# Finite-Time Stabilization of Continuous-Time Systems with Sampled Control

Frédéric Mazenc, Michael Malisoff

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Frédéric Mazenc, Michael Malisoff. Finite-Time Stabilization of Continuous-Time Systems with Sampled Control. ACC 2025 - American Control Conference, Jul 2025, Denver, United States. hal-05009489

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# Finite-Time Stabilization of Continuous-Time Systems with Sampled Control

Frédéric Mazenc

Michael Malisoff

**Abstract**— We prove finite-time stabilization properties for continuous-time time-varying linear systems, using sampled controls. Our main results yield finite-time input-to-state stability, where the upper bounding supremum of the uncertainty is over a time interval of constant finite length. Our work includes output feedback stabilization and input delays. We use our results to prove novel global exponential input-to-state estimates for nonlinear systems with state delays, including systems with outputs, using a trajectory based approach. We illustrate our work using a pendulum dynamics with poorly known friction.

**Index Terms**— Finite-time, stabilization, input-to-state stability, sampling

## I. INTRODUCTION

Input-to-state stability (or ISS) is a generalization of global asymptotic stability, and it provides the foundation for considerable ongoing research [1], [2]. In its basic form, ISS provides an upper bound for the norm of the state of the system at each time t that is expressible as the sum of two upper bounding terms, namely, a term that asymptotically decays to zero as $t \to +\infty$ uniformly in the norm of the initial state, plus an increasing function of the supremum of the locally bounded uncertainty from the initial time through the current time. Analogously to global asymptotic stability, ISS is often proven using Lyapunov function methods [2]. In many cases, ISS is achieved using sample data feedback controls [3]–[6], which are implemented in a zero-order hold sense, and then the uncertainty can represent either model uncertainty or the uncertain impacts of the feedback control; see [7], [8].

In this letter, we use sampled control laws, in order to prove a finite-time ISS result for time-varying continuous-time linear systems, which has a fading property whereby the upper bound on the norm of the state at each time only depends on the values that are taken by the uncertainty on an interval of a constant finite length, provided that the current time is sufficiently large; see our finite-time ISS definition below. For time-varying nonlinear systems with state delays using the same sampled control, we also prove global exponential ISS with a rate of convergence that is proportional to the log of a suitable multiple of a Lipschitz constant for the uncertain nonlinear vector fields in the systems. This fading effect, and the speed of convergence, are key advantages of our control.

F. Mazenc is with INRIA EPI DISCO, L2S-CNRS-CentraleSupélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France (e-mail: frederic.mazenc@l2s.centralesupelec.fr).

M. Malisoff is with Department of Mathematics, 303 Lockett Hall, 175 Field House Drive, Louisiana State University, Baton Rouge, LA 70803, USA (e-mail: malisoff@lsu.edu).

Supported by US National Science Foundation Grant 2308282 and Office of Naval Research Grant N00014-22-1-2135.

A key idea uses the controllability of the systems. We adapt a result from [9] to determine robust feedbacks. We also use a trajectory based approach that provides a useful alternative to more standard Lyapunov function methods and which can be more conducive to systems with uncertain delays and uncertain vector fields, and which has been developed, e.g., [10] and [11]. For the problems we solve in this letter, the Lyapunov approach cannot be used. We also prove a dynamic output feedback stabilization result using finite-time observers.

We first provide a definition which plays a prominent role in the rest of this letter. To the best of our knowledge, this definition is new. A delayed system [5] of the form

$$ \dot{x}(t) = f(t, x_t, \delta(t)) \quad (1) $$

with $x$ valued in $\mathbb{R}^n$ and the piecewise continuous functions $\delta$ being locally bounded is called finite-time input-to-state stable (which is also abbreviated by ISS), provided there are a function $\varphi \in \mathcal{K}$ and constants $T_2 \ge 0$ and $T_1 \ge T_2$ such that for each initial time $t_0 \le T_1$ and each absolutely continuous initial function and all choices of $\delta$, the corresponding solution $x: [t_0, +\infty) \to \mathbb{R}^n$ of (1) is uniquely determined and satisfies

$$ |x(t)| \le \varphi \left( \sup_{m \in [t-T_2, t]} |\delta(m)| \right) \quad (2) $$

for all $t \ge T_1$. Here $\mathcal{K}$ denotes the set of all strictly increasing continuous functions $\alpha: [0, +\infty) \to [0, +\infty)$ such that $\alpha(0) = 0$. Unlike standard ISS, in (2), only values taken by $\delta$ on a time interval of length $T_2$ are present. Hence, the values taken by $\delta$ outside a time interval of a constant finite length have no effect on the bound on the current value of $|x(t)|$.

In Section II, we prove a foundational finite-time ISS result for time-varying linear systems, which we combine with the trajectory based approach from [10] to prove an analog for nonlinear systems in Section III. We provide generalizations for systems with outputs in Section IV and input delayed systems in Section V. A key ingredient to cover systems with outputs will be an artificial delays approach, where the original systems might not contain delays, but where the control uses delayed measurements. This artificial delays approach is inspired by literature that observed that delays can improve convergence rates for linear systems. In particular, [12] showed that adding delays to a linear system can improve the rate of convergence of its solutions. We then illustrate the main ideas in an example for pendulum systems with unknown friction.

We use the following other notation. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise, $|\cdot|$ is the standard Euclidean 2 norm and corresponding matrix operator norm, and $|f|_{\mathcal{I}}$ (resp., $|f|_{\infty}$) is the supremum of any
function $f$ in this norm over any interval $\mathcal{I}$ in its domain (resp., its entire domain). When $r$ represents a time variable, we use the standard notation $g_r(\ell) = g(r + \ell)$ for functions $g$ and all $\ell \le 0$ and $r \ge 0$ such that $r+\ell$ is in the domain of $g$. Set $\mathbb{Z}_0 = \{0, 1, 2, ...\}$. Let $C([-\tau, 0], \mathbb{R}^n)$ be the set of all continuous $\mathbb{R}^n$-valued functions defined on $[-\tau, 0]$ for any $\tau > 0$. We let $C_{\text{in}}(\tau)$ be the set of all absolutely continuous functions $\phi \in C([-\tau, 0], \mathbb{R}^n)$. We let $\mathcal{I}$ denote the identity matrix, $|\mathbf{x}|$ (resp., $|\mathbf{x}|$) denote the floor (resp., ceiling) function, and $\Phi_A$ denote the fundamental solution associated to a continuous matrix valued function $A: [0, \infty) \to \mathbb{R}^{n \times n}$; see [13, Appendix C.4] for basic properties of fundamental solutions used in this letter. For simplicity, we assume that the initial times for all of our solutions of our dynamics are $t_0 = 0$.

## II. FOUNDATIONAL FINITE-TIME STABILIZATION RESULT

In this section, we use a controllability approach from [9] to find a feedback that renders a linear continuous-time system

$$ \dot{x}(t) = A(t)x(t) + B(t)u(t) + \delta(t) \quad (3) $$

finite-time ISS with respect to an additive disturbance $\delta$, where $x$ is valued in $\mathbb{R}^n$, the input $u$ is valued in $\mathbb{R}^p$, $A: [0, +\infty) \to \mathbb{R}^{n \times n}$ and $B: [0, +\infty) \to \mathbb{R}^{n \times p}$ are continuous, and $\delta$ is a locally bounded piecewise continuous function. In terms of a constant $\nu > 0$ that we fix in what follows, and the sequence

$$ t_k = k\nu \quad (4) $$

indexed by $k \in \mathbb{Z}_0$, and the sequence

$$ W(k) = \int_{t_k}^{t_{k+1}} \Phi_A(t_k, s) B(s) B^\top(s) \Phi_A^\top(t_k, s) ds, \quad (5) $$

we introduce the following assumption:

**Assumption 1:** The functions $A$ and $B$ are bounded. Also, the constant $\nu > 0$ is such that $W(k)$ is invertible for each $k \in \mathbb{Z}_0$, and such that $W^{-1}(k)$ is a bounded sequence. $\square$

As noted in [13, Section 3.5], Assumption 1 holds for time invariant systems for any $\nu > 0$ provided $(A, B)$ is a constant controllable pair, and in that case, $W$ is constant. Therefore, Assumption 1 holds when $A$ and $B$ have the forms $A(t) = A_* + \Delta_A(t)$ and $B(t) = B_* + \Delta_B(t)$ where $(A_*, B_*)$ is a controllable pair and the sup norms of the continuous functions $\Delta_A$ and $\Delta_B$ are small enough. Let us introduce the functions

$$ g(\nu) = \sup \left\{ \int_{r-\nu}^{r} |\Phi_A(r, \ell)| d\ell \left( |\Phi_A(t, r)| + \int_{r}^{t} |\Phi_A(t, \ell) B(\ell) B^{\top}(\ell) \Phi_A^{\top}(r, \ell)| d\ell \overline{W}(\nu) \right) + \int_{r}^{t} |\Phi_A(t, \ell)| d\ell : t \ge 0, \nu \le r \le t \le r + \nu \right\} \quad (6) $$
and $\overline{W}(\nu) = \sup \{|W^{-1}(k)| : k \in \mathbb{Z}_0\}.$$

The following result renders (3) finite-time ISS with respect to $\delta$ (but see Remark 3 for other formulas for the gain $g(\nu)$):

**Theorem 1:** Let the system (3) satisfy Assumption 1. Then all solutions $x: [0, +\infty) \to \mathbb{R}^n$ of this system for all choices of $\delta$, in closed loop with the feedback defined by

$$ u(\ell) = -B^\top(\ell) \Phi_A^\top(t_k, \ell) W^{-1}(k) x(t_k) \quad (7) $$
for all $\ell \in [t_k, t_{k+1})$ and for all $k \in \mathbb{Z}_0,$

satisfy

$$ |x(t)| \le g(\nu) \sup_{\ell \in [t-2\nu, t]} |\delta(\ell)| \quad (8) $$

for all $t \ge 2\nu$.

*Proof:* The variation of parameters formula (e.g., in [13, Appendix C.4, Equation (C.26)]) gives

$$ x(t) = \Phi_A(t, t_k)x(t_k) + \int_{t_k}^{t} \Phi_A(t, \ell)B(\ell)u(\ell)d\ell \quad (9) $$

for all $k \in \mathbb{Z}_0$ and $t \ge 0$. It therefore follows from the semigroup property of fundamental solutions that

$$ x(t_{k+1}) = \Phi_A(t_{k+1}, t_k) [x(t_k) + \int_{t_k}^{t_{k+1}} \Phi_A(t_k, \ell) B(\ell) u(\ell) d\ell] + \int_{t_k}^{t_{k+1}} \Phi_A(t_{k+1}, \ell) \delta(\ell) d\ell \quad (10) $$

for all $k \in \mathbb{Z}_0$. Hence, by our choice (7) of the feedback and our formula (5) for $W$, we get

$$ x(t_{k+1}) = \int_{t_k}^{t_{k+1}} \Phi_A(t_{k+1}, \ell) \delta(\ell) d\ell \quad (11) $$

