# Stabilization of time-varying continuous-time systems with a small delay in the input

Frederic Mazenc, Catherine Bonnet

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Frederic Mazenc, Catherine Bonnet. Stabilization of time-varying continuous-time systems with a small delay in the input. Joint IFAC Conference 2025 - IFAC Symposium on System Structure and Control, 19th IFAC Workshop on Time Delay Systems and 2nd IFAC Workshop on Control of Complex Systems, Jun 2025, Gif sur Yvette, France. hal-05318036

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# Stabilization of time-varying continuous-time systems with a small delay in the input

F. Mazenc* C. Bonnet**

* Université Paris Saclay, Inria, CNRS, CentraleSupelec, L2S, 91190 Gif-sur-Yvette, France, (e-mail: frederic.mazenc@centralesupelec.fr).

** Université Paris Saclay, Inria, CNRS, CentraleSupelec, L2S, 91190 Gif-sur-Yvette, France, (e-mail: catherine.bonnet@inria.fr).

**Abstract:** Two control designs of exponentially stabilizing feedbacks for two families of time-varying continuous-time systems with a pointwise delay in the input are proposed. A smallness condition is imposed on the delay. In contrast to the classical reduction model approach, the results do not necessitate knowledge of the state transition matrix.

**Keywords:** delay, stabilization, time-varying, control

## 1. INTRODUCTION

Stabilizing systems with input delay is one of the most important problems of the theory of the systems with delay because input delays arise from measurement and transport phenomena in engineering. For the stabilization of systems of the family

$$ \dot{x}(t) = A(t, x(t))x(t) + Bu(t - \tau), \quad (1) $$

where $x$ is valued in $\mathbb{R}^n$, the input $u$ is valued in $\mathbb{R}^p$, $\tau \ge 0$ is a constant delay, there exist several approaches.

First, when the function $A$ does not depend on $x$ and the state transition matrix of $A(t)$ is known, it is possible to use prediction or reduction model approaches to achieve exponential stability of (1) for any arbitrarily large known delay. This has been demonstrated in contributions such as Artstein (1982), Kwon and Pearson (1980) and Manitius and Olbrot (1979). However, these approaches are not applicable when $A(t, x)$ depends on $x$ or when $A$ does not depend on $x$ and the state transition matrix of $A(t)$ is unknown.

Second, in some case, the system (1) can be stabilized for any delay $\tau$ by using predictors or subpredictors with pointwise delays, as explained in particular in Najafi et al. (2013), Mazenc and Malisoff (2017), Rojas-Ricca et al. (2024) and Lei and Khalil (2016). However, these approaches apply only when the function $f(t, x) = A(t, x)x$ is known and is globally Lipschitz with respect to $x$, or when $f$ admits a special structure. Moreover, in some cases the necessity for multiple, potentially cumbersome, subpredictors is problematic.

Third, when the system (1) is stabilizable in the absence of delay and $A$ is bounded, a stabilizing control of the form

$$ u(t - \tau) = K(t, x(t - \tau))x(t - \tau) \quad (2) $$

can generally be determined, provided that a smallness condition on $\tau$ is satisfied.

In this context, a question arises: what is the largest value $\tau_M > 0$ such that for any $\tau \in [0, \tau_M)$, there is a function $K$ for which the system

$$ \dot{x}(t) = A(t, x(t))x(t) + BK(t, x(t - \tau))x(t - \tau) \quad (3) $$

is exponentially stable? To the best of our knowledge, this challenging question does not have a general answer, even in the case where $A$ does not depend on $x$. However, using Lyapunov approaches similar to those presented in (Fridman, 2014, Chapt. 4) and (Gu et al., 2003, Chapt. 5.3), it is often possible to determine a constant $\bar{\tau} \in (0, \tau_M)$ such that the origin of the system (3) is globally exponentially stable when $\tau \in [0, \bar{\tau})$. However, these Lyapunov approaches are complicated and yield a conservative value for the constant $\bar{\tau}$.

This leads us to consider the following question: is there a feedback, not necessarily of the form (2), that can stabilize the system in scenarios where the delay exceeds the value $\bar{\tau}$ mentioned earlier, even in cases where the state transition matrix of $A$ is unknown and $f(t, x) = A(t, x)x$ is not globally Lipschitz?

The remarks above motivate the present work. To achieve satisfactory allowable delays in situations where the state transition matrix of $A$ is not available, we propose two designs for control laws that stabilize the system when the delay $\tau$ is smaller than a constant defined by an explicit formula. The first design addresses the scenario where $B$ is invertible and $A$ is known, while the second design pertains to cases where $B$ is not invertible and $A$ is poorly characterized. The proposed designs draw inspiration from the reduction model approach, incorporating distributed terms that depend on the input within their formulations. Our proofs rely on the theory of the positive matrices, the trajectory based approach Mazenc et al. (2017) and an extension of the Halanay's inequality result (let us recall that the Halanay's inequality has been introduced in Halanay (1966)). Let us observe that we believe that they are simpler than those relying on Lyapunov-Krasovskii functions which are commonly used to study the system (3) (see for instance Mazenc et al. (2013)). To take into account the possible presence of uncertainties, we have studied systems with an additive disturbance and derived inequalities of Input-to-State-Stable (ISS)
type (the celebrate notion ISS for systems with delay is presented in particular in Chaillet et al. (2023)).

Finally, we would like to indicate that we have assessed the efficiency of our design by comparing our proposed results with those derived from a frequency domain approach.

The paper is organized as follows. Section 2 is devoted to the global exponential stabilization of a nonlinear system with a known vector field. Section 3 proposes stabilizing feedbacks for systems with uncertainties in their vector field. Concluding remarks are drawn in Section 4.

**Notation.** We use notation which we simplify when no confusion would arise. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise. The standard Euclidean norm and induced matrix norm are denoted by $|\cdot|$. I denotes the identity matrix.

## 2. STABILIZATION OF A KNOWN NONLINEAR SYSTEM

### 2.1 Theoretical result

We consider the nonlinear system

$$ \dot{x}(t) = A(t, x(t))x(t) + Bu(t - \tau) + \delta(t), \quad (4) $$

where $x$ is valued in $\mathbb{R}^n$, $u$ is valued in $\mathbb{R}^n$ and $\tau \ge 0$ and where $A$ is a piecewise continuous function with respect to its first argument and Lipschitz continuous with respect to its second argument. The function $\delta$ is piecewise continuous. It represents a disturbance.

In this section, our goal is to construct a stabilizing control law, provided that the delay $\tau$ is sufficiently small. We focus on the scenario where the matrix $A$ is bounded, the matrix $B$ is invertible, and the system described by equation (4) is stabilizable in the absence of delay.

Let us introduce the constant

$$ \bar{B} = |B| \quad (5) $$

and assumptions:

**Assumption 1.** There is constant $\bar{A} \ge 0$ such that

$$ |A(t, x)| \le \bar{A} \quad (6) $$

for all $t \ge 0$ and $x \in \mathbb{R}^n$. Moreover, the matrix $B$ is invertible.

**Assumption 2.** There are a function $K$ piecewise continuous with respect to its first argument and Lipschitz continuous with respect to its second and a constant $\bar{K} \ge 0$ such that

$$ |K(t, x)| \le \bar{K} \quad (7) $$

for all $t \ge 0$ and $x \in \mathbb{R}^n$ and there are constants $c_i > 0$, $i = 1$ to 3, such that, for any continuous function $\xi: [0, +\infty) \to \mathbb{R}^n$ and any piecewise continuous function $\zeta: [0, +\infty) \to \mathbb{R}^n$, the solutions of

$$ \dot{r}(t) = [A(t, \xi(t)) + BK(t, \xi(t))]r(t) + \zeta(t) \quad (8) $$

satisfy

$$ |r(t)| \le c_1 e^{-c_2(t-s)} |r(s)| + c_3 \sup_{\ell \in [s, t]} |\zeta(\ell)| \quad (9) $$

for all $t \ge s \ge 0$.

Assumption 1 implies that the function

$$ R(t, x) = B^{-1}A(t, x)B \quad (10) $$

is well-defined and

$$ |R(t, x)| \le \bar{R} \quad (11) $$

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

$$ \bar{R} = \bar{A}|B^{-1}|\bar{B}. \quad (12) $$

We can now introduce a third assumption:

**Assumption 3.** The inequality

$$ s_{\tau} < 1 \quad (13) $$

with

$$ s_{\tau} = \tau \bar{R} \quad (14) $$

holds.

We are ready to state and prove the following result:

**Theorem 1.** Let the system (4) satisfy Assumptions 1 to 3. Consider the solutions $x(t)$ of the system (4) in closed-loop with control law defined by

$$ u(t) = K(t, x(t)) \left[ x(t) + B \int_{t-\tau}^{t} u(m) dm \right] + R(t, x(t)) \int_{t-\tau}^{t} u(m) dm \quad (15) $$

for all $t \ge 0$ with

$$ u(t) = 0 \quad (16) $$

for all $t \in [-\tau, 0)$. Then there are constants $k_i > 0$, $i = 1$ to 3 and $T > 0$ such that

$$ \begin{aligned} |x(t)| &\le k_1 \sup_{m \in [0, T]} |x(m)| e^{\frac{\ln(1+s\tau)}{T}t} \\ &\quad + k_2 \sup_{m \in [0, T]} |u(m)| e^{\frac{\ln(1+s\tau)}{T}t} + k_3 \sup_{m \in [0, t]} |\delta(m)| \end{aligned} \quad (17) $$

for all $t \ge T$.

**Remark.**

1. The inequality (17) is similar to an ISS inequality. Let us also observe in particular that Assumption 3 ensures that $e^{\frac{\ln(1+s\tau)}{T}t}$ goes exponentially to zero.

2. From the proof provided below, we can derive explicit values for the constants $k_i$ and $T$.

3. Let us observe that Assumptions 1 and 2 do not imply that the functions $A$ and $K$ are globally Lipschitz.

**Proof.** First, let us observe that Assumption 1 ensures that the finite escape time phenomenon does not occur for the system (4) in closed-loop with the feedback defined in (15)-(16). Now, let us introduce

$$ \Gamma(t) = B \int_{t-\tau}^{t} u(m) dm \quad (18) $$
and

$$z(t) = x(t) + \Gamma(t) \quad (19)$$

for all $t \ge 0$. One can prove that

$$\begin{aligned} \dot{z}(t) = & [A(t, x(t)) + BK(t, x(t))]z(t) \\ & -A(t, x(t))B \int_{t-\tau}^{t} u(m)dm \\ & +BR(t, x(t)) \int_{t-\tau}^{t} u(m)dm + \delta(t) \end{aligned} \quad (20)$$

for all $t \ge 0$. From the definition of $R$ in (10), it follows that

$$\dot{z}(t) = [A(t, x(t)) + BK(t, x(t))]z(t) + \delta(t). \quad (21)$$

Now, we observe that Assumption 2 ensures

$$|z(t)| \le c_1 e^{-c_2(t-s)} |z(s)| + c_3 \sup_{\ell \in [s,t]} |\delta(\ell)| \quad (22)$$

for all $t \ge s \ge 0$. Since $c_2 > 0$, one can find a constant $T \ge \tau$ such that

$$|z(t)| \le \frac{s_{\tau} + 1}{4} |z(t-T)| + c_3 \sup_{\ell \in [t-T, t]} |\delta(\ell)| \quad (23)$$

for all $t \ge T$, where $s_{\tau}$ is the constant defined in (14).

On the other hand, the definition of $u$ in (15) gives, for all $t \ge 0$,

$$\begin{aligned} |u(t)| & \le |R(t, x(t))| \left| \int_{t-\tau}^{t} u(m) dm \right| + |K(t, x(t))| |z(t)| \\ & \le \bar{R} \left| \int_{t-\tau}^{t} u(m) dm \right| + \bar{K} |z(t)|. \end{aligned} \quad (24)$$

where the last inequality is a consequence of (11) and (7).

Taking into account the definition of $s_{\tau}$ in (14), we obtain

$$|u(t)| \le s_{\tau} \sup_{m \in [t-\tau, t]} |u(m)| + \bar{K} |z(t)| \quad (25)$$

for all $t \ge 0$.

The inequalities (23) and (25) give the vector inequality

$$\left( \begin{array}{c} |u(t)| \\ |z(t)| \end{array} \right) \le M_{\tau} \left( \begin{array}{c} \sup_{m \in [t-T, t]} |u(m)| \\ \sup_{m \in [t-T, t]} |z(m)| \end{array} \right) + \Delta_s(t) \quad (26)$$

for all $t \ge T$ with

$$M_{\tau} = \begin{bmatrix} s_{\tau} & \bar{K} \\ \frac{1-s_{\tau}^2}{8\bar{K}} & \frac{s_{\tau}+1}{4} \end{bmatrix} \quad (27)$$

and

$$\Delta_s(t) = \begin{pmatrix} 0 \\ c_3 \sup_{\ell \in [t-T, t]} |\delta(\ell)| \end{pmatrix} \quad (28)$$

because $\frac{1-s_{\tau}^2}{8\bar{K}} > 0$. Assumption 3 ensures that the matrix

$M_{\tau}$ and the vector $V_{\tau} = \begin{pmatrix} 1 \\ \frac{1-s_{\tau}}{2\bar{K}} \end{pmatrix}$ are positive. Moreover,

$$M_{\tau} V_{\tau} = \frac{1 + s_{\tau}}{2} V_{\tau}. \quad (29)$$

From the Perron-Frobenius theorem see Ngoc (2006), we deduce that $M_{\tau}$ is Schur stable. Then the trajectory based approach (Mazenc and Malisoff, 2022, Theorem 2) ensures that $u$ and $z$ satisfy the vector ISS inequality:

$$\left( \begin{array}{c} |u(t)| \\ |z(t)| \end{array} \right) \le S_{u,z}(T) e^{\frac{\ln\left(\frac{1+s_{\tau}}{2}\right)}{T}(t-T)} V_{\tau} \\ -(-I + M_{\tau})^{-1} \left( \begin{array}{cc} 0 & c_3 \sup_{\ell \in [0,t]} |\delta(\ell)| \\ c_3 \sup_{\ell \in [0,t]} |\delta(\ell)| & 0 \end{array} \right) \quad (30)$$

with

$$S_{u,z}(T) = \sup_{m \in [0,T]} |u(m)| + \sup_{m \in [0,T]} |z(m)| \quad (31)$$

for all $t \ge T$.

Using this inequality and the definition of $z$ in (19), one can prove that

$$\begin{aligned} |x(t)| & \le \left( \frac{1-s_{\tau}}{2\bar{K}} + \tau\bar{B} \right) \sup_{m \in [0,T]} |x(m)| e^{\frac{\ln\left(\frac{1+s_{\tau}}{2}\right)}{T}(t-\tau-T)} \\ & \quad + \left( \frac{1-s_{\tau}}{2\bar{K}} + \tau\bar{B} \right) (1 + (T + \tau)\bar{B}) \\ & \quad \times \sup_{m \in [-\tau, T]} |u(m)| e^{\frac{\ln\left(\frac{1+s_{\tau}}{2}\right)}{T}(t-\tau-T)} \\ & \quad + (\tau\bar{B} + 1) \left| (-I + M_{\tau})^{-1} \right| c_3 \sup_{m \in [0,T]} |\delta(m)| \end{aligned} \quad (32)$$

for all $t \ge T$. This allows us to conclude.

## 2.2 Efficiency of Theorem 1

In this part, we demonstrate the effectiveness of Theorem 1 by applying it to a scalar system characterized by a function $A$ that is independent of $x$. We will compare the performance of the control law derived from this theorem with that of a control law that does not incorporate a distributed term.

Let us begin by considering the system

$$\dot{x}(t) = x(t) - (k+1)x(t-\tau), \quad (33)$$

where $x$ is valued in $\mathbb{R}$ and $\tau \ge 0$. When $k > 0$, the delay-free closed-loop is stable. Using the the Walton and Marshall method Walton and Marshall (1987), it is straightforward to determine that the smallest value of the delay that destabilizes the system is given by

$$\tau_{\max} = \frac{\arctan(\sqrt{(k+1)^2 - 1})}{\sqrt{(k+1)^2 - 1}}. \quad \text{This implies that for any}$$

$\tau \in [0, 1]$, there exists a constant $k > 0$ such that the system (33) is exponentially stable. However, for $\tau \ge 1$, the system (33) is not exponentially stable, regardless of the value of $k$.

Now, let us consider the system

$$\dot{x}(t) = A(t)x(t) - (k(t) + 1)x(t-\tau), \quad (34)$$

where $x$ is valued in $\mathbb{R}$, $\tau \ge 0$, $A$ is such that $A(t) \in [-1, 1]$ for all $t \ge 0$ and both $A$ and $k$ are piecewise continuous. Then, to the best of our knowledge, it remains uncertain whether for any $\tau \in [0, 1)$ there exists a function $k$ such
that the system (34) is exponentially stable. We conjecture that all the methods that provide functions $k$ and delay $\tau$ such that (34) is exponentially stable (notably the one of Mazenc et al. (2013)) give values of $\tau$ strictly smaller than 1.

On the other hand, when the function $A$ is locally Lipschitz and such that $A(t, x) \in [-1, 1]$ for all $t \ge 0$ and $x \in \mathbb{R}$, we prove below that, for any delay $\tau \in [0, 1)$, Theorem 1 provides with a control law which exponentially stabilizes the system

$$ \dot{x}(t) = A(t, x(t))x(t) + u(t - \tau). \quad (35) $$

This highlights the benefits of the control law presented in Theorem 1 compared to memoryless controls.

Now, let us apply Theorem 1 to the system (35) with

$$ \bar{A} = 1. \quad (36) $$

Then Assumption 1 is satisfied and, with the notation of the previous section, $B = 1$,

$$ R(t, x) = A(t, x) \quad (37) $$

and

$$ \bar{R} = 1. \quad (38) $$

Since

$$ A(t, x) + BK(t, x) = A(t, x) + K(t, x), \quad (39) $$

Assumption 2 is satisfied with

$$ K(t, x) = -k - A(t, x), \quad (40) $$

where $k > 0$ is a tuning parameter.

Since $\bar{R} = 1$, Assumption 3 is satisfied when

$$ \tau < 1. \quad (41) $$

It follows that, in this case, Theorem 1 applies. We conclude that the system (35) is exponentially stabilized by the control law

$$ u(t) = (-k - A(t, x(t))) \left( x(t) + \int_{t-\tau}^{t} u(m)dm \right) + A(t, x(t)) \int_{t-\tau}^{t} u(m)dm \quad (42) $$

when (41) holds.

### 3. STABILIZATION OF SYSTEMS WITH UNCERTAIN TERMS

Theorem 1 applies to the systems (4) when the matrix $B$ is invertible and the function $A$ is perfectly known. However, in practice, this is often not the case. This motivates this section. We propose a result for which $B$ is not necessarily invertible and $A$ admits a decomposition $A_1 + A_2$ where $A_1$ is known and constant and $A_2$ is poorly known and time-varying.

For the sake of simplicity, we consider the case where the function $A$ does not depend on $x$, but an extension to the case where it does can straightforwardly be obtained. For the sake of generality, we let $B$ be a time-varying function.

#### 3.1 Theoretical result

We consider the system

$$ \dot{x}(t) = A_1x(t) + A_2(t)x(t) + B(t)u(t - \tau) + \delta(t), \quad (43) $$

where $x$ is valued in $\mathbb{R}^n$, $u$ is valued in $\mathbb{R}^p$, $\tau \ge 0$, where $A_2$ and $B$ are piecewise continuous functions such that there are two constants $\bar{A}_2 \ge 0$ and $\bar{B} \ge 0$ such that

$$ |A_2(t)| \le \bar{A}_2 \quad (44) $$

and

$$ |B(t)| \le \bar{B} \quad (45) $$

for all $t \ge 0$ and where $\delta$ is a piecewise continuous function which represents a disturbance.

Our goal is to stabilize the system (43) when $A_2$ can be poorly known and when $B$ is not necessarily invertible. However, we will not treat the term $A_2(t)x(t)$ as a mere perturbation without any available information. Instead, our approach leverages potentially known properties of $A_2$, as demonstrated in Section 3.2.3. It is also worth pointing out that $A_2$ is incorporated in Assumption 4 through the function $H$ defined below. Consequently, the stabilizability assumption we will impose depends on $A_2$. Let us observe also that in the system (43), the term $A_1x(t)$ is supposed to be known. We have decomposed the function $A$ of (1) into $A_1 + A_2(t)$ to reduce the conservativeness of our approach. Let us observe that we could let $A_1$ depend on $t$ and be such that its state transition matrix is known. For the sake of simplicity, we consider the case where $A_1$ is constant.

Now, let us introduce the function

$$ H(t) = A_1 + A_2(t) + e^{-\tau A_1} B(t + \tau) K(t), \quad (46) $$

where $K$ is a piecewise-continuous function such that there is a constant $\bar{K} \ge 0$ such that

$$ |K(t)| \le \bar{K} \quad (47) $$

for all $t \ge 0$. Let us also introduce the assumption:

**Assumption 4.** There are constants $c > 0$, $\bar{p} > 0$ and $\underline{p} > 0$ and a function $P: [0, +\infty) \to \mathbb{R}^{n \times n}$ of class $C^1$, such that $P(t)$ is symmetric and positive definite for all $t \ge 0$ and such that

$$ |P(t)| \le \bar{p} \quad (48) $$

for all $t \ge 0$ and the LMIs

$$ \dot{P}(t) + P(t)H(t) + H(t)^{\top}P(t) \le -cP(t) \quad (49) $$

and

$$ \underline{p}I \le P(t) \quad (50) $$

are satisfied for all $t \ge 0$.

Let us introduce the function

$$ J(\tau) = 2 \frac{\bar{p}}{\underline{p}} \bar{A}_2 \bar{B} K \int_{0}^{\tau} |e^{-A_1 m}| dm. \quad (51) $$

Finally, we introduce an assumption which is a restriction imposed on the size of $\tau$:
**Assumption 5.** The inequality

$$J(\tau) < \frac{3c}{4} \qquad (52)$$

is satisfied.

Assumption 5 ensures that there exists $\lambda > 0$ such that

$$\lambda + J(\tau)e^{\lambda\tau} = \frac{3c}{4}. \qquad (53)$$

We are ready to state and prove the following result:

**Theorem 2.** Let the system (43) satisfy Assumptions 4 and 5. Let

$$u(t) = K(t)(x(t) + \int_{t-\tau}^{t} e^{A_1(t-m-\tau)}B(m+\tau)u(m)dm) \qquad (54)$$

for $t \ge 0$ and

$$u(t) = 0 \qquad (55)$$

for all $t \in [-\tau, 0)$. There are constants $q_i > 0$ such that

$$|x(t)| \le q_1 \sup_{\ell \in [0, \tau]} |x(\ell)| e^{-\frac{\lambda}{2}t} + q_2 \int_0^\tau |u(m)| dme^{-\frac{\lambda}{2}t} \\ + q_3 \sup_{\ell \in [0, t]} |\delta(\ell)| \qquad (56)$$

for all $t \ge \tau$

**Remark.**

1. The inequality (56) is similar to an ISS inequality. Let us observe in particular that the function $e^{-\frac{\lambda}{2}t}$ goes exponentially to zero when the time goes to the infinity.

2. From the proof provided below, we can derive explicit values for the constants $q_i$. Notice also that a positive lower bound for $\lambda$ can be found easily.

**Proof.** The proof is omitted.

### 3.2 Illustration

To illustrate Theorem 2 and demonstrate its efficiency, we consider the system (35) across several scenarios. In the first case, we show that when the decomposition $A_1 = 0$, $A_2 = A$ is chosen, Theorem 2 is less efficient than Theorem 1. In the second and the third cases, we show how an appropriate decomposition of A can make Theorem 2 more efficient than Theorem 1.

**Unknown function A** In this part, we consider the particular case where $A_1 = 0$, $A_2 = A$ and $\bar{A}_2 = 1$. Then, in this particular case,

$$H(t) = A_2(t) + K(t). \qquad (57)$$

We choose $K(t) = -\bar{K}$ with $\bar{K}$ such that $\bar{K} > 1$.

$$\bar{K} > 1. \qquad (58)$$

Then Assumption 4 is satisfied with

$$P = 1 \qquad (59)$$

because

$$H(t) + H(t)^{\top} = 2A_2(t) - 2\bar{K} \le 2 - 2\bar{K} < 0. \qquad (60)$$

Thus, we can choose

$$c = 2\bar{K} - 2 \quad \text{and} \quad \underline{p} = \bar{p} = 1 \qquad (61)$$

and obtain

$$J(\tau) = 2\bar{K}\tau. \qquad (62)$$

It follows that Assumption 5 is satisfied only when

$$\tau < \frac{3(\bar{K} - 1)}{4\bar{K}}. \qquad (63)$$

Thus, for any value $\bar{K}$, if Assumption 5 is satisfied, then necessarily,

$$\tau < \frac{3}{4}. \qquad (64)$$

Since Theorem 1 gives 1 as maximum allowable delay, it follows that, as expected, in this case Theorem 1 outperforms Theorem 2.

**Unknown function A**$_2$ In this part, we consider the scenario where $A_1 = \frac{8}{9}$ and $A_2$ is unknown but smaller in absolute value than $\frac{1}{9}$. Then $|A_1 + A_2(t)| \le 1$ for all $t \ge 0$. We will demonstrate that in this case Theorem 2 outperforms Theorem 1. This highlights the significance of decomposing A into $A_1 + A_2$ where $A_2$ is as small as possible and $A_1$ is known and constant.

We consider the system

$$\dot{x}(t) = \frac{8}{9}x(t) + A_2(t)x(t) + u(t - \tau), \qquad (65)$$

where $x$ and $u$ are valued in $\mathbb{R}$, $A_2$ is piecewise continuous and $\tau \ge 0$.

$$H(t) = \frac{8}{9} + A_2(t) + e^{-\frac{8}{9}\tau}K(t). \qquad (66)$$

Then by choosing

$$K(t) = -2e^{\frac{8}{9}\tau}, \qquad (67)$$

we obtain $\bar{K} = 2e^{\frac{8}{9}\tau}$ and

$$H(t) = -\frac{10}{9} + A_2(t). \qquad (68)$$

Let us choose

$$P(t) = 1 \qquad (69)$$

for all $t \ge 0$. Then

$$\dot{P}(t) + P(t)H(t) + H(t)^{\top}P(t) = -\frac{20}{9} + 2A_2(t). \qquad (70)$$

Since $\bar{A}_2 = \frac{1}{9}$, Assumption 4 is satisfied with $c = 2$, $\underline{p} = \underline{p} = 1$. Then

$$J(\tau) = \frac{4}{9}e^{\frac{8}{9}\tau} \int_{0}^{\tau} e^{-\frac{8}{9}m} dm = \frac{1}{2}e^{\frac{8}{9}\tau} (1 - e^{-\frac{8}{9}\tau}). \qquad (71)$$
Thus Assumption 5 is satisfied when the inequality

$$e^{\frac{8}{5}\tau} \left(1 - e^{-\frac{8}{9}\tau}\right) < 3 \qquad (72)$$

is satisfied. This is the case in particular for some values of $\tau > 1$. Since $A_1 + \bar{A}_2 = 1$, we deduce that in this case, Theorem 2 outperforms Theorem 1 because this theorem gives 1 as maximum allowable delay. Bearing in mind Section 2.2, we also note that the control law established by Theorem 2 stabilizes the system (65) for delays such that there is no value $p$ such that

$$\dot{x}(t) = \frac{8}{9}x(t) + A_2(t)x(t) - px(t - \tau) \qquad (73)$$

is exponentially stable for any piecewise continuous function $A_2$ smaller in absolute value than $\frac{1}{9}$.

### Unknown function $A_2$ with a known sign
In this part, we consider the system (65) in a new scenario: we assume that the sign of $A_2$ is known:

$$A_2(t) \in \left[-\frac{1}{9}, 0\right] \qquad (74)$$

for all $t \ge 0$. We will show that we can stabilize this system with larger delays than in the case where the sign of $A_2$ is unknown.

By choosing

$$K(t) = -\frac{17}{9}e^{\frac{8}{5}\tau}, \qquad (75)$$

we obtain $\bar{K} = \frac{17}{9}e^{\frac{8}{5}\tau}$ and

$$H(t) = -1 + A_2(t). \qquad (76)$$

We choose again the function $P$ defined in (69). Then

$$\dot{P}(t) + P(t)H(t) + H(t)^{\top}P(t) = -2 + 2A_2(t) \le -2, \quad (77)$$

where the last inequality is a consequence of the negativity of $A_2$. It follows that we can choose $c = 2$. Then $J(\tau) = \frac{17}{36}e^{\frac{8}{5}\tau} \left(1 - e^{-\frac{8}{9}\tau}\right)$. Thus Assumption 5 is satisfied when the inequality

$$e^{\frac{8}{5}\tau} \left(1 - e^{-\frac{8}{9}\tau}\right) < \frac{54}{17} \qquad (78)$$

is satisfied. Since $\frac{54}{17} > 3$, the condition (78) is less restrictive than (72).

## 4. CONCLUSION

To stabilize continuous-time nonlinear time-varying systems that have a known constant pointwise delay in the input, we have proposed control laws incorporating distributed terms. The stability we have obtained is global, exponential, and robust against additive disturbances. Much remains to be done. In particular, we plan to explore the following areas: (i) extending our results to the case where the systems have several delays or time-varying delays or delays with disturbances; (ii) achieving the local stabilization of nonlinear systems which do not satisfy Assumption 1, (iii) stabilizing switched systems, (iv) addressing output feedback stabilization problems. We also plan to apply our results to the pendulum model studied in Mazenc et al. (2014) and to a model of a wheeled mobile robot studied in Jiang et al. (2001) and Mazenc et al. (2013).

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