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

Denis Efimov, Ilya Kolmanovsky

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Denis Efimov, Ilya Kolmanovsky. On homogeneity with respect to a part of variables and accelerated regulation. IEEE Transactions on Automatic Control, inPress. hal-05102779

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

Denis Efimov, Ilya Kolmanovsky

*Abstract*—This 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 uses the implicit Lyapunov function framework.

## 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. 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 [1]–[3] and references therein. 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 [4] for the nonlinear time-varying systems (where asymptotic non-exponential stabilizers were converted to exponential stabilizers) and in [5]–[9] for linear autonomous models (where finite/fixed-time convergence has been obtained). In [4], [6]–[9] the implicitly defined Lyapunov functions have been used.

In this work, the method of [4] 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. 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. 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 [10], [11].

The paper is organized as follows. The 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.

Preliminary results have been included into our conference work [12]. As compared to [12], 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. However, the simulation example is given only in [12], and the analysis of digital implementation is included in preprint [13].

### Notation

* $\mathbb{R}_+ = \{x \in \mathbb{R} : x \ge 0\}$, $\mathbb{R}$ 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 $\|d\|_{[t_0, t_1)} = \ess\sup_{t \in [t_0, t_1]} \|d(t)\|$, then $\|d\|_{\infty} = \|d\|_{[0, +\infty)}$ and the set of $d$ with the property $\|d\|_{\infty} < +\infty$ we further denote as $\mathcal{L}_{\infty}^s$.

* A continuous function $\alpha: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}$ if $\alpha(0) = 0$ and it is strictly increasing; $\alpha \in \mathcal{K}$ belongs to the class $\mathcal{K}_{\infty}$ if 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 $s > 0$.

* 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, ..., $n$ is denoted by $\overline{1, n}$.

## II. PRELIMINARIES

### A. Uniform output stability

Consider a non-autonomous differential equation:

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

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.
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 $d \in D \subseteq \mathcal{L}_\infty$ ($D$ 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.** [10] 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.** [10] 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 for all $x_0 \in \mathbb{R}^n$ and $d \in D$:

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

**Theorem 1.** [10] 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:

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

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

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

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

The given Lyapunov conditions have been developed in [11] for the case of non-UBIBS systems. In the papers [14], [15] 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.

**Definition 3.** [15] 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 $\sup_{\rho>0} T_\rho = +\infty$. The respective Lyapunov characterizations of the output stability properties with accelerated convergence can be found in [14]–[16]. These stability concepts can also be defined on a finite interval of time $[0, T(x_0))$.

## B. Homogeneity

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

**Definition 4.** [3], [17] 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.

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 for some $\underline{\sigma}_r, \bar{\sigma}_r \in \mathcal{K}_\infty$ [18]:

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

## 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:

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

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 $j = 0, \dots, m$. If $X_0(y) = 0$, then (2) becomes driftless, such models frequently appear in the study
of mechanical systems with nonholonomic constraints [4]. 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 [19], hence, frequently discontinuous [20] 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 [4], [21], [22]. 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 $x(t) = (y(t), z(t))$.

In [4] 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 [23], [5], or using implicit Lyapunov functions as exemplified in later works [6]–[9]. 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 $\mathbf{r} = [r_1, \dots, r_p]$ with $r_i > 0$, $i = \bar{1}, \bar{p}$ such that

$$ F(\Lambda_r(\lambda)y, z) = \lambda^\nu \Lambda_r(\lambda) F(y, z), \quad G(\Lambda_r(\lambda)y, z) = \lambda^\nu G(y, z) $$ 
 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 [24], 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 $\tilde{\beta}(q, t) = \underline{\sigma}_r^{-1} \circ \beta(\bar{\sigma}_r(q), t)$ 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 [24], a compact set $\mathcal{A} \subset \mathbb{R}^p$ is called partially positively invariant for the system (3) if $y_0 \in \mathcal{A}$ and $z_0 \in \mathbb{R}^s$ implies that $Y(t, y_0, z_0) \in \mathcal{A}$ for all $t \ge 0$. Moreover, $\mathcal{A}$ is called strictly partially positively invariant if $Y(t, y_0, z_0) \in \text{int}(\mathcal{A})$ for all $t > 0$.