for all $k \in \mathbb{Z}_0$. Thus,

$$ x(t_k) = \int_{t_{k-1}}^{t_k} \Phi_A(t_k, \ell) \delta(\ell) d\ell \quad (12) $$

for all $k \ge 1$. We can then combine (7), (9), and (12) to get

$$ x(t) = [\Phi_A(t, t_k) - \mathcal{M}_k(t)] \int_{t_{k-1}}^{t_k} \Phi_A(t_k, \ell) \delta(\ell) d\ell + \int_{t_k}^{t} \Phi_A(t, \ell) \delta(\ell) d\ell \quad (13) $$

for all $t \in [t_k, t_{k+1})$ and $k \ge 1$, where

$$ \mathcal{M}_k(t) = \int_{t_k}^{t} \Phi_A(t, \ell) B(\ell) B^{\top}(\ell) \Phi_A^{\top}(t_k, \ell) d\ell W^{-1}(k). \quad (14) $$

As an immediate consequence,

$$ |x(t)| \le \left( \left[ |\Phi_A(t, t_k)| + |\mathcal{M}_k(t)| \right] \int_{t_{k-1}}^{t_k} |\Phi_A(t_k, \ell)| |\delta(\ell)| d\ell + \int_{t_k}^{t} |\Phi_A(t, \ell)| |\delta(\ell)| d\ell \right) \quad (15) $$

holds for all $t \in [t_k, t_{k+1})$ and $k \ge 1$. Hence, our conclusion (8) of the theorem follows by choosing $r = t_k$ in the supremum that we used in the definition of $g(\nu)$ in (6). $\square$

**Remark 1:** The inequality (8) implies that (7) stabilizes (3) to the origin in the fixed finite time $2\nu$, when $\delta$ is the zero function. Different sample rates $\nu$ produce different control formulas (7) and finite-time ISS bounds (8). We illustrate the effects of changing the value of $\nu$ in Section VI below. $\square$

**Remark 2:** Although our feedback control (7) is expressed in terms of fundamental solutions (which are usually not available in closed form, except for one-dimensional or upper triangular systems), we can use a factoring approach to numerically compute the required fundamental solution values. In fact, using [13, Lemma C.4.1], we can write $\Phi_A(t, s) = \alpha_A(t)\beta_A(s)$ where $\alpha_A$ and $\beta_A$ are the unique solutions of

$$ \dot{\alpha}_A(t) = A(t)\alpha_A(t) \text{ and } \dot{\beta}_A(s) = -\beta_A(s)A(s) \quad (16) $$

that satisfy $\alpha_A(0) = \beta_A(0) = I$. By applying the preceding factoring approach, we can compute the fundamental solution values to implement the controls throughout this letter. $\square$

**Remark 3:** Our proof of Theorem 1 shows that the theorem remains true if we replace $g(\nu)$ by

$$ g_{\nu}(\nu) = \sup \left\{ \int_{t_{k-1}}^{t_k} |\Phi_A(t_k, \ell)| d\ell \left| \Phi_A(t, t_k) - \int_{t_k}^{t} \Phi_A(t, \ell) B(\ell) B^{\top}(\ell) \Phi_A^{\top}(t_k, \ell) d\ell W^{-1}(k) \right| + \int_{t_k}^{t} |\Phi_A(t, \ell)| d\ell : k \ge 1, t \in [t_k, t_{k+1}] \right\}, \quad (17) $$
where $t_k = k\nu$ for all $k \ge 1$, as before. Since $g_b(s) \le g(s)$ for all $s \ge \nu$, this can lead to a less conservative upper bound in our conclusion (8) of Theorem 1, by replacing $g$ by $g_b$ in (8). On the other hand, by using the triangle inequality and the fact that $|\Phi_A(t, s)| \le e^{|A|\infty(t-s)}$ for all $t \ge s$ and all $s \ge 0$, we can check that $g(\nu) \le g_\#(\nu)$, where

$$g_\#(\nu) = (e^{2|A|\infty\nu} + \nu e^{3|A|\infty\nu} |BB^\top|_\infty \overline{W}(\nu) + e^{|A|\infty\nu}) \nu. \quad (18)$$

However, it can be beneficial to use the smaller bounding values $g_b(\nu)$ or $g(\nu)$ instead of the simpler formula for $g_\#(\nu)$ in (18). This is because the reciprocal of the bounding values will be an upper bound for the allowable growth rates of the nonlinearities in the next section. Therefore, using the smaller values $g_b(\nu)$ or $g(\nu)$ (instead of the larger and more conservative values $g_\#(\nu)$) can enable us to cover a larger class of possible growth rates for our nonlinearities. $\square$

### III. NONLINEAR SYSTEMS

We next prove an analog for systems

$$\dot{x}(t) = A(t)x(t) + B(t)u(t) + f(t, x_t) + \kappa(t) \quad (19)$$

that can contain nonlinearities $f$ in the state and state delays, where $x(t)$ is valued in $\mathbb{R}^n$, the initial functions are absolutely continuous, Assumption 1 is satisfied, $u$ is valued in $\mathbb{R}^p$, $\kappa$ is a piecewise continuous locally bounded function, and $f$ is such that all solutions of (19) are well-defined, under the following assumption on $f$, where $g$ is defined in (6) as before, which ensures that the finite escape time phenomenon does not occur:

