# On Hyperexponential Stabilization of a Class of Nonlinear Systems

Moussa Labbadi and Denis Efimov

*Abstract*— This article studies the problem of hyperexponential stabilization for a class of nonlinear systems subject to perturbations, where the nonlinear functions are unknown but satisfy a linear growth condition. Both continuous-time and discrete-time control designs are proposed. In the continuous-time setting, we employ parametric Lyapunov equations incorporating a time-varying function that diverges to infinity as $t \to \infty$. A linear time-varying controller is designed, and the closed-loop system stability is established using Lyapunov quadratic functions. In the discrete-time setting, we utilize a semi-implicit Euler method that preserves the properties of the continuous-time dynamics.

**Keywords**— Hyperexponential convergence; Stabilization of nonlinear systems; Semi-implicit Euler method

## I. INTRODUCTION

Accelerated convergence to equilibrium is crucial in many applications [1] and is typically achieved via nonlinear sliding mode or finite/fixed-time control laws, where the most solutions are obtained for linear systems in different canonical forms. If the model contains nonlinear unknown functions, then ensuring accelerated convergence presents greater challenges compared to traditional exponential, finite-time, or fixed-time convergence in controlled-linear systems [2]. Nonlinear unknown functions, which act on channels other than the control input, are common in various applications, such as sampled-data control [3], time-delay systems [4], multiagent systems [5], and systems with input saturation and time-delay [6].

In practice, accelerated stabilization problems have received significant attention in recent years. Various methods have been developed, including sliding mode control [7], [8], Lyapunov differential inequalities [9], [10], and homogeneous approaches [11], [12]. Recently, prescribed-time feedback control has emerged as a promising approach for linear and nonlinear systems, where time-varying gains grow unbounded within a fixed duration [13], [14], [15]. This method ensures uniform convergence in the presence of matched perturbations, akin to sliding mode control. Alternative concepts, such as hyperexponential convergence [16], [17], provide faster convergence rates compared to linear feedback, while avoiding the issue related with finite-time escape of the control gains to infinity. Hyperexponential convergence ensures that equilibrium is reached asymptotically at a rate faster than any exponential dynamics. As

shown in [18], output-based accelerated stabilization can be achieved for a chain of integrators. For hyperexponential and fixed-time stability of time-delay systems, see [19]. The concept of hyperexponential stability and stabilization is further discussed in [20].

For instance, hyperexponential convergence for the double integrator system was studied in [21], and extended to nth-order integrator chains with matched perturbations in [2]. However, in the presence of unknown nonlinear functions, the stability of this control was not proven. Additionally, due to the complexity of stability analysis and feedback construction, it remains unclear how to quantitatively evaluate the limitations of this approach or extend its applicability. In addition, the implicit Euler method used for discretization in [2] cannot be applied to nonlinear systems (in linear case we can get an explicit solution).

Motivated by these challenges, this paper extends the study of stabilizing feedback for a linear system subjected to perturbations, as proposed in [2], where convergence occurs faster than any exponential rate. The previous work demonstrated uniform convergence with respect to matched disturbances; however, the effect of presence of uncertain Lipschitz nonlinearities in continuous or discrete time was not addressed. In this paper, we extend the results of [2] to a class of nonlinear systems. Specifically, we adopt the approach of [22], which uses a class of time-varying parametric Lyapunov equations (PLEs), to solve the hyperexponential stabilization problem for a nonlinear system. In continuous time, we prove that a uniform hyperexponential convergence is achieved for the studied system. In discrete time, we employ a semi-implicit Euler discretization scheme to achieve hyperexponential convergence for state variables. This approach preserves the key properties of the continuous-time counterpart while ensuring boundedness for the entire state vector. Furthermore, we present a numerical example to validate the proposed control law and demonstrate its effectiveness.