**Lemma 2.** Under Assumption 1, if the system (3) has a compact partially positively invariant set $\mathcal{A} \subset \mathbb{R}^p$ with $0 \in \text{int}(\mathcal{A})$, 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 $\mathcal{A}$ 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 $\sigma \in \mathcal{K}$ and $\sigma_0 \ge 0$, then it is uOS.

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

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

for all $\tilde{y}_0 \in \mathcal{R}$ and $z_0 \in \mathbb{R}^s$, where $R = \sup_{y \in \mathcal{A}} \|y\|_r$ by partial positive invariance of $\mathcal{A}$. 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:

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

which implies that the following estimate is satisfied:

$$ \|Y(t, y_0, z_0)\|_r \le \kappa \|y_0\|_r, \forall t \ge 0, \kappa = \frac{R}{\rho} $$

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

$$ \|Y(t, \tilde{y}_0, z_0)\|_a < \|\tilde{y}_0\|_a, \forall t \ge 0 $$

for all $\tilde{y}_0 \in \partial\mathcal{A}$ and $z_0 \in \mathbb{R}^s$ for a suitably defined norm $\|\cdot\|_a$, and utilizing this contraction and dilating those initial conditions by $\lambda \in (0, 1)$ we get that the same property is satisfied for all $\tilde{y}_0 \in \mathcal{A} \setminus \{0\}$, which 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 [3], [24], [25]. The same characteristics 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 $\sigma \in \mathcal{K}$ 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 $\mathcal{KL}$, there exists a (locally bounded) map $T: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$ such that

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

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 $\lambda > 0$,

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

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

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

which 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 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.):

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

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 a compact set $\mathcal{Z} \subset \mathbb{R}^s$, then global output $y$ stability and convergence are preserved but for the initial conditions in $\mathcal{Z}$. Moreover, the only properties of $z(t)$ are existence of solutions and their boundedness, then requirements on continuity of $G$ can be relaxed.

The results above extend [24] to the case of partial homogeneity formulated in Assumption 1. It is also possible to extend the result of [17], [26] on the existence of a partially homogeneous Lyapunov function:
**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 $c_1, c_2, c_3 > 0, \mu > kr_{\max}$ and all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$:

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

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

Inversely, if (3) is SIuOS, then according to [10], [14], 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 $\alpha_3 \in \mathcal{K}$,

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

hold for all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$ (the norms $\|\cdot\|$ and $\|\cdot\|_r$ are interchangeable). Following [26], 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 $y \in \mathbb{R}^p$, $z \in \mathbb{R}^s$. 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 $z \in \mathbb{R}^s$,

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

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 [26]. 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 [21]. A similar 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 $z \in \mathbb{R}^s$:

$$\begin{aligned} & \tilde{\alpha}_1(\|y\|) \le W(y, z) \le \tilde{\alpha}_2(\|x\|), \\ & W(y, z) \ne 0 \Rightarrow \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) < 0, \\ & W(\Lambda_r(\lambda)y, z) = \lambda^\mu W(y, z), \forall \lambda > 0, \end{aligned}$$

where $x = [y^\top z^\top]^\top$ and $\mu > kr_{\max}$.

*Proof.* The existence of such a Lyapunov function $W$ implies uOS (see Lemma A.5 in [10]).
Inversely, if the system (3) is uOS and UBIBS, then according to [10], 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 $\alpha_3 \in \mathcal{K}\mathcal{L}$,

$$ \alpha_1 (\|y\|) \le V(y, z) \le \alpha_2 (\|x\|), \\ \frac{\partial V(y, z)}{\partial y} F(y, z) + \frac{\partial V(y, z)}{\partial z} G(y, z) \le -\alpha_3 (V(y, z), \|x\|) $$

hold for all $y \in \mathbb{R}^p$, $z \in \mathbb{R}^s$ and $x = [y^\top z^\top]^\top \in \mathbb{R}^n$. Following [26], define $a : \mathbb{R} \to \mathbb{R}_+$ as in (5). Take any $\mu > \max\{0, -\nu\}$ and consider a candidate Lyapunov function:

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

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

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

for any $y \in \mathbb{R}^p \setminus \{0\}$ and $z \in \mathbb{R}^s$. Since $\lim_{\|y\| \to +\infty} \bar{l}_y = 0$ and $W(y, z) = 0$ whenever $V(z, y) = 0$, there are $\tilde{\alpha}_1, \tilde{\alpha}_2 \in \mathcal{K}_\infty$ such that $\tilde{\alpha}_1 (\|y\|) \le W(y, z) \le \tilde{\alpha}_2 (\|x\|)$ for all $y \in \mathbb{R}^p$, $z \in \mathbb{R}^s$ and $x = [y^\top z^\top]^\top \in \mathbb{R}^n$. For all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$,

$$ \begin{aligned} & \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) \\ &= \int_{0}^{+\infty} \frac{a'(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+\nu+1}} \left[ \frac{\partial V(\Lambda_r(\lambda)y, z)}{\partial y} \lambda^\nu \Lambda_r(\lambda) F(y, z) \right. \\ & \qquad \left. + \frac{\partial V(\Lambda_r(\lambda)y, z)}{\partial z} \lambda^\nu G(y, z) \right] d\lambda \\ &= \int_{0}^{+\infty} \frac{a'(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+\nu+1}} \left[ \frac{\partial V(\Lambda_r(\lambda)y, z)}{\partial y} F(\Lambda_r(\lambda)y, z) \right. \\ & \qquad \left. + \frac{\partial V(\Lambda_r(\lambda)y, z)}{\partial z} G(\Lambda_r(\lambda)y, z) \right] d\lambda \\ & \le - \int_{0}^{+\infty} \frac{a'(V(\Lambda_r(\lambda)y, z))}{\lambda^{\mu+\nu+1}} \alpha_3(V(\Lambda_r(\lambda)y, z), \\ & \qquad \qquad \| (\Lambda_r(\lambda)y, z) \|) d\lambda < 0 \end{aligned} $$

for $V(y, z) \neq 0$, which is equivalent to $W(y, z) \neq 0$. The remaining steps are the same as in proof of Theorem 3. $\square$

## V. CONTROL DESIGN

We now exploit theoretical findings of Section IV to accelerate time-varying stabilizing feedbacks (considering non-zero homogeneity degree) for a class of systems (2). The proposed approach extends the ideas of [4]. To this end we need the following assumptions:

**Assumption 2.** There exists a vector of weights $\mathbf{r} = [r_1, ..., r_p]$ with $r_i > 0$, $i = \bar{1}, \bar{p}$ such that the property

$$ X_j(\Lambda_r(\lambda)y) = \lambda^{\nu_j} \Lambda_r(\lambda) X_j(y), \quad \nu_j \ge -r_{\min}, \quad j = \overline{0, m} $$

holds for all $\lambda > 0$ and $y \in \mathbb{R}^p$.

Under this assumption, each vector field $X_j$, $j = \overline{0, m}$ in (2) is $\mathbf{r}$-homogeneous of a possibly different degree $\nu_j$.

**Assumption 3.** There is a locally Lipschitz continuous vector field $g : \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}^s$ such that the system

$$ \dot{z}(t) = g(y(t), z(t)), \quad t \ge 0, \qquad (6) $$

with $z(t) \in \mathbb{R}^s$ and $y \in \mathcal{L}_\infty$, admits a compact invariant set $\mathcal{Z} \subset \mathbb{R}^s$.

A signal generator (6) defined in Assumption 3 will be used to induce the time dependence in the controls:

**Assumption 4.** There exist locally Lipschitz continuous functions $U_i : \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}$, $i = \bar{1}, \bar{m}$ and a smooth function $V : \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$ such that for $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ and $\alpha_3 \in \mathcal{K}$,

$$ \alpha_1 (\|y\|) \le V(y, z) \le \alpha_2 (\|y\|), \\ \frac{\partial V(y, z)}{\partial y} f(y, z) + \frac{\partial V(y, z)}{\partial z} g(y, z) \le -\alpha_3 (\|y\|) $$

hold for all $y \in \mathcal{X} \subset \mathbb{R}^p$ (with $0 \in \text{int}(\mathcal{X})$), $z \in \mathcal{Z}$, where $f(y, z) = X_0(y) + \sum_{i=1}^m X_i(y) U_i(y, z)$.

This assumption states that for the system (2), (6) there exist non-autonomous (dependent on $z$) control laws $U_i$, $i = \bar{1}, \bar{m}$ that render the closed-loop dynamics SIuOS with respect to the output $y$ at least locally in $\mathcal{X} \times \mathcal{Z}$ with a known Lyapunov function $V$.

Without loosing generality, assume that the set $\mathcal{V} = \{y \in \mathbb{R}^p, z \in \mathcal{Z} : V(y, z) \le 1\}$ belongs to the interior of $\mathcal{X} \times \mathcal{Z}$. Define a map $\psi : \Omega \to (0, +\infty)$, where $\Omega = \mathbb{R}^p \setminus \{0\} \times \mathcal{Z}$, as a solution of the equation:

$$ V(\Lambda_r(\psi(y, z))y, z) = 1. \qquad (7) $$

Such a $\psi(y, z)$ always exists since $V$ is assumed to be positive definite and radially unbounded in $y$, while the transformation $\Lambda_r(\psi(y, z))y$ linearly scales the homogeneous norm of $y$ (i.e., $\|\Lambda_r(\psi(y, z))y\|_r = \psi(y, z)\|y\|_r$) and the homogeneous norm is equivalent to the standard one $\| \cdot \|$. According to the Implicit function Theorem, $\psi$ is locally uniquely defined provided that

$$ \sum_{k=1}^{p} \frac{\partial V(\Lambda_r(\psi(y, z))y, z)}{\partial y_k} r_k \psi(y, z)^{r_k-1} y_k \neq 0 $$

for all $(y, z) \in \Omega$, which is equivalent to the condition

$$ \frac{\partial V(y, z)}{\partial y} \mathcal{H} y > 0, \forall (y, z) \in \partial \mathcal{V} \qquad (8) $$

with $\mathcal{H} = \text{diag}\{r_k\}_{k=1}^p$, and obviously $\psi(y, z) = 1$ for $(y, z) \in \partial \mathcal{V}$ (the sign > is chosen without loosing generality since $-\mathcal{H}$ is a Hurwitz matrix). Under (8), for any $(y, z) \in \Omega$ there is a unique continuously differentiable solution $\psi(y, z)$ since this condition implies global monotonicity of $V(\Lambda_r(\psi)y, z)$ in $\psi$. 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., $\psi^{-1}$ 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

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

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 accelerator 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). To calculate the value $\psi(y, z)$, a bisection algorithm from [27] can be used. Typically, the required number of iterations to achieve a reasonable accuracy in $\psi$ 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 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). 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 equation $V(\Lambda_r(W^{-1}(y, z))y, z) = 1$ as in [23], [4], [6], [7]). 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)):

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

for all $(y, z) \in \Omega$. Since $\frac{\partial V(\xi, z)}{\partial y} \mathcal{H}\xi \leq \eta(\|\xi\|)$ 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 [4] 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 [3], [28].

Another way of development consists of considering in Assumption 4 the Lyapunov functions $V$ with semi-definite derivatives as in [4].

**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 $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$,

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

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 $v_0 < 0$ or unFxTOS for $v_0 > 0$ 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 [29], 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). □

## VI. CONCLUSION

In this paper, several results extending the conventional theory of homogeneity to systems that are homogeneous only with respect to a part of the state variables were obtained addressing the scaling of trajectories, global stability implications from local estimates or from existence of (strictly) positively invariant set, existence of a homogeneous Lyapunov functions, and finite-time or nearly fixed-time convergence for negative or positive degrees of homogeneity, respectively. The method from [4] has been extended to obtain a feedback ensuring global finite/fixed-time convergences from a locally asymptotically stabilizing control law.

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