\|), \\
& W(y, z) \ne 0 \Rightarrow \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) < 0, \\
& W(\Lambda_r(\lambda)y, z) = \lambda^\mu W(z, y), \quad \forall \lambda > 0, \\
& \text{where } x = [y^\top z^\top]^\top \text{ and } \mu > kr_{\max}.
\end{aligned}
$$

*Proof.* The existence of such a Lyapunov function $W$ (with the first two properties) implies uOS (see Lemma A.5 in [21]).

Inversely, if the system (3) is uOS and UBIBS, then according to the results given in [21], there is a smooth function $V: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$ such that for some $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ and $\alpha_3 \in \mathcal{K}$,

$$ 
\begin{aligned}
& \alpha_1(\|y\|) \le V(y, z) \le \alpha_2(\|x\|), \\
& \frac{\partial V(y, z)}{\partial y} F(y, z) + \frac{\partial V(y, z)}{\partial z} G(y, z) \le -\alpha_3(V(y, z), \|x\|)
\end{aligned}
$$
since $-\mathcal{H}$ is a Hurwitz matrix). Under (8), for any $(y, z) \in \Omega$
there is a unique solution $\psi(y, z)$ since this condition implies
global monotonicity of $V(\Lambda_r(\psi)y, z)$ in $\psi$. Using the implicit
differentiation rule, we get:

$$
\left[ \begin{array}{cc} \frac{\partial \psi}{\partial y} & \frac{\partial \psi}{\partial z} \end{array} \right] = -\psi \frac{\left[ \frac{\partial V(\Lambda_r(\psi)y, z)}{\partial y} \Lambda_r(\psi) \quad \frac{\partial V(\Lambda_r(\psi)y, z)}{\partial z} \right]}{\frac{\partial V(\Lambda_r(\psi)y, z)}{\partial y} \mathcal{H} \Lambda_r(\psi)y}
$$

that is well defined under the restriction (8) for all $(y, z) \in \Omega$.
Hence, if (8) is verified, then $\psi$ is a continuously differen-
tiable function in $\Omega$. Due to these regularity properties and
positive definiteness of $V$ with respect to $y$, it follows that

$$
\lim_{\|y\| \to +\infty} \psi(y, z) = 0, \quad \lim_{\|y\| \to 0} \psi(y, z) = +\infty.
$$

Moreover, by construction

$$
\psi^{-1}(\Lambda_r(\lambda)y, z) = \lambda\psi^{-1}(y, z)
$$

for all $\lambda > 0$, all $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$, i.e., $\psi^{-1}$ is $\mathbf{r}$-homogeneous of degree 1 considering $z$ as a constant, and it can be considered as a partially homogeneous norm with respect to $y$.

In order to use the results of Section IV, and recalling
assumptions 2, 3 and 4, let us define the controls in (2) as

$$
u_i(y, z) = \begin{cases} K_i(y, z) & \text{if } y \neq 0, \\ 0 & \text{if } y = 0, \end{cases} \quad i = \overline{1, m}, \quad (9)
$$

$$
K_i(y, z) = \psi^{-\nu_0 + \nu_i}(y, z) U_i(\Lambda_r(\psi(y, z))y, z),
$$

and the accelerated generator dynamics as

$$
\dot{z}(t) = G(y(t), z(t)) \quad (10)
$$

$$
= \begin{cases} \psi^{-\nu_0}(y(t), z(t))g(\Lambda_r(\psi(y, z))y, z(t)) & \text{if } y \neq 0 \\ 0 & \text{if } y = 0. \end{cases}
$$

Note that the control laws (9) are locally Lipschitz continuous
on $\Omega$ provided that

$$
\nu_0 \geq \max_{i=1,m} \nu_i \quad (11)
$$

and partially homogeneous:

$$
K_i(\Lambda_r(\lambda)y, z) = \lambda^{\nu_0 - \nu_i} K_i(y, z)
$$

for all $\lambda > 0$ and all $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$. Similarly, the acceler-
ator dynamics (10) are also locally Lipschitz continuous on
$\Omega$ and partially homogeneous:

$$
G(\Lambda_r(\lambda)y, z) = \lambda^{\nu_0} G(y, z)
$$

for all $\lambda > 0$, $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$.