## NOTATION

* The set of nonnegative real numbers is denoted as $\mathbb{R}^+ = \{x \in \mathbb{R} \mid x \ge 0\}$, where $\mathbb{R}$ represents the set of real numbers, $\mathbb{Z}$ denotes the set of integers, and $\mathbb{Z}^+ = \mathbb{Z} \cap \mathbb{R}^+$ represents the set of non-negative integers.

* The absolute value of a scalar $x \in \mathbb{R}$ is denoted by $|x|$, while $|| \cdot ||$ represents the Euclidean norm in $\mathbb{R}^n$.

* For a Lebesgue-measurable function $d: \mathbb{R}^+ \to \mathbb{R}^m$, the essential supremum norm is defined as $||d||_\infty = \ess \sup_{t \ge 0} ||d(t)||$. The space of functions $d$ satisfying $||d||_\infty < +\infty$ is denoted by $L_m^\infty$.

<footnote>*The first author gratefully acknowledges the support of the CNRS through the project ROC-MAS LA-SE DÉVELOP RELAT-O INTERNA-TIO.</footnote>

<footnote>M. Labbadi is with the Aix-Marseille University, LIS UMR CNRS 7020, 13013 Marseille, France (e-mail: moussa.labbadi@lis-lab.fr).</footnote>

<footnote>D. Efimov is with the Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France (e-mail: denis.efimov@inria.fr).</footnote>
• Given a sequence $d_k \in \mathbb{R}^m$, $k \in \mathbb{Z}^+$, we define its supremum norm as $|d|_\infty = \sup_{k \in \mathbb{Z}^+} \|d_k\|$. The set of sequences satisfying $|d|_\infty < +\infty$ is denoted as $\ell_m^\infty$.

• A continuous function $\alpha : \mathbb{R}^+ \to \mathbb{R}^+$ belongs to the class $\mathcal{K}$ if it is strictly increasing and satisfies $\alpha(0) = 0$. A function $\alpha : \mathbb{R}^+ \to \mathbb{R}^+$ is in the class $\mathcal{K}_\infty$ if $\alpha \in \mathcal{K}$ and $\alpha(s) \to \infty$ as $s \to \infty$. A continuous function $\beta : \mathbb{R}^+ \times \mathbb{R}^+ \to \mathbb{R}^+$ belongs to the class $\mathcal{KL}$ if $\beta(\cdot, \cdot) \in \mathcal{K}$ for each fixed $t \in \mathbb{R}^+$ and $\beta(s, \cdot)$ is strictly decreasing and satisfies $\lim_{t \to \infty} \beta(s, t) = 0$ for all $s > 0$.

• For any matrix $X$, we use $X^\top$ to denote its transpose, and $\lambda_i(X)$ to represent its $i$th eigenvalue (if $X$ is square). For a symmetric matrix $P$, the largest and smallest eigenvalues are denoted by $\lambda_{\max}(P)$ and $\lambda_{\min}(P)$, respectively. Moreover, $P > 0$ (resp. $P \ge 0$) indicates that $P$ is positive definite (resp. positive semidefinite).

• The identity matrix of size $n \times n$ is denoted as $I_n$.

• $\exp(1) = e$.

## II. PROBLEM STATEMENT

In this paper, we consider the following single-input nonlinear time-varying system in the presence of matched perturbation:

$$ 
\begin{aligned}
\dot{x}_1(t) &= x_2(t) + g_1(t, x, u), \\
\dot{x}_2(t) &= x_3(t) + g_2(t, x, u), \\
\vdots \\
\dot{x}_{n-1}(t) &= x_n(t) + g_{n-1}(t, x, u), \\
\dot{x}_n(t) &= u(t) + g_n(t, x, u) + d(t),
\end{aligned}
\quad (1) 
$$

where $x(t) = [x_1(t) \ x_2(t) \ \dots \ x_n(t)]^\top \in \mathbb{R}^n$ is the state vector, and $u(t) \in \mathbb{R}$ is the input. The functions $g_i(t, x, u)$ are unknown and continuous for $i = 1, \dots, n$, with $t \ge 0$. The function $d(t) \in \mathbb{R}$ represents an external disturbance acting on the system. The stabilization of the nonlinear system (1) with the nonlinearities $g_i$ satisfying for the linear growth rate condition in the state was first investigated in [23], where it was demonstrated that exponential global stabilization can be achieved via linear state feedback. Additionally, observer-based output feedback stabilization was addressed. More recently, global prescribed-time stabilization was studied in [24].

Similar to [23], we impose the following assumption:

**Assumption 1:** There exist known constants $c_{ij} \ge 0$ for $j = 1, \dots, i$ and $i = 1, \dots, n$ such that

$$ |g_i(t, x, u)| \le \sum_{j=1}^{i} c_{ij} |x_j|, \quad i = 1, \dots, n. $$

for all $t \ge 0$, $x \in \mathbb{R}^n$ and $u \in \mathbb{R}$.

Additionally, we assume:

**Assumption 2:** The disturbance $d$ is bounded, i.e., $\|d\|_\infty < +\infty$.

We first address the control design problem in continuous time before extending the approach to discrete time.

For future reference, we rewrite the system as:

$$ 
\begin{aligned}
\dot{x}(t) &= Ax(t) + b(u(t) + d(t)) + g(t, x, u), \\
\text{where } g(t, x, u) &= [g_1(t, x, u), \dots, g_n(t, x, u)]^\top, \text{ and}
\end{aligned}
 $$

$$ b = \begin{bmatrix} 0_{(n-1)\times 1} \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0_{(n-1)\times 1} & I_{n-1} \\ 0 & 0_{1\times(n-1)} \end{bmatrix}. $$

It is required to design a feedback control $u(t) = u(x(t), t)$ that ensures stabilization of $x(t)$ at the origin with a hyperexponential convergence rate, uniformly for all $d \in L_\infty^1$ and the functions $g_i(t, x, u)$ satisfying Assumption 1. Note that any controllable nonlinear single-input, single-output system can be expressed in this canonical form [23], [25], where controllability is essential for achieving accelerated stabilization.

## III. PRELIMINARIES

Standard stability concepts for non-autonomous nonlinear systems, such as asymptotic stability and input-to-state stability, are detailed in [26]. Below, we recall less conventional notions, an auxiliary lemma used in this paper, and key properties of parametric Lyapunov equations.

### A. Uniform hyperexponential stability

Consider the following non-autonomous system:

$$ \frac{dx(t)}{dt} = f(t, x(t), d(t)), \quad t \ge t_0, \quad t_0 \in \mathbb{R}^+ \quad (2) $$

where $x(t) \in \mathbb{R}^n$ is the state, $d(t) \in \mathbb{R}^m$ represents the external disturbances, $d \in L_m^\infty$, and $f : \mathbb{R}^+ \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ is a continuous function with respect to $x$ and $d$, and piecewise continuous with respect to $t$. Additionally, $f(t, 0, 0) = 0$ for all $t \in \mathbb{R}^+$. The solution of system (2) with an initial condition $x_0 \in \mathbb{R}^n$ at time $t_0 \in \mathbb{R}^+$ and a disturbance $d \in L_m^\infty$ is denoted as $\varphi(t, t_0, x_0, d)$. It is assumed that $f$ ensures the existence of solutions $\varphi(t, t_0, x_0, d)$, at least locally in forward time. The following definition introduces the uniform stability property for (2) with a hyperexponential rate of convergence.

**Definition 1:** [17], [2] Given a set $D \subset L_m^\infty$, the system (2) is termed uniformly hyperexponentially stable if there exist constant $\kappa_0 \in \mathbb{R}^+$, functions $\rho, \kappa \in \mathcal{K}_\infty$ and $\beta \in \mathcal{KL}$ such that:

$$ 
\begin{aligned}
\|\varphi(t, t_0, x_0, d)\| \le & e^{-(\kappa(t-t_0)+\kappa_0)(t-t_0)} \rho(\|x_0\|) \\
& + \beta(\|d\|_\infty, t-t_0), \quad \forall t \ge t_0
\end{aligned}
\quad (3) 
$$

for all $x_0 \in \mathbb{R}^n$, $t_0 \in \mathbb{R}^+$, and $d \in D$.

An illustrative example is:

$$ \dot{x}(t) = -(1+t)x(t) + d(t), \quad t \ge 0 \quad (4) $$

where $x(t), d(t) \in \mathbb{R}$. Let $t_0 = 0$, the solutions to (4) can be bounded by:

$$ |x(t)| \le e^{-(\frac{t^2}{2}+t)} |x(0)| + \frac{2\|d\|_\infty}{1+(\frac{t^2}{2}+t)} \quad (5) $$

for all $t \ge 0$, any $x(0) \in \mathbb{R}$, and $d \in L_m^\infty$. Thus, we can define $\rho(s) = s$, $\kappa(s) = \frac{s}{2}$, $\kappa_0 = 1$, and $\beta(s, t) = \frac{2s}{1+(\frac{t^2}{2}+1)t}$.
In this definition, the hyperexponential rate of convergence is required only for the initial conditions, whereas uniformity is understood as independence from the exogenous input $d \in D$ and the initial time $t_0$. Although $D \subset L_m^\infty$ is assumed, any other suitable class of inputs can be considered. Similarly, the uniform hyperexponential stability can be defined with respect to an output or a state component by replacing in the left-hand side of (3) the norm of the state trajectory with the corresponding quantity.

### B. Auxiliary properties

In this paper, the following lemma will be used.

**Lemma 2:** [27] For all $\tau \ge 0$, and $a, \alpha > 0$ such that $\alpha a > 1$, the following holds:

$$ \int_0^\tau e^{s-\tau} \frac{ds}{(as+1)^\alpha} \le \frac{r_{a,\alpha}}{(a\tau+1)^\alpha}, $$

where

$$ r_{a,\alpha} = (a\alpha)^\alpha \int_0^{\frac{a\alpha-1}{a}} e^{s-\frac{a\alpha-1}{a}} \frac{ds}{(as+1)^\alpha} + (a\alpha+1) + \left(\frac{a\alpha+1}{a\alpha}\right)^\alpha \left(1 - e^{-\frac{1}{a}}\right). $$

### C. Properties of parametric Lyapunov equations

Consider the following parametric Lyapunov equation (PLE) [28], [24], [22], [29]:

$$ A^\top P_\psi + P_\psi A - P_\psi b b^\top P_\psi = -\psi P_\psi, \quad (6) $$

where $\psi$ is a (possibly time-varying) parameter to be determined later.

**Lemma 3:** [28], [24], [22], [29] Let $(A, b) \in \mathbb{R}^{n \times n} \times \mathbb{R}^{n \times m}$ be controllable. Then, the following statements hold:

i) The PLE (6) admits a unique positive definite solution $P_\psi$ if and only if

$$ \psi > -2 \min_{i=1,2,...,n} \{\text{Re}\{\lambda_i(A)\}\}. \quad (7) $$

When this condition is satisfied, the solution is explicitly given by $P_\psi = W_\psi^{-1}$, which is a polynomial function of $\psi$ verifying the Lyapunov equation

$$ \left(A + \frac{\psi}{2} I_n\right) W_\psi + W_\psi \left(A + \frac{\psi}{2} I_n\right)^\top = b b^\top. $$

ii) If $\psi$ satisfies (7), then $P_\psi$ exhibits the following properties:

$$ \frac{dP_{\psi}}{d\psi} > 0, \quad (8) $$

ensuring that $P_\psi$ is a strictly increasing function of $\psi$. Additionally, its trace satisfies

$$ \text{tr}(b^\top P_\psi b) = 2\text{tr}(A) + n\psi. $$

iii) If all eigenvalues of $A$ lie on the imaginary axis, then

$$ \lim_{\psi \to 0^+} P_\psi = 0. $$

iv) When all eigenvalues of $A$ are zero, there exists a constant $d_c \ge 1$, independent of $\psi$, such that