**Assumption 2:** The function $f$ is continuous in its first argument and locally Lipschitz in its second argument. Also, there are known constants $\bar{c} > 0$ and $f_0 > 0$ such that

$$g(\nu)f_0 \in (0, 1) \quad (20)$$

holds, and such that

$$|f(t, \phi)| \le f_0 \sup_{m \in [-\bar{c}, 0]} |\phi(m)| \quad (21)$$

holds for all $\phi \in C_{in}(\bar{c})$ and all $t \ge 0$. $\square$

A key feature of our Assumption 2 is that $f$ can be unknown, as long as we know constants $\bar{c}$ and $f_0$ satisfying the conditions of the assumption. We then prove the following, which we refer to as an almost finite-time ISS result because instead of finite-time convergence as we had in the linear case in Theorem 1, the rate of convergence

$$R(f_0) = -\frac{\ln(g(\nu)f_0)}{2\nu + \bar{c}} \quad (22)$$

of the first right side term in the exponential ISS estimate (23) to zero is such that $\lim_{f_0 \to 0} R(f_0) = +\infty$, so smaller growth constants $f_0$ on $f$ (for given values of $g(\nu)$ and $\bar{c}$) correspond to faster convergence in the exponential ISS estimate:

**Theorem 2:** Let the system (19) satisfy Assumptions 1-2, and choose the control $u$ in (7). Then for all solutions $x : [0, +\infty) \to \mathbb{R}^n$ of the closed loop system given by (19) and (7), with initial functions in $C_{in}(\bar{c})$, we have

$$|x(t)| \le |x|_{[0,2\nu+\bar{c}]} e^{\frac{\ln(g(\nu)f_0)(t-2\nu-\bar{c})}{2\nu+\bar{c}}} + \frac{g(\nu)|\kappa|_{[0,t]}}{(1-g(\nu)f_0)^2} \quad (23)$$

for all $t \ge 2\nu + \bar{c}$.

**Proof:** Since the feedback defined in (7) is used, the inequality (8) applied with $\delta(\ell) = f(\ell, x_\ell) + \kappa(\ell)$ gives

$$|x(t)| \le g(\nu) \sup_{\ell \in [t-2\nu, t]} |f(\ell, x_\ell) + \kappa(\ell)| \quad (24)$$

for all $t \ge 2\nu$. It follows from Assumption 2 that

$$|x(t)| \le g(\nu)f_0 \sup_{m \in [t-2\nu-\bar{c}, t]} |x(m)| + g(\nu) \sup_{\ell \in [t-2\nu, t]} |\kappa(\ell)| \quad (25)$$

for all $t \ge 2\nu + \bar{c}$. Hence, by replacing $t$ by $t + 2\nu + \bar{c}$ in (25), it follows that the function $w(t) = x(t + 2\nu + \bar{c})$ satisfies

$$|w(t)| \le g(\nu)f_0|w|_{[t-2\nu-\bar{c}, t]} + d(t) \quad (26)$$

for all $t \ge 0$, where $d(t) = g(\nu)|\kappa|_{[t+\bar{c}, t+2\nu+\bar{c}]}$. Since $g(\nu)f_0 \in (0, 1)$, it follows from applying [10, Lemma 1] to the function $w$ that

$$|w(t)| \le |w|_{[-2\nu-\bar{c}, 0]} e^{\frac{\ln(g(\nu)f_0)t}{2\nu+\bar{c}}} + \frac{g(\nu)|\kappa|_{[0, t+2\nu+\bar{c}]} (1-g(\nu)f_0)^2}{(1-g(\nu)f_0)^2} \quad (27)$$

for all $t \ge 0$. Then (23) follows from our choice of $w$ and from replacing $t$ by $t - 2\nu - \bar{c}$ in (27). $\square$

### IV. SYSTEMS WITH OUTPUTS

In this section, we prove a finite-time ISS result under output feedback stabilization, using the finite-time observer proposed in [12]. Consider the system

$$\begin{cases} \dot{x}(t) = A(t)x(t) + B(t)u(t) + \delta(t) \\ y(t) = C(t)x(t) \end{cases} \quad (28)$$

where $x$ is valued in $\mathbb{R}^n$, the input $u$ is valued in $\mathbb{R}^p$, the output $y$ is valued in $\mathbb{R}^q$, and $\delta$ is a locally bounded piecewise continuous function. In terms of the sequence $t_k$ as defined in (4), and any constant $\tau > 0$, and the function

$$E(t) = \int_{t-\tau}^{t} \Phi_{A}^{\top}(s,t)C^{\top}(s)C(s)\Phi_{A}(s,t)ds, \quad (29)$$

we assume that Assumption 1 and the following hold (but see Remark 5 below for analogs that cover nonlinear systems):

**Assumption 3:** The function $C : [0, +\infty) \to \mathbb{R}^{q \times n}$ is continuous and bounded. Also, the constant $\tau > 0$ is such that $E(t)$ is invertible for each $t \ge \tau$ and such that $E^{-1} : [\tau, +\infty) \to \mathbb{R}^{n \times n}$ is a bounded function. $\square$