Remark 3. The controller (9), (10) is algorithmically defined since for its implementation we need to calculate the function $\psi$ that is implicitly given in (7). Such kind of algorithmically defined regulators are widely used in practice, e.g., in model predictive control, and different approximation tools can be applied for their explicit realization. To calculate the value $\psi(y, z)$, a bisection algorithm from [42] can be used.

Typically, the required number of iterations to achieve a
reasonable accuracy in $\psi$ and the associated computational
load are small.

Denote the closed-loop dynamics (2), (9) as

$$
F(y, z) = X_0(y) + \sum_{i=1}^{m} X_i(y) K_i(y, z),
$$

then due to the established above symmetry relations for $K_i$
and Assumption 2,

$$
F(\Lambda_r(\lambda)y, z) = \lambda^{\nu_0} \Lambda_r(\lambda) F(y, z)
$$

for all $\lambda > 0$, and all $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$. Therefore, Assumption 1 is verified for (2), (9), (10). Further, it is easy to check that $\mathcal{V}$ is a strictly partially positively invariant set for this system under Assumption 4 (since $F(y, z) = f(y, z)$ and $G(y, z) = g(y, z)$ for $(y, z) \in \partial \mathcal{V}$), and by Lemma 2 the closed-loop system is uOS (for $\nu_0 < 0$ considering the time interval where $y(t) \neq 0$). In addition, Theorem 2 helps to substantiate uFTOS or unFxTOS properties for (2), (9), (10) with initial conditions in $\mathbb{R}^p \setminus \{0\} \times \mathcal{Z}$ for $\nu_0 < 0$ or $\nu_0 > 0$, respectively.

Finally, define

$$
W(y, z) = \begin{cases} \psi^{-1}(y, z) & \text{if } y \neq 0, \\ 0 & \text{if } y = 0 \end{cases} \quad (12)
$$

as a Lyapunov function candidate (i.e., $W(y, z)$ is
an implicit Lyapunov function defined by the equa-
tion $V(\Lambda_r(W^{-1}(y, z))y, z) = 1$ as in [15], [1], [11],
[12]). Its time derivative along the trajectories of (2),
(9), (10) takes the form for $\xi = \Lambda_r(\psi)y \in \partial\mathcal{V}$
(hence, $\psi(\xi, z) = 1$ and $F(\xi, z) = X_0(\xi) +$
$\sum_{i=1}^m X_i(\xi)\psi^{-\nu_0+\nu_i}(\xi, z)U_i(\Lambda_r(\psi(\xi, z))\xi, z) = f(\xi, z)$,
similarly $G(\xi, z) = g(\xi, z)$):

$$
\begin{align*}
& \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) \\
&= \psi^{-1-\nu_0}(y, z) \frac{\frac{\partial V(\xi, z)}{\partial y} F(\xi, z) + \frac{\partial V(\xi, z)}{\partial z} G(\xi, z)}{\frac{\partial V(\xi, z)}{\partial y} \mathcal{H} \xi} \\
&= \psi^{-1-\nu_0}(y, z) \frac{\frac{\partial V(\xi, z)}{\partial y} f(\xi, z) + \frac{\partial V(\xi, z)}{\partial z} g(\xi, z)}{\frac{\partial V(\xi, z)}{\partial y} \mathcal{H} \xi} \\
&\leq -\psi^{-1-\nu_0}(y, z) \frac{\alpha_3 (\|\xi\|)}{\frac{\partial V(\xi, z)}{\partial y} \mathcal{H} \xi}
\end{align*}
$$

for all $(y, z) \in \Omega$. Since $\frac{\partial V(\xi, z)}{\partial y} \mathcal{H} \xi \le \eta (\|\xi\|)$ due to the
properties of $V$ for some $\eta \in \mathcal{K}$, the latter estimate implies
SIuOS property for (2), (9), (10).

Therefore, the following result has been proven:

**Theorem 5.** Let (8) and assumptions 2 with (11) and 3, 4 be satisfied. Then the system (2) under the dynamic control (9), (10) is SIuOS, and uFTOS for $\nu_0 < 0$ or unFxTOS for $\nu_0 > 0$ in $\mathbb{R}^p \times \mathcal{Z}$ (by considering $y$ as the output).
In [1] such a theorem was obtained for $\nu_0 = 0$ only (leading to exponential stabilization, see Theorem 2).

*Remark 4.* If $X_0(y) = 0$, combining uFTOS property for $\psi^{-1}(y, z) < 1$ and unFxTOS property for $\psi^{-1}(y, z) > 1$ by switching the homogeneity degree $\nu_0$ in the control (9), (10) it is possible to get uFxTOS with the settling time uniformly bounded in $\mathbb{R}^p \times \mathcal{Z}$ (preserving the continuity of the control).

The condition (8) can be relaxed by using the concept of generalized linear homogeneity [7], [43] instead of the weighted one (all proofs can be repeated with minor changes).

Using the Lyapunov function $W$ it is possible to extend these results to the systems with exogenous inputs and input-to-output stability [21]. The dynamic extensions of (2) can also be investigated; this extension is important as models of nonholonomic mechanical systems can be transformed into dynamic extensions of drift-free systems that typically only capture the kinematics [44]. Another way of development consists of considering in Assumption 4 the Lyapunov functions $V$ with semi-definite derivatives as in [1].

**Assumption 5.** There exist locally Lipschitz continuous functions $U_i: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}$, $i = \overline{1,m}$ and a smooth function $V: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$ such that for some $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$,

$$ \alpha_1(\|y\|) \le V(y, z) \le \alpha_2(\|y\| + \|z\|), \\ \frac{\partial V(y, z)}{\partial y} f(y, z) + \frac{\partial V(y, z)}{\partial z} g(y, z) \le 0 $$

hold for all $y \in \mathcal{X}$, $z \in \mathcal{Z}$, and the closed-loop system is uOS with respect to the output $y$ at least locally in $\mathcal{X} \times \mathcal{Z}$.

There are two key differences here in comparison with Assumption 4: the upper bound of $V$ depends on $z$, and the derivative of $V$ is negative semi-definite. These relaxations allow us to substantiate uOS property (as opposite to SIuOS previously) for the controlled system.

**Theorem 6.** Let (8), Assumption 2 with (11) and assumptions 3, 5 be satisfied. Then the system (2) under the dynamic control (9), (10) having unique solutions is uOS, and uFTOS for $\nu_0 < 0$ or unFxTOS for $\nu_0 > 0$ in $\mathbb{R}^p \times \mathcal{Z}$ (by considering $y$ as the output).

*Proof.* All steps and constructions of the proof of Theorem 5 can be repeated with the following differences: i) the condition (8) saves its role in the well-posedness of $\psi$, but now $\lim_{\|y\| \to 0} \psi(y, z) \le +\infty$; ii) the implicitly defined Lyapunov function $W$ in (12) is positive definite in $y$, but radially unbounded in the full state vector including $y$ and $z$; iii) due to Assumption 5, we have

$$ \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) \le 0 $$

for all $(y, z) \in \Omega$, therefore, all solutions of (2), (9), (10) are bounded and defined for $t \ge 0$ considering initial conditions in $\mathbb{R}^p \times \mathcal{Z}$. Assume that $y(t) \ne 0$, then any trajectory admits $\omega$-limit set [4], and $W(y(t), z(t)) = \ell > 0$ for all $t \ge 0$

on such a $\omega$-limit set. Dilating this solution as $\Lambda_r(\ell^{-1})y(0)$ we get that the new one lives in $\partial\mathcal{V}$, hence, having $y(t)$ asymptotically converging to zero. The same holds for the original trajectory on $\omega$-limit set due to (4). $\square$

## VI. CONTROL IMPLEMENTATION

It is a well-known fact that for simulation or implementation, the finite/fixed-time converging systems (controllers or observers) need special discretization approaches [45]–[48]. For the control laws based on implicit Lyapunov functions, efficient sampled realization schemes were proposed in [11], [12]; such schemes can also be developed for the control laws proposed in this paper.

To this end, assume that a sequence of times $0 = t_0, t_1, \dots, t_j, \dots$ is given such that $t_{j+1} > t_j$ for all $j \ge 0$ with $\lim_{j \to +\infty} t_j = +\infty$, and the function $\psi$ is calculated (sampled) only at these instants such that the system (2), (9), (10) is implemented as follows:

$$ \begin{aligned} \dot{y}(t) &= X_0(y(t)) + \sum_{i=1}^{m} X_i(y(t)) u_{i,j}(t), \quad t \in [t_j, t_{j+1}), \\ u_{i,j}(t) &= \begin{cases} \psi_j^{-\nu_0+\nu_i} U_i(\Lambda_r(\psi_j)y(t), z(t)) & \text{if } y(t) \ne 0, \\ 0 & \text{if } y(t) = 0 \end{cases}, \quad (13) \\ \dot{z}(t) &= \begin{cases} \psi_j^{-\nu_0} g(\Lambda_r(\psi_j)y(t), z(t)) & \text{if } y(t) \ne 0, \\ 0 & \text{if } y(t) = 0, \end{cases} \end{aligned} $$

where $\psi_j = \psi(y(t_j), z(t_j))$ is obtained as a solution of the equation (7), i.e.,

$$ V(\Lambda_r(\psi_j)y(t_j), z(t_j)) = 1, $$

for all $(y(t_j), z(t_j)) \in \Omega$. A part of the results of Theorem 5 is preserved for this system:

**Theorem 7.** Let (8) and assumptions 2 with (11) and 3, 4 be satisfied with $\mathcal{X} = \mathbb{R}^p$. Then the system (13) is SIuOS.

*Proof.* On each time interval $[t_j, t_{j+1})$, $j \ge 0$, define for (13) an auxiliary explicit Lyapunov function

$$ V_j(y, z) = V(\Lambda_r(\psi_j)y, z), $$

which, due to Assumption 4, is positive definite and radially unbounded in $y$. Its time derivative along the trajectories of the system (13) takes the form for $y \ne 0$:

$$ \begin{aligned} \dot{V}_j(y, z) &= \frac{\partial V_j(y, z)}{\partial y} \dot{y} + \frac{\partial V_j(y, z)}{\partial z} \dot{z} \\ &= \frac{\partial V(\Lambda_r(\psi_j)y, z)}{\partial z} \psi_j^{-\nu_0} g(\Lambda_r(\psi_j)y, z) \\ &\quad + \frac{\partial V(\Lambda_r(\psi_j)y, z)}{\partial y} \Lambda_r(\psi_j) \\ &\quad \times [X_0(y) + \sum_{i=1}^{m} X_i(y) \psi_j^{-\nu_0+\nu_i} U_i(\Lambda_r(\psi_j)y, z)] \end{aligned} $$
$$ = \psi_j^{-\nu_0} \left( \frac{\partial V(\Lambda_r(\psi_j)y, z)}{\partial z} g(\Lambda_r(\psi_j)y, z) + \frac{\partial V(\Lambda_r(\psi_j)y, z)}{\partial y} \times [X_0(\Lambda_r(\psi_j)y) + \sum_{i=1}^{m} X_i(\Lambda_r(\psi_j)y)U_i(\Lambda_r(\psi_j)y, z)] \right). $$

Recall that $(\Lambda_r(\psi_j)y(t_j), z(t_j)) \in \partial\mathcal{V} \subset \mathcal{X} \times \mathcal{Z}$, hence $\dot{V}_j(y(t_j), z(t_j)) \le -\psi_j^{-\nu_0} \alpha_3 (\|\Lambda_r(\psi_j)y(t_j)\|) < 0$, and the property $(\Lambda_r(\psi_j)y(t), z(t)) \in \mathcal{X} \times \mathcal{Z}$ holds for all $t \in [t_j, t_{j+1})$ and results in

$$ \dot{V}_j \le -\psi_j^{-\nu_0} \alpha_3 (\|\Lambda_r(\psi_j)y\|) \le -\psi_j^{-\nu_0} \alpha_3 \circ \alpha_2^{-1}(V_j), $$

which implies strict decreasing of $V_j$ on the interval $[t_j, t_{j+1})$. Therefore,

$$ \begin{aligned} 1 &= V(\Lambda_r(\psi(y(t), z(t)))y(t), z(t)) \\ &= V(\Lambda_r(\psi(y(t_j), z(t_j)))y(t_j), z(t_j)) = V_j(y(t_j), z(t_j)) \\ &> V_j(y(t), z(t)) = V(\Lambda_r(\psi(y(t_j), z(t_j)))y(t), z(t)) \end{aligned} $$

for all $t \in (t_j, t_{j+1})$ by definition of $\psi$ and $V_j$. Hence, due to (8), the relations $\psi(y(t), z(t)) > \psi(y(t_j), z(t_j))$ or $W(y(t), z(t)) < W(y(t_j), z(t_j))$, where $W$ is defined by (12), are satisfied for all $t \in (t_j, t_{j+1})$. Consequently, $W(y(t_j), z(t_j))$ is a monotonously decreasing sequence for all $j \ge 0$ for the $\psi$-sampled system (13).

It is left to show that $W(y(t_j), z(t_j))$ is decreasing to zero. By contradiction, assume that there exists $W_* > 0$ such that $W(y(t_j), z(t_j)) \ge W_*$ for all $t \ge 0$ (and without loss of generality let $W_*$ be the supremum of such $W_*$). It implies that for any $\epsilon > 0$ there exists $j_\epsilon > 0$ such that $|\psi_j^{-1} - W_*| < \epsilon$ for all $j \ge j_\epsilon$. We need to demonstrate that for any such level $W_*$ the respectively defined local Lyapunov function $V_*(y, z) = V(\Lambda_r(W_*^{-1})y, z)$ is decreasing for a sufficiently small $\epsilon$ and big $j \ge j_\epsilon$, hence, $W_* > 0$ does not exist. For this purpose note that $g$ and the controls $U_i$ and $K_i$, $i = 1, \dots, m$ are continuous functions in $\mathbb{R}^p \times \mathcal{Z}$, hence, for any $\delta > 0$ there is $\epsilon_\delta > 0$ such that for all $(y, z) \in \Omega_*:

$$ \begin{aligned} |g(\Lambda_r(\psi_j)y, z) - g(\Lambda_r(W_*^{-1})y, z)| &\le \delta, \\ |U_i(\Lambda_r(\psi_j)y, z) - U_i(\Lambda_r(W_*^{-1})y, z)| &\le \delta \end{aligned} $$

for all $j \ge j_\epsilon$ and all $i = 1, \dots, m$, where $\Omega_* = \{(y, z) \in \Omega : 1 \le V_*(y, z) \le 2\}$ is a compact set. Thus, for any $j \ge 0$ with $(y(t_j), z(t_j)) \in \Omega_*$, i.e., in a vicinity of the set $W(y, z) = W_*$, we get

$$ \begin{aligned} \dot{V}_* &= \frac{\partial V(\Lambda_r(W_*^{-1})y, z)}{\partial z} \psi_j^{-\nu_0} g(\Lambda_r(\psi_j)y, z) \\ &\quad + \frac{\partial V(\Lambda_r(W_*^{-1})y, z)}{\partial y} \Lambda_r(W_*^{-1}) \\ &\quad \times [X_0(y) + \sum_{i=1}^{m} X_i(y) \psi_j^{-\nu_0+\nu_i} U_i(\Lambda_r(\psi_j)y, z)] \\ &\le -W_*^{\nu_0} \alpha_3 \circ \alpha_2^{-1}(V_*) + e_j(y, z), \end{aligned} $$

$$ \begin{aligned} e_j(y, z) &= \frac{\partial V(\Lambda_r(W_*^{-1})y, z)}{\partial z} (\psi_j^{-\nu_0} g(\Lambda_r(\psi_j)y, z) \\ &\quad -W_*^{\nu_0} g(\Lambda_r(W_*^{-1})y, z)) \\ &\quad + \frac{\partial V(\Lambda_r(W_*^{-1})y, z)}{\partial y} \Lambda_r(W_*^{-1}) \sum_{i=1}^{m} X_i(y) \\ &\quad \times [\psi_j^{-\nu_0+\nu_i} U_i(\Lambda_r(\psi_j)y, z) - W_*^{\nu_0-\nu_i} U_i(\Lambda_r(W_*^{-1})y, z)] \end{aligned} $$

for $t \in [t_j, t_{j+1})$. Note that by definition $V_*(y(t), z(t)) \ge 1$ while $W(y(t), z(t)) \ge W_*$, then $-W_*^{\nu_0} \alpha_3 \circ \alpha_2^{-1}(V_*) \le -W_*^{\nu_0} \alpha_3 \circ \alpha_2^{-1}(1) < 0$ and there exist $\epsilon > 0$ and $\delta > 0$ sufficiently small such that $|e_j(y, z)| \le \frac{W_*^{\nu_0} \alpha_3 \circ \alpha_2^{-1}(1)}{2}$ for all $j \ge \max\{j_\epsilon, j_{\epsilon_\delta}\}$ with $(y, z) \in \Omega_*$. It implies that $V_*(y(t), z(t))$ is strictly decreasing for all such intervals $[t_j, t_{j+1})$ and the level $W_*$ has to be transgressed by $W(y(t_j), z(t_j))$. $\square$

Note that above there is no constraint on the lower or upper bound of sampling intervals $t_{j+1} - t_j, j \ge 0$. The intuition behind the absence of the upper bound is that $(\Lambda_r(\psi(y(t_j), z(t_j)))y(t), z(t)) \in \mathcal{V}$ by definition for $t = t_j$, while for $t > t_j$ this inclusion holds due to decay of $V_j$ on the interval $[t_j, t_{j+1})$.

The main drawback of the considered sampled implementation (13) consists in the need of calculation of $\psi_j$, which can be computationally costly. According to the established homogeneity of $\psi^{-1}$, the following relations are satisfied:

$$ \chi_1 \|y\|_r \le \psi^{-1}(y, z) \le \chi_2 \|y\|_r $$

for all $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$, where $\chi_1 = \inf_{(y,z) \in S_r \times \mathcal{Z}} \psi^{-1}(y, z)$ and $\chi_2 = \sup_{(y,z) \in S_r \times \mathcal{Z}} \psi^{-1}(y, z)$ with $S_r = \{y \in \mathbb{R}^p : \|y\|_r = 1\}$. Then, it is straightforward to check that all steps of the proof of Theorem 7 can be repeated if $\psi_j$ is replaced by its lower estimate $\frac{1}{\chi_2 \|y(t_j)\|_r}$, which is much easier to calculate.

**Corollary 1.** Let all conditions of Theorem 7 be verified and $\psi_j = \frac{1}{\chi_2 \|y(t_j)\|_r}$ in (13) for $t_{j+1} \in \{t > t_j : \|y(t)\|_r < \|y(t_j)\|_r\}$, then this system is SIuOS.

Developing the results of [49], it may be possible to show that (13) has actually accelerated convergence rates for $\nu_0 < 0$ close to the origin and for $\nu_0 > 0$ far outside, but for the sampled system it will be a hyperexponential convergence (that means an asymptotic convergence, but faster than any exponential). Completing this analysis is an interesting line for future research.

## VII. CONTROL OF A NONHOLONOMIC INTEGRATOR

For illustration purposes consider an example of (2) with $p = 3$ and $m = 2$:

$$ \begin{aligned} X_0(y) &= 0, X_1(y) = \begin{bmatrix} 1 & 0 & y_2 \end{bmatrix}^\top, \\ X_2(y) &= \begin{bmatrix} 0 & 1 & 0 \end{bmatrix}^\top, \end{aligned} $$
which verifies Assumption 2 for $\mathbf{r} = [1, 1, 2]$, $\nu_1 = \nu_2 = -1$ and $\nu_0 \ge -1$. Take

$$g(y, z) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} z$$

with $z \in \mathbb{R}^2$ satisfying Assumption 3 for $\mathcal{Z} = \{z \in \mathbb{R}^2 : \|z\| = 1\}$. In [35] (see also [1]) it has been shown that the control law

$$\begin{aligned} U_1(y, z) &= -y_1 - y_2(y_3 - (y_2 - y_3z_1)z_1), \\ U_2(y, z) &= -y_2 + y_3(z_1 + z_2) \end{aligned}$$

stabilizes the system at the origin globally with asymptotic convergence rate. The global Lyapunov function is given by

$$V(y, z) = \frac{1}{2} [y_1^2 + (y_2 - y_3z_1)^2 + y_3^2],$$

where

$$\dot{V} = -(y_2 - y_3z_1)^2 - (y_1 + y_2(y_3(1 + z_1^2) - y_2z_1))^2.$$

Then Assumption 4 is true for $\mathcal{X} = \mathbb{R}^3$ together with the condition (8):

$$\frac{\partial V(y, z)}{\partial y} = \begin{bmatrix} y_1 & y_2 - y_3z_1 & (1 + z_1^2)y_3 - z_1y_2 \end{bmatrix},$$

$$\mathcal{H} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix},$$

$$\frac{\partial V(y, z)}{\partial y} \mathcal{H} y = y_1^2 + y_2^2 + 2(1 + z_1^2)y_3^2 - 3z_1y_2y_3 \\ \ge y_1^2 + \frac{1}{4}y_2^2 + (2 - z_1^2)y_3^2 > 0, \forall y \in \mathcal{X} \setminus \{0\}, z \in \mathcal{Z}.$$

Therefore, the conditions of Theorem 5 are verified, for $\nu_0 = -0.5$ providing uFTOS or for $\nu_0 = 0.5$ ensuring unFxTOS, for the control (9), (10) given for $y \ne 0$:

$$K_1(y, z) = \psi^{-\nu_0}(y, z)(-y_1 - \psi(y, z)y_2)(\psi(y, z)y_3 - (y_2 - \psi(y, z)y_3z_1)z_1)),$$

$$K_2(y, z) = \psi^{-\nu_0}(y, z)(-y_2 + \psi(y, z)y_3(z_1 + z_2)),$$

$$G(y, z) = \psi^{-\nu_0}(y, z) \begin{bmatrix} z_2 \\ -z_1 \end{bmatrix},$$

where $\psi = \psi(y, z)$ is the solution of the equation:

$$\psi^2[y_1^2 + (y_2 - \psi y_3 z_1)^2 + \psi^2 y_3^2] = 2.$$

The results of simulations in MATLAB of the closed-loop sampled system (13) by applying the explicit Euler method with the discretization step $h = 0.01$, with the sampling instants $t_j = k_s hj$ for $j \in \mathbb{Z}$ and $k_s \in \{5, 50\}$, are presented in Figs. 1 and 2 for $\nu_0 = -0.5$ and $\nu_0 = 0.5$, respectively. On the plots, the state norm $\|y\|_r$ (with $\varpi = 1$) in logarithmic scale, the generator output $z_1$ and the controls $u$ are shown versus the time $t$. In order to guarantee the numeric stability of the Euler method, the maximal value of $\psi$ has been bounded by $h^{-2}$. As we can conclude from these results, the state norm convergence is accelerated and the frequency

<figure><img src="image_1.png" /><figcaption>Figure 1: The results of finite-time stabilization for $\nu_0 = -0.5: \|y\|_r, z_1$ and $u$ versus time $t$</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Figure 2: The results of nearly fixed-time stabilization for $\nu_0 = 0.5: \|y\|_r, z_1$ and $u$ versus time $t$</figcaption></figure>

of oscillation of the generator variables is increasing with time while the controls are converging to zero.

Finally, the fixed-time stabilizing control was tested, then the sign of $\nu_0 \in \{-0.5, 0.5\}$ was changed depending on the value of $\psi$ being smaller than one or not, and the results are shown in Fig. 3 for the noise-free setting, and in Fig. 4 for the case when the measurements of the variable $y$ are available with a uniformly distributed noise of amplitude $\frac{1}{20k_s}$.

## VIII. CONCLUSION

Considering systems that are homogeneous with respect to a part of the state variables, several results extending the conventional theory of homogeneous systems to such a scenario are proposed: scaling of trajectories, global stability implications from local estimates or from existence

<figure><img src="image_3.png" /><figcaption>Figure 3: The results of noise-free fixed-time stabilization for $\nu_0 \in \{-0.5, 0.5\}: \|y\|_r, z_1$ and $u$ versus time $t$</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 4: The results of fixed-time stabilization with noise for $v_0 \in \{-0.5, 0.5\}: \|y\|_r, z_1$ and $u$ versus time $t$</figcaption></figure>

of (strictly) positively invariant set, existence of a homogeneous Lyapunov functions, finite-time or nearly fixed-time convergence for negative or positive degrees of homogeneity, respectively. Next, the method from [1] is extended to obtain a feedback ensuring finite/fixed-time convergences from a locally asymptotically stabilizing counterpart. For a sampled discretization or implementation of the control a simple method is given. The obtained results are illustrated with a nonholonomic integrator. Extensions of these findings to the classes of systems lacking Lipschitz continuity not only at the origin in the output variables, and also analysis of robustness with respect to external disturbances and delays, can be considered as directions for future research.

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<figure><img src="image_5.png" /></figure>

**Ilya V. Kolmanovsky** received his Ph.D. degree in 1995 from the Department of Aerospace Engineering at the University of Michigan where he is presently a full professor. Before joining the University of Michigan in January 2010, he was with Ford Research and Advanced Engineering in Dearborn, Michigan for close to 15 years. He is a Fellow of IEEE, IFAC, and National Academy of Inventors. He is also an Associate Fellow of AIAA, a past recipient of the Donald P. Eckman Award of American Automatic Control Council, of 2002 and 2016 IEEE Transactions on Control Systems Technology Outstanding Paper Awards and of several technical achievement, innovation and publication awards of Ford Research and Advanced Engineering. He presently serves as an Editor-in-Chief of IEEE Transactions on Control Systems Technology.

<figure><img src="image_6.png" /></figure>

**Denis Efimov** received a Ph.D. degree in Automatic Control from the Saint-Petersburg State Electrical Engineering University (Russia) in 2001 and a Dr.Sc. degree in 2006 from the Institute for Problems in Mechanical Engineering RAS (Saint-Petersburg, Russia). From 2006 to 2011, he worked in the L2S CNRS (Supelec, France), the Montréal Institute (University of Liege, Belgium), and the IMS CNRS lab (University of Bordeaux, France). In 2011, he joined Inria. Since 2018 he has been the scientific head of the Valse team

of Inria. He co-authored over 200 scientific articles in top peer-reviewed journals and coordinated several scientific projects. He is a member of several IFAC TCs and a Senior Member of IEEE. He is an Associate Editor for IEEE Transactions on Automatic Control, IFAC Journal on Nonlinear Analysis: Hybrid Systems and Automatica.