$$ \frac{P_{\psi}}{n\psi} \le \frac{dP_{\psi}}{d\psi} \le d_c \frac{P_{\psi}}{n\psi}, \quad \forall \psi > 0. \quad (9) $$

v) Under these conditions, the unique solution of (6) is expressed as

$$ P_{\psi} = \psi \Gamma(\psi) P \Gamma(\psi), \quad (10) $$

where $P = P_1$ ($P_1$ is a positive matrix defined for $\psi = 1$), and the matrix $\Gamma(\psi)$ is defined as

$$ \Gamma(\psi) = \text{diag}(\psi^{n-1}, \psi^{n-2}, \dots, 1). \quad (11) $$

Moreover, it satisfies the differential relation

$$ \frac{dP_{\psi}}{d\psi} = \frac{P_{\psi}}{\psi} + \Gamma(\psi)(HP + PH)\Gamma(\psi), \quad (12) $$

where $H = \text{diag}\{n-1, n-2, \dots, 0\}$.

**Lemma 4:** Let Assumption 1 be satisfied and $\Gamma(\psi)$ be defined in (11). Then, for any $\psi > 1$, any $x \in \mathbb{R}^n$, $u \in \mathbb{R}$, and any $t \ge 0$, we have:

$$ \begin{aligned} G(x) &:= (\Gamma(\psi)g(t, x, u))^{\top} (\Gamma(\psi)g(t, x, u)) \\ &\le \delta(\Gamma(\psi)x)^{\top} \Gamma(\psi)x \end{aligned} $$

where $\delta \ge 0$ is defined by

$$ \delta = n \max_{1 \le i \le n} \left( \max_{1 \le j \le i} c_{ij}^2 \right). \quad (13) $$

## IV. CONTROL IN CONTINUOUS TIME

Consider a control law for (1) chosen in the following form (the same control but written differently was used in [2]):

$$ u(t) = -\frac{1}{2} b^\top P_\psi(t) x(t), \quad (14) $$

where $\psi(t) = \lambda + \alpha t$, $\lambda \ge 1$, $\alpha > 0$ are tuning parameters and $P_\psi(t) = P_\psi(t)$ is the unique positive definite solution to the PLE (6) (it is given in (10) with the matrix $P$).

**Remark 5:** Any continuous, strictly increasing function $\psi: \mathbb{R}^+ \to \mathbb{R}^+$ with $\psi(0) \ge 1$ and an unbounded integral can be utilized in (14). For instance, $\psi(t) = ae^{\alpha t}$ or $\psi(t) = bt^t$, where $a > 0$, $b > 0$ are suitable defined, and $\alpha > 0$. The closed-loop system of (1) and (14) is given by

$$ \dot{x}(t) = \left(A - \frac{1}{2} b b^{\top} P_{\psi}(t)\right)x(t) + bd(t) + g(t, x, u). \quad (15) $$

The following result is presented:

**Theorem 6:** Under assumptions in 1 and 2, let $d_c$ from (9), $\delta$ from (13), $\lambda$ and $\alpha$ satisfy the following condition:

$$ \frac{1}{2} > \frac{\alpha d_c}{n\lambda^2} + \frac{2\delta\Lambda}{\lambda} \quad \text{for} \quad \Lambda = \frac{\lambda_{\max}(P_1)}{\lambda_{\min}(P_1)}. \quad (16) $$

Then, the linear time-varying state feedback (14) renders the closed-loop system (1) uniformly hyperexponential stable.
V. DISCRETIZATION OF THE CONTROLLED SYSTEM (15)

In this section, we investigate a consistent discretization
of (15). The system (15) is a nonlinear time-varying system
with external input $d(t)$. Since the control gain $P_\psi$ grows
unbounded over time and nonlinear function $g$ appears in
the right-hand side, the implicit or explicit Euler methods
are not suitable for all $t \ge 0$. However, the semi-implicit
Euler method can provide a stable discretization approach
for such a class of systems [30], [31].