Assumptions 1 and 3 will both be satisfied for each choice of $\tau > 0$ and each $\nu > 0$ in the time invariant case (i.e., when $A, B$, and $C$ are all constant) if $\Sigma = (A, B, C)$ is observable in the sense of [13, Section 6.1]. Hence, they will also be satisfied if $A, B$, and $C$ are time-varying functions having the form $A(t) = A_* + \Delta_A(t)$, $B(t) = B_* + \Delta_B(t)$, and $C(t) = C_* + \Delta_C(t)$ provided $\Sigma = (A_*, B_*, C_*)$ is observable and provided the continuous time-varying parts $\Delta_A, \Delta_B$, and $\Delta_C$ have small enough sup norms. In terms of the functions

$$\begin{align} J_u(t, \ell) &= \int_{\ell}^{t} \Phi_A(\ell, m) B(m) u(m) dm, \nonumber \\ \bar{E}(\tau) &= \sup\{|E^{-1}(t)| : t \ge \tau\}, \tag{30} \\ \xi(\tau, \nu) &= g(\nu)(|B|_{\infty} \bar{\varpi}(\tau, \nu)\tau + 1), \text{ and} \nonumber \\ \bar{\varpi}(\tau, \nu) &= |B|_{\infty} e^{|A|\infty\nu} \bar{W}(\nu) \bar{E}(\tau) \tau e^{|A|\infty\tau} |C^{\top}C|_{\infty} \nonumber \end{align}$$

and our other notation above, we prove the following result:
**Theorem 3:** Let the system (28) satisfy Assumptions 1 and 3. Then each solution $x: [0, +\infty) \to \mathbb{R}^n$ of the system (28), in closed loop with the control $u$ that is defined by

$$u(\ell) = -B^{\top}(\ell)\Phi_{A}^{\top}(t_k, \ell)W^{-1}(k)\hat{x}(t_k) \quad (31)$$
for all $\ell \in [t_k, t_{k+1})$ and for all $k \in \mathbb{Z}_0$

where $t_k = \nu k$ for all $k \in \mathbb{Z}_0$ and where

$$\hat{x}(t) = E^{-1}(t) \int_{t-\tau}^{t} \Phi_{A}^{\top}(\ell, t) C^{\top}(\ell) y(\ell) d\ell + E^{-1}(t) \int_{t-\tau}^{t} \Phi_{A}^{\top}(\ell, t) C^{\top}(\ell) C(\ell) J_{u}(t, \ell) d\ell \quad (32)$$

for all $t \ge \tau$ and $\hat{x}(t) = 0$ when $t \in [0, \tau)$, satisfies

$$|x(t)| \le \xi(\tau, \nu)|\delta|_{[t-3\nu-\tau, t]} \quad (33)$$

for all $t \ge \nu(3 + \lceil \tau/\nu \rceil)$.

**Proof:** By applying the variation of parameters formula to the $x$ dynamics on any interval $[s, t]$ with $s \ge 0$, we get

$$x(t) = \Phi_A(t, s)x(s) + \int_s^t \Phi_A(t, \ell)[B(\ell)u(\ell) + \delta(\ell)]d\ell. \quad (34)$$

By left multiplying the inequality (34) through by the $n \times n$ matrix $\Phi_A^\top(s, t)C^\top(s)C(s)\Phi_A(s, t)$ and then using the semi-group property of fundamental solutions, we get

$$\begin{aligned} & \Phi_A^\top(s, t)C^\top(s)C(s)\Phi_A(s, t)x(t) \\ & = \Phi_A^\top(s, t)C^\top(s)y(s) \\ & + \Phi_A^\top(s, t)C^\top(s)C(s)\int_s^t \Phi_A(s, \ell)[B(\ell)u(\ell) + \delta(\ell)]d\ell. \end{aligned} \quad (35)$$

By integrating (35) over $s \in [t-\tau, t]$, and then left multiplying both sides of the result by $E^{-1}(t)$, and then using the formula (32) for $\hat{x}(t)$, we obtain

$$\begin{aligned} x(t) = \hat{x}(t) \\ + E^{-1}(t) \int_{t-\tau}^{t} \Phi_{A}^{\top}(\ell, t) C^{\top}(\ell) C(\ell) \delta^{\sharp}(t, \ell) d\ell \end{aligned} \quad (36)$$

for all $t \ge \tau$, where

$$\delta^{\sharp}(t, \ell) = \int_{\ell}^{t} \Phi_{A}(\ell, m) \delta(m) dm. \quad (37)$$

Hence, for each integer $k \ge \frac{\tau}{\nu}$, the feedback from (31) satisfies

$$u(\ell) = -B^{\top}(\ell)\Phi_{A}^{\top}(t_k, \ell)W^{-1}(k)x(t_k) + w(\ell, t_k) \quad (38)$$

for all $\ell \in [t_k, t_{k+1})$, where

$$w(\ell, t_k) = B^{\top}(\ell)\Phi_{A}^{\top}(t_k, \ell)W^{-1}(k)G(t_k) \quad (39)$$

and

$$G(t) = E^{-1}(t) \int_{t-\tau}^{t} \Phi_{A}^{\top}(\ell, t) C^{\top}(\ell) C(\ell) \delta^{\sharp}(t, \ell) d\ell. \quad (40)$$

Then the closed loop system can be rewritten as

$$\begin{aligned} \dot{x}(t) = & A(t)x(t) + B(t)w(t, \sigma(t)) + \delta(t) \\ & -B(t)B^{\top}(t)\Phi_{A}^{\top}(\sigma(t), t)W^{-1}(k)x(\sigma(t)) \end{aligned} \quad (41)$$

for all $t \ge \nu \lceil \tau/\nu \rceil$, where $\sigma(t)$ is defined by $\sigma(t) = t_k$ when $t \in [t_k, t_{k+1})$ for each $k \in \mathbb{Z}_0$. By the reasoning that we used to prove (8) in Theorem 1, we deduce that

$$|x(t)| \le g(\nu) \sup_{\ell \in [t-2\nu, t]} |B(\ell)w(\ell, \sigma(\ell)) + \delta(\ell)| \quad (42)$$

for all $t \ge \nu(2 + \lceil \tau/\nu \rceil)$. Consequently,

$$|x(t)| \le g(\nu) |B|_{\infty} \sup_{\ell \in [t-2\nu, t]} |w(\ell, \sigma(\ell))| + g(\nu) |\delta|_{[t-2\nu, t]} \quad (43)$$

for all $t \ge \nu(2 + \lceil \tau/\nu \rceil)$. Also, our formulas for $w$ and $\varpi$ in (30) and (39) give

$$|w(\ell, \sigma(\ell))| \le \varpi(\tau, \nu) \int_{\sigma(\ell)-\tau}^{\sigma(\ell)} |\delta(m)| dm \quad (44)$$

for all $\ell \ge \tau + \nu$. As an immediate consequence,

$$\begin{aligned} |x(t)| & \le g(\nu) |B|_{\infty} \varpi(\tau, \nu) \sup_{\ell \in [t-2\nu, t]} \int_{\sigma(\ell)-\tau}^{\sigma(\ell)} |\delta(m)| dm \\ & \qquad + g(\nu) \sup_{\ell \in [t-2\nu, t]} |\delta(\ell)| \\ & \le g(\nu) |B|_{\infty} \varpi(\tau, \nu) \tau \sup_{m \in [t-3\nu-\tau, t]} |\delta(m)| \\ & \qquad + g(\nu) \sup_{\ell \in [t-2\nu, t]} |\delta(\ell)| \end{aligned} \quad (45)$$

for all $t \ge \nu(3 + \lceil \tau/\nu \rceil)$. This allows us to conclude. ■

**Remark 4:** We can use a variation of parameters argument to rewrite the formula (32) for $\hat{x}(t)$ without integrals. To see how this can be done, first notice that $\Phi_A^\top(t, s) = \Phi_{-A^\top}(s, t)$ holds for all $s \ge 0$ and $t \ge 0$; this follows by first noting that $\frac{d}{dt}\Phi_A^\top(t, s) = \Phi_A^\top(t, s)A^\top(t)$ holds for all $t \ge 0$ and $s \ge 0$, and by then using [13, Lemma C.4.1 (d)] and the uniqueness properties of fundamental solutions. Therefore, we can reason as in [14, Remark 1] (with $f_1$ and $f_2$ in [14, Remark 1] chosen to be $Bu(t)$ and $0$, respectively) except using the variation of parameters formula from [13, Equation C.26] (instead of the version of the variation of parameters formula for constant $A$), to rewrite (32) as

$$\begin{aligned} \hat{x}(t) = & E^{-1}(t)[L_1(t) - \Phi_{-A^\top}(t, t-\tau)L_1(t-\tau)] \\ & - E^{-1}(t)[L_3(t) - \Phi_{-A^\top}(t, t-\tau)L_3(t-\tau)] + L_2(t) \end{aligned} \quad (46)$$

for all $t \ge \nu[\tau/\nu]$, where $L_1$, $L_2$, and $L_3$ are the states of

$$\begin{aligned} \dot{L}_1(t) &= -A^\top(t)L_1(t) + C^\top(t)y(t) \\ \dot{L}_2(t) &= A(t)L_2(t) + B(t)u(t) \\ L_3(t) &= -A^\top L_3(t) + C^\top(t)C(t)L_2(t) \end{aligned} \quad (47)$$

for all $t \ge 0$, and where the new formula

$$\begin{aligned} u(\ell) = & -B^{\top}(\ell)\Phi_{A}^{\top}(t_k, \ell)W^{-1}(k)\{L_2(t_k) \\ & + E^{-1}(t_k)[L_1(t_k) - \Phi_{-A^{\top}}(t_k, t_k-\tau)L_1(t_k-\tau)] \\ & - E^{-1}(t_k)[L_3(t_k) - \Phi_{-A^{\top}}(t_k, t_k-\tau)L_3(t_k-\tau)]\} \end{aligned} \quad (48)$$

for all $\ell \in [t_k, t_{k+1})$ and $k \ge 1$

is obtained by substituting (46) into (31) and using the formula $J_u(t, \ell) = -(L_2(\ell) - \Phi_A(\ell, t)L_2(t))$. $\square$

**Remark 5:** By using the approach from Theorem 2, we can generalize Theorem 3 to nonlinear systems with outputs

$$\begin{cases} \dot{x}(t) = A(t)x(t) + B(t)u(t) + f(t, x_t) + \delta(t) \\ y(t) = C(t)x(t) \end{cases} \quad (49)$$

provided that the function $f$ is continuous in its first argument and locally Lipschitz in its second argument and admits constants $\bar{c} > 0$ and $f_0 > 0$ such that for the function $\xi(\tau, \nu)$ defined in (30), we have

$$\xi(\tau, \nu) f_0 \in (0, 1), \quad (50)$$

and such that (21) holds for all $\phi \in C_{\text{in}}(\bar{c})$. This generalization is obtained by replacing $\delta(m)$ in the part of the proof of
Theorem 3 up through (45) by $\delta(m) + f(m, x_m)$. With this replacement, we instead obtain the conclusion

$$|x(t)| \le |x|_{[0, \nu(3 + [\tau/\nu]) + \bar{c}]} e^{\frac{\ln(\xi(\tau, \nu)f_0)(t - \nu(3 + [\tau/\nu]) - \bar{c})}{\nu(3 + [\tau/\nu]) + \bar{c}}} \\ + \frac{\xi(\tau, \nu)}{(1 - \xi(\tau, \nu)f_0)^2} |\delta|_{[0, t]} \quad (51)$$

for all $t \ge \nu(3 + [\tau/\nu]) + \bar{c}$, by applying [10, Lemma 1]. □

## V. DELAYED SYSTEMS

The preceding two sections cover systems with state delays, using the nonlinear term $f(t, x_t)$ in the dynamics (19) and (49) for the cases of state and output feedback, respectively. On the other hand, the feedback control (7) in Section II requires measurements $x(t_k)$ of the state at each sample time $t_k = k\nu$ in order to compute the value of the control on the interval $[t_k, t_{k+1})$. However, in some cases, the sample values $x(t_k)$ might not be immediately available, and then we must instead use sample values $x(\sigma(t) - h)$ at each time $t \ge 0$, where $\sigma(t) = \nu|t/\nu|$ is the last sample time before time $t \ge 0$ and where the constant $h > 0$ represents a delay.

We can cover such cases, by instead using the feedback

$$u(\ell) = -B^{\top}(\ell)\Phi_{A}^{\top}(t_k, \ell)W^{-1}(k)\left[\Phi_{A}(t_k, t_k-h)x(t_k-h) + \int_{t_k-h}^{t_k} \Phi_{A}(t_k, m)B(m)u(m)dm\right] \quad (52)$$

for all $\ell \in [t_k, t_{k+1})$

for all $k \ge \lceil h/\nu \rceil$, with $u(\ell) = 0$ otherwise, where we continue the notation introduced in Section II. With the new feedback (52), we then obtain this variant of Theorem 1:

**Theorem 4:** Let (3) satisfy Assumption 1, and $h > 0$ and $\nu > 0$ be constants. Then all solutions $x: [0, +\infty) \to \mathbb{R}^n$ of (3) in closed loop with the feedback from (52) satisfy

$$|x(t)| \le g(\nu) \left[ |B|_{\infty}^2 e^{|A|_{\infty}(3\nu+h)} \overline{W}(\nu)h + 1 \right] |\delta|_{[t-2\nu-h, t]} \quad (53)$$

for all $t \ge \nu(3 + \lceil h/\nu \rceil)$.

*Proof:* The variation of parameters formula gives

$$x(t) = \Phi_A(t, t-h)x(t-h) + \int_{t-h}^{t} \Phi_A(t, m)Bu(m)dm + \int_{t-h}^{t} \Phi_A(t, m)\delta(m)dm \quad (54)$$

for all $t \ge h$. Setting $t = t_k$ in (54) and using the result to replace the quantity in squared brackets in (52) gives

$$u(\ell) = -B^{\top}(\ell)\Phi_{A}^{\top}(t_k, \ell)W^{-1}(k)x(t_k) + B^{\top}(\ell)\Phi_{A}^{\top}(t_k, \ell)W^{-1}(k)\Delta(t_k) \quad (55)$$

where

$$\Delta(t_k) = \int_{t_k-h}^{t_k} \Phi_A(t_k, m) \delta(m) dm. \quad (56)$$

Since the first term on the right side of (55) coincides with the control (7) from Theorem 1, it follows from the argument that led us to (8) that

$$|x(t)| \le g(\nu) \sup_{\ell \in [t-2\nu, t]} \left| B(\ell)B^{\top}(\ell)\Phi_{A}^{\top}(\sigma(t), \ell)W_{*}(t)\Delta(t_k) \right| + g(\nu)|\delta|_{[t-2\nu, t]} \quad (57)$$

for all $t \ge \nu(2 + \lceil h/\nu \rceil)$, where $W_*(t) = W^{-1}(k)$ where $k\nu = \sigma(t)$. Consequently, we can reason as in the last part of the proof of Theorem 1 to upper bound the supremum in (57), to obtain the conclusion (53). ■

## VI. ILLUSTRATION

We illustrate Theorem 2. Consider the equations

$$\begin{cases} \dot{x}_1(t) = x_2(t) \\ \dot{x}_2(t) = -\frac{G}{L}\sin(x_1(t)) - \left(\frac{K}{M} + \epsilon(t)\right)x_2(t) + v(t) \end{cases} \quad (58)$$

of the pendulum with a poorly known friction term, where $x_1$ and $x_2$ are valued in $\mathbb{R}$, $G$ is the gravitational constant, $K > 0$ is a constant, $M$ is the mass, $L$ is the pendulum length, $v$ is the input, and the continuous function $\epsilon(t)$ is unknown but such that $\epsilon(t) \in [0, \bar{c}]$ for all $t \ge 0$, where $\bar{c} > 0$ is a known constant.

The change of feedback

$$v(t) = \frac{G}{L} \sin(x_1(t)) + \frac{K}{M} x_2(t) + u(t) \quad (59)$$

gives:

$$\left\{ \begin{aligned} \dot{x}_1(t) &= x_2(t) \\ \dot{x}_2(t) &= u(t) - \epsilon x_2(t) \end{aligned} \right. \quad (60)$$

Then, with the notation of the previous sections, we choose

$$A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad (61)$$

and

$$f(t, x_t) = \begin{bmatrix} 0 \\ -\epsilon(t)x_2(t) \end{bmatrix} \quad (62)$$

Since $(A, B)$ is a controllable pair, Assumption 1 and the requirement (21) from Assumption 2 are satisfied, with $f_0 = \bar{c}$. Since the definition of the matrix exponential gives

$$e^{A\ell} = \begin{bmatrix} 1 & \ell \\ 0 & 1 \end{bmatrix} \quad (63)$$

for all $\ell \in \mathbb{R}$, our formula (5) gives

$$W(k) = \int_{0}^{\nu} \left[ \begin{matrix} \ell^2 & -\ell \\ -\ell & 1 \end{matrix} \right] d\ell = \begin{bmatrix} \frac{1}{3}\nu^3 & -\frac{\nu^2}{2} \\ -\frac{\nu^2}{2} & \nu \end{bmatrix} \quad (64)$$

for all $k \in \mathbb{Z}_0$. Then

$$W^{-1}(k) = 12 \begin{bmatrix} \frac{1}{\nu^3} & \frac{1}{2\nu^2} \\ \frac{1}{2\nu^2} & \frac{1}{3\nu} \end{bmatrix} \quad (65)$$

for all $k \in \mathbb{Z}_0$. Then the feedback given by (7) is

$$u(\ell) = -12[0 \ 1] \begin{bmatrix} 1 & 0 \\ \sigma(\ell) - \ell & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{\nu^3} & \frac{1}{2\nu^2} \\ \frac{1}{2\nu^2} & \frac{1}{3\nu} \end{bmatrix} x(\sigma(\ell)) \quad (66)$$

where $\sigma(\ell) = \nu|\ell/\nu|$ is the largest of the sample times $t_k = k\nu$ on the interval $[0, \ell]$ for each $\ell \ge 0$.

The expression (66) simplifies to

$$u(\ell) = 12 \left[ \frac{\ell - \sigma(\ell)}{\nu^3} - \frac{1}{2\nu^2}, \frac{\ell - \sigma(\ell)}{2\nu^2} - \frac{1}{3\nu} \right] x(\sigma(\ell)). \quad (67)$$

Also, by applying the formula

$$\int_a^b (a - \ell)(b - \ell)d\ell = -\frac{1}{6}(b - a)^3 \quad (68)$$
with $a = t_k$ and $b = t$ for each integer $k \ge 2$ and each $t \in [t_k, t_{k+1})$, it follows that for the preceding pendulum dynamics, we have

$$ \int_{t_k}^{t} \Phi(t, \ell) B(\ell) B^{\top}(\ell) \Phi_{A}^{\top}(t_k, \ell) d\ell = \begin{bmatrix} -\frac{(t-t_k)^3}{6} & \frac{(t-t_k)^2}{2} \\ -\frac{(t-t_k)^2}{2} & t-t_k \end{bmatrix} \quad (69) $$

for all $k \ge 2$ and $t \in [t_k, t_{k+1})$. Hence, the function $g_b$ in (17) is

$$ g_b(\nu) = \max_{s \in [0, \nu]} \left[ \int_0^s |e^{A\ell}| d\ell + \begin{bmatrix} 1 & s \\ 0 & 1 \end{bmatrix} - 12 \begin{bmatrix} -\frac{s^3}{6} & \frac{s^2}{2} \\ -\frac{s^2}{2} & s \end{bmatrix} \begin{bmatrix} \frac{1}{\nu^3} & \frac{1}{2\nu^2} \\ \frac{1}{2\nu^2} & \frac{1}{3\nu} \end{bmatrix} \int_0^{\nu} |e^{A\ell}| d\ell \right] \quad (70) $$

and the maximum in (70) occurs at $s = \nu$. It follows from Remark 3 that the conclusion of Theorem 2 will be satisfied if $\nu > 0$ is such that $g_b(\nu)\bar{\epsilon} < 1$. By the formula (70), it follows that larger values of $\nu$ can produce smaller allowable bounds $\bar{\epsilon}$ on the uncertainty parameter $\epsilon$ in the pendulum dynamics. We illustrate this point in Table I, which shows values of the bound $1/g_b(\nu)$ on $\epsilon$ for different values of $\nu > 0$. Then, in

<table><caption>TABLE I
RELATIONSHIP BETWEEN <i>ν</i> AND ALLOWABLE BOUND ON <i>ε</i></caption><tr><td>Sample Rate <i>ν</i> in <i>u</i></td><td>1</td><td>0.5</td><td>0.25</td><td>0.125</td></tr><tr><td>Bound 1/g<sub>b</sub>(<i>ν</i>) on <i>ε</i></td><td>0.118485</td><td>0.260579</td><td>0.546051</td><td>1.1174</td></tr></table>

Figs. 1-3, we plot Mathematica simulations of solutions of (58) in closed loop with the sample feedback (67), for three values of the sample rate $\nu$ and using the choice $\epsilon = 0.99/g_b(\nu)$ in each case. For each simulation, we chose $K = L = M = 1$ and $G = 9.81$. The figures illustrate how enlarging $\nu$ can slow the convergence but reduce the oscillation of $x_2(t)$.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Closed Loop Solution Component $x_1(t)$ (resp., $x_2(t)$) in Solid Red (resp., Dashed Blue) for $\nu = 0.25$ and $\epsilon = 0.540591$</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Closed Loop Solution Component $x_1(t)$ (resp., $x_2(t)$) in Solid Red (resp., Dashed Blue) for $\nu = 0.5$ and $\epsilon = 0.257973$</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Closed Loop Solution Component $x_1(t)$ (resp., $x_2(t)$) in Solid Red (resp., Dashed Blue) for $\nu = 0.75$ and $\epsilon = 0.164024$</figcaption></figure>

## VII. CONCLUSIONS

We used the property of controllability in combination with the trajectory based approach to determine robust feedbacks which stabilize time-varying continuous time systems in finite-time (for linear systems) or in almost finite-time (for nonlinear systems). We allowed state delays, input delays, and systems with outputs. Many extensions can be expected. For instance, since quantized and switched systems [3], [4] are important in applications, we aim to prove generalizations for switched systems with quantization.

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