Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Jérôme Harmand

Michael Malisoff
email: malisoff@lsu.edu

Jérome Harmand
email: jerome.harmand@supagro.inra.fr

Michael Maliso↵

Stabilization in a Chemostat with Sampled and Delayed Measurements and Uncertain Growth Functions

Keywords: output feedback, stabilization, delay, sampling ?

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

This work continues our search for controls that stabilize componentwise positive equilibria in chemostat models, under the incomplete state measurements and model uncertainties that usually occur in biotechnology laboratories, and so is strongly motivated by the ubiquity of the chemostat in a plethora of biological and engineering settings that are of compelling interdisciplinary interest, in which stabilization of componentwise positive equilibria is needed to ensure persistence of species. The chemostat is used for the continuous culture of microorganisms. It was first studied in [START_REF] Monod | La technique de culture continue: Théorie et applications[END_REF] and [START_REF] Novick | Description of the chemostat[END_REF]. It is regarded in biotechnology, ecology, and microbiology as an ideal way to represent cell or microorganism growth, wastewater treatment, or natural environ-ments like lakes; see [START_REF] Beauthier | Input/state invariant LQ-optimal control: application to competitive coexistence in a chemostat[END_REF]; [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]; [START_REF] Fritsch | A modeling approach of the chemostat[END_REF]; [START_REF] Gouzé | Feedback control for nonmonotone competition models in the chemostat[END_REF]; [START_REF] Lemesle | A simple unforced oscillatory growth model in the chemostat[END_REF], and [START_REF] Robledo | Global stability for a model of competition in the chemostat with microbial inputs[END_REF]. The variables are the microorganism and substrate concentrations, whose dynamics are based on mathematical models, e.g., mass-balance equations; see [START_REF] Mazenc | Further results on stabilization of periodic trajectories for a chemostat with two species[END_REF] and [START_REF] Smith | The Theory of the Chemostat[END_REF]. Two challenges in designing controls for chemostats are their nonlinearity and their lack of online actuators and sensors; see [START_REF] Cougnon | On-line optimization of fedbatch bioreactors by adaptive extremum seeking control[END_REF].

Moreover, when online devices are available to measure biomass and substrate concentrations, they usually only provide delayed discrete measurements. It is common to design controls using continuous time models, which are then discretized before being applied. However, to prove that continuous time controllers ensure that the desired stability objectives are met, one must show robustness with respect to discretization. Chemostats are also subjected to uncertainty in the growth functions, which should also be taken into account in the control design. To the best of our knowledge, no rigorous theoretical analysis in the literature has addressed the delay, robustness, and sampling problems that we consider here. The work [START_REF] Robledo | Feedback stabilization for a chemostat with delayed output[END_REF] assumes that the measurements are continuous.

The preceding remarks motivated Mazenc et al. (2013a) and this work, which solves a complementary problem to the ones in Mazenc et al. (2013a). Here we consider the classical chemostat model in [START_REF] Smith | The Theory of the Chemostat[END_REF] that contains one substrate and one species, except here we also include delays, sampling, and uncertainties, which are three features that are not contained in the classical chemostat model. We assume that the input substrate concentration is constant, and that the growth rate is of Haldane type (which has a growth limitation for low substrate concentrations, and inhibition at high concentrations). The dilution rate is the control, and uses delayed and sampled observations. Controlling this system is di cult, for two reasons. First, works such as Mazenc et al. (2013b) that prove global asymptotic stability under delay and sampling use state feedbacks. Since our work has output feedbacks, it is outside the scope of Mazenc et al. (2013b).

Second, chemostats with non-monotonic growth rates generally have multiple equilibria, under constant dilution rates. One is unstable, while another is locally exponentially stable. The work Mazenc et al. (2013a) stabilized points of the second type, but here we stabilize points of the first type in cases where the growth rate is uncertain and not necessarily monotone. Our stabilizing controller only requires measurements of the substrate, which are piecewise constant and delayed. Under suitable bounds on the delay size and on the sampling interval, our control provides global asymptotic stability to a componentwise positive equilibrium when the growth function is known, and input-to-state stability (or ISS) (as defined in [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]) with respect to uncertainties in the growth functions. This di↵ers from Mazenc et al. (2013a), where no constraints on the delay and sampling intervals were used. We believe that these extra constraints are needed because under constant dilution rates, the equilibrium that we stabilize in this paper would have been unstable. Mazenc et al. (2013b), the barrier functions that we use here allow us to certify ISS, which was not considered in Mazenc et al. (2013b). The main result of Mazenc et al. (2013b) does not apply here, even in the special case where the growth functions are known. Our proof also di↵ers from [START_REF] Mazenc | Stabilization of a chemostat model with Haldane growth functions and a delay in the measurement[END_REF], which assumes that species measurements are available. When there are no perturbations, our results contrast with [START_REF] Gouzé | Robust control for an uncertain chemostat model[END_REF] and other works that do not include delays or sampling or ISS. Our new work also improves on our conference version (i.e., [START_REF] Mazenc | Stabilization in a chemostat with sampled and delayed measurements[END_REF]), which did not allow uncertainties in the growth functions, because here, we prove ISS with respect to the uncertainties in the growth functions under arbitrarily large uncertainty bounds and positivity constraints on the states. See Section 3 below for our main result, Section 4 for its proof, and Section 5 for an illustration including simulations.