Let $h > 0$ be the time step, and define the discretization
time instants as $t_k = hk$, $k \in \mathbb{Z}^+$. Applying the semi-
implicit Euler method to (15) results in the discrete-time
system:

$$
\xi_{k+1} = Z^{-1}(t_{k+1}) (\xi_k + hbd_{k+1} + hg_{k+1}), \quad (17)
$$

where the matrix $Z(t)$ is given by:

$$
Z(t) = I_n - h\left(A - \frac{1}{2}bb^\top P_\psi(t)\right). \quad (18)
$$

and $\xi_k \in \mathbb{R}^n$ is a discrete approximation of $x_k = x(t_k)$,
ensuring that $\xi_k \to x_k$ as $h \to 0$. The inputs are defined
as: $g_{k+1} = g(t_{k+1}, x_k, u_k)$, $d_k = d(t_k)$. We assume that
perturbation signal remain bounded, i.e., $d \in \ell_1^\infty$.

Directly, we can calculate the matrix Z:

$$
Z(t) = \begin{pmatrix}
1 & -h & 0 & 0 & \cdots & 0 & 0 \\
0 & 1 & -h & 0 & \cdots & 0 & 0 \\
0 & 0 & 1 & -h & \cdots & 0 & 0 \\
0 & 0 & 0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \cdots & 1 & -h \\
a_1 & a_2 & a_3 & a_4 & \cdots & a_{n-1} & a_n
\end{pmatrix}
$$

where for $\psi_k = \psi(t_k)$,

$$
a_j = P_{n,j} \frac{h \psi^{n-j+1}(t)}{2} \quad \text{for } j = 1, \dots, n-1,
$$

$$
\text{and } a_n = P_{n,n} \frac{h\psi(t)}{2} + 1.
$$

We can define $Z^{-1}(t) = \frac{D(t)}{O(t)}$, where

$$
O(t) = \sum_{i=1}^{n} \frac{P_{n,i} \psi^{n-i+1}(t) h^{n-i+1}}{2} + 1.
$$

As we can conclude from these computations, the discrete
state transition matrix $Z^{-1}(t)$ is nonsingular since $O(t) > 0$
for all $t \ge 0$. This is because $O(t)$ is a polynomial function
of $\psi(t)$ of order $n$, and $O(t)$ the determinant of $Z(t)$. Thus,
the matrix $D(t)$ has the following form:

$$
D(t) =
\begin{pmatrix}
1 + \sum_{k=1}^{n-1} a_k h^{k-1} & h + \sum_{k=2}^{n-1} a_k h^{k-1} & \cdots & h^{n-1} \\
-a_1 h^{n-2} & 1 + \sum_{k=2}^{n-1} a_k h^{k-2} & \cdots & h^{n-2} \\
-a_1 h^{n-3} & -a_2 h^{n-3} & \cdots & h^{n-3} \\
\vdots & \vdots & \ddots & \vdots \\
-a_1 & -a_2 & \cdots & 1
\end{pmatrix}
$$

Moreover,

$$
\lim_{t \to \infty} Z^{-1}(t) =
\begin{pmatrix}
h^{1-n} & 0 & 0 & \cdots & 0 \\
-h^{-n} & 0 & 0 & \cdots & 0 \\
-h^{-n} & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
-h^{-n} & 0 & 0 & \cdots & 0
\end{pmatrix}
$$

which implies that asymptotically the linear dynamics of
(17) approaches a static linear system with a nilpotent matrix
of index $n$. Obviously, the effect of the disturbance $d$ is still
asymptotically annihilated by the control with

$$
h Z^{-1}(t) b = \frac{h}{O(t)} \begin{pmatrix} D_{1n} \\ D_{2n} \\ \vdots \\ D_{nn} \end{pmatrix} = \frac{h}{O(t)} \begin{pmatrix} h^{n-1} \\ h^{n-2} \\ \vdots \\ 1 \end{pmatrix}
$$

So:

$$
\lim_{t \to +\infty} h Z^{-1}(t) b = \lim_{t \to +\infty} \frac{1}{O(t)} \begin{pmatrix} h^n \\ h^{n-1} \\ \vdots \\ h \end{pmatrix} = 0.
$$

It remains to demonstrate that the hyperexponential rate of
convergence is also preserved in (17). Moreover, the effect
of the nonlinearities $g$ is still asymptotically annihilated by
the control, which intuitively follows the structure of the
respective term:

$$
h Z^{-1}(t) g = \begin{pmatrix} \sum_{i=1}^{n} \frac{g_i h \cdot P_{i1}(\psi, h)}{O(t)} \\ \sum_{i=1}^{n} \frac{g_i h \cdot P_{i2}(\psi, h)}{O(t)} \\ \vdots \\ \sum_{i=1}^{n} \frac{g_i h \cdot P_{in}(\psi, h)}{O(t)} \end{pmatrix}
$$

where $P_{ij}(\psi, h)$ represents polynomial terms in $\psi$ and $h$.

VI. NUMERICAL EXAMPLE

To validate the theoretical analysis, we simulate the
closed-loop system (17) under the proposed controller for
$n = 2$. The nonlinear system

$$
\begin{cases}
\dot{x}_1 = x_2 + x_1 \sin(x_2^2), \\
\dot{x}_2 = u + x_1^{2/3} x_2^{1/3} + d(t), \\
y = x_1,
\end{cases}
\quad (19)
$$

is considered with disturbance input $d(t)$. The nonlinear
bounds are satisfied with coefficients

$$
c_{11} = 1, \quad c_{21} = \frac{2}{3}, \quad c_{22} = \frac{1}{3},
$$

which yield $\delta = 2$.

The following Lyapunov matrix is a solution for (6):

$$
P_1 = \begin{pmatrix} 0.9571 & 0.1000 \\ 0.1000 & 0.9000 \end{pmatrix},
$$

whose eigenvalues are approximately 1.0325 and 0.8246
with $\Lambda = 1.0325$. In this case, the controller parameters
are chosen as

$$
K = [-1.3438, -2.5000], \quad \lambda = 11, \quad \alpha = 0.5.
$$
The simulation is carried out with step size $h = 0.001$ and initial condition

$$ \xi_0 = [7 - 5]^T. $$

Figure 1 illustrates the time evolution of: (a) the state variables, (b) the state norm, (c) the control signal, and (d) the perturbation input.

From the simulation results, we observe that:

* the closed-loop trajectories converge with hyperexponential rate, in agreement with the theoretical guarantees;

* the control input remains bounded throughout the simulation;

* the effect of the perturbation $d(t)$ is successfully attenuated.

Note that the system states $\xi_i$ in Figure 1(a) converge to zero faster than the hyperexponential function $10e^{-t^2}$ after about 2 s, illustrating the achievement of hyperexponential convergence.

<figure><img src="image_1.png" /><figcaption>Fig. 1. Simulation results: (a) state variables and state norm, (b) control signal, and (c) perturbation signal under the proposed controller.</figcaption></figure>

## VII. CONCLUSION

This article has studied the hyperexponential stabilization problem for a class of nonlinear systems subject to perturbations, where the nonlinear functions, possibly unknown, satisfy a linear growth condition. A linear time-varying feedback controller, where the gain is designed through a solution to a class of parametric Lyapunov equations and approaches infinity as $t \to \infty$, was proposed. A Lyapunov function was constructed to prove the closed-loop system stability. Both continuous-time and discrete-time designs were considered, with the discrete-time design preserving the properties of the continuous-time approach through a semi-implicit Euler method. A numerical example was provided to illustrate the effectiveness of the proposed methods, demonstrating the hyperexponential convergence of the system states to the equilibrium and rejecting matched perturbations.

In future work, we will establish the stability of the proposed discrete-time system and also address the problem of mismatched perturbations.

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