While reminiscent of

Model and Notation

Our basic chemostat model is

( ṡ(t) = D[s in s(t)] µ(s(t))x(t) ẋ(t) = [µ(s(t)) D]x(t) (1)
(where we used the standard technique of scaling the species level x(t) in order to eliminate the constant yield) but see below for generalizations where the growth function µ can be uncertain. The states x and s are positive valued (and represent the species and substrate levels, respectively), the substrate input concentration s in > 0 is a constant, the dilution rate D is a positive valued control that we will specify, and the growth function µ satisfies:

Assumption 1 The function µ is of class C 1 and µ(0) = 0. Also, there is a constant s M > 0 such that µ 0 (s) > 0 for all s 2 [0, s M ) and µ 0 (s)  0 for all s 2 [s M , 1). Finally, µ(s) > 0 for all s > 0.

⇤

By C 1 , we mean continuously di↵erentiable. Assumption 1 holds for all functions of the form

µ(s) = k1s 1+k2s+k3s 2 , (2)
for any constants k i > 0 for i = 1 to 3, with s M = 1/ p k 3 . Functions of the form (2) are called Haldane functions. In Fig. 1, we plot the special case of (2) and s in where prove the next lemma, where a function ↵ : [0, 1) ! [0, 1) is defined to be of class K 1 provided ↵(0) = 0 and ↵ is continuous, strictly increasing, and unbounded; and µ 0 1 (0) is the derivative from the right.

µ(s) = 0.
Lemma 1 If Assumption 1 holds, then we can construct a function µ

1 2 C 1 \ K 1 and a nondecreasing C 1 function : R ! [0, 1) such that (m) = 0 for all m  0, µ(s) = µ1(s) 1+ (s) for all s 0, (4)
µ 0 1 (s) > 0 for all s 0, and 0 (s) > 0 for all s s M . ⇤

Remark 1 If µ 0 (s) < 0 for all s > s M (which holds for (2)), and s in > s M , and the dilution rate D is a constant D 2 (µ(s in ), µ(s M )) ✓ (0, 1), then the system (1) has a locally unstable positive equilibrium point of the form (s ⇤ , s in s ⇤ ) and the locally stable equilibrium (s in , 0), where s ⇤ 2 (s M , s in ) and D = µ(s ⇤ ). Our work Mazenc et al. (2013a) globally stabilized an equilibrium that can be locally exponentially stabilized by a constant dilution rate. ⇤

To explain our sampling control goals, fix any two constants

✏ 1 > 0 and ✏ 2 > 0 such that ✏ 2 > ✏ 1 , and let {t i } be a sequence of real numbers such that 0 < ✏ 1  t i+1 t i  ✏ 2 for all i 2 N [ {0},
where t 0 = 0 and N = {1, 2, . . .}. Given any constant ⌧ f 0, we define the function ⌧ as follows:

⌧ (t) = ( ⌧ f , t2 [0, ⌧ f ) ⌧ f + t t j ,t 2 [t j + ⌧ f , t j+1 + ⌧ f ) and j 0
This is reminiscent of the representation of sampling in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]. For all j 0 and t 2

[t j + ⌧ f , t j+1 + ⌧ f ), we have t ⌧ (t) = t (⌧ f + t t j ) = t j ⌧ f , so t ⌧ (t) is piecewise constant.
In the special case where ⌧ f = 0, we also have t ⌧ (t) = t j for all t 2 [t j , t j+1 ) and j 0. Moreover, for all t 0, we have

0  ⌧ (t)  ⌧ M , where ⌧ M = 2⌧ f + ✏ 2 . ( 5
)
We assume that s(t ⌧ (t)) is the only available measurement. Our control D will be computed in terms of the delayed sampled values s(t ⌧ (t)) of the substrate, so when ⌧ f = 0, the control values will be computed from the sequence of observations {s(t j )} at the sample times; see (12). When µ is known, our goal is asymptotic stabilization of E ⇤ = (s ⇤ , s in s ⇤ ) for any constant s ⇤ 2 (0, s in ), using our positive valued dilution rate feedback. Then the components of E ⇤ are positive, and E ⇤ is an equilibrium of (1) if and only if D takes the value µ(s ⇤ ) when s = s ⇤ .

Main Result

Let Assumption 1 hold with s in s M , and fix any constant s ⇤ 2 (0, s in ) and any functions and µ 1 that satisfy the requirements from Lemma 1. We use the constants

µ a = µ 1 (s ⇤ )s in , (6)
$ s = inf s2[0,sin] µ 0 1 (s) , $ l = sup s2[0,sin] µ 0 1 (s) , ⇢ l = sup s2[0,sin] 0 (s), (7)
and

⇢ m = ⇢ 2 l 2$ s max l2[0,sin] µ 2 1 (l + 1.1µ a ⌧ M ) 1 + (l) (8)
where ⌧ M satisfies (5). The preceding constants are all positive, by the properties of µ 1 and . To model uncertainties, we study the more general model

( ṡ(t) = D[s in s(t)] (1 + (t))µ(s(t))x(t) ẋ(t) = [(1 + (t))µ(s(t)) D]x(t) (9)
where the unknown measurable essentially bounded function : [0, 1) ! [d, 1) admits a known constant lower bound d 2 ( 1, 0]. We also assume that d and ⌧ M satisfy:

Assumption 2 The constants ⌧ M and d are such that (1 + d)µ 1 (s in ) 1 + (s in ) µ 1 (s ⇤ ) 1 + (s in µ a ⌧ M ) > 0 , (10)
⌧ M < 1 2 p 2⇢ m $ l s in , and ⌧ M < 1 2⇢ l s in µ 1 (s in ) (11)
are all satisfied. ⇤ Note for later use that since is nondecreasing, (10) gives

(1 + d)µ 1 (s in ) µ 1 (s ⇤ ).
In Fig. 2, we illustrate how the 's in (9) appropriately address the uncertainty in the uptake function µ, without imposing any upper bounds on the range of . For simplicity, we took constant 's in Fig. 2, but another valuable feature of ( 9) is that it allows time varying 's, without any monotonicity or nonnegativity requirements on the 's.

Since µ

1 strictly increases and

s ⇤ < s in , we have µ 1 (s in ) > µ 1 (s ⇤ )
. Therefore, it is usually easy to determine constants d ⇤ 2 ( 1, 0) and ⌧ > 0 such that ( 10)-( 11) hold for all d 2 (d ⇤ , 0] and ⌧ M 2 (0, ⌧ ); see our illustration in Section 5. We say that ( 9) is input-to-state stable (or ISS) with respect to (D, E, S) for sets D ✓ R and S ✓ R 2 and the point E = (s ⇤ , s in s ⇤ ) provided that there are functions ¯ 2 KL and ¯ 2 K 1 such that for all solutions (s(t), x(t)) of ( 9) whose initial functions are valued in S, and for all choices of : [0, 1) ! D, we have

|(s(t), x(t))| E  ¯ (|(s, x)| E,[ ⌧ M ,0] , t)+¯ (| | [0,t] ) for all t 0, where |(s(t), x(t))| E = |(s(t), x(t)) E| is the distance to the equilibrium E, | • | [0,t] is the essential supremum over [0, t], |(s, x)| E,[ ⌧ M ,0] is the essential supremum of |(s(t), x(t))| E over [ ⌧ M , 0]
, K 1 was defined in the preceding section, and KL is the set of all continuous functions ¯ : [0, 1) ⇥ [0, 1) ! [0, 1) such that (i) for each t 0, the function f (s) = ¯ (s, t) is of class K 1 and (ii) for each s 0, the function g(t) = ¯ (s, t) is nonincreasing and satisfies lim t!1 g(t) = 0. In our theorem, we assume that the initial functions are constant valued, but see Remark 3 for more general cases. Our theorem is:

Theorem 1 If Assumption 1 holds, and if s in s M , s ⇤ 2 (0, s in ), and d 2 ( 1, 0] are any constants such that Assumption 2 holds, then for any functions µ 1 and that satisfy the requirements of Lemma 1 and all constants s s in , x > 0, and d 0, the system (9), in closed loop with

D(t) = µ1(s⇤) 1+ (s(t ⌧ (t))) , (12) is ISS with respect to ([ d, d], (s ⇤ , s in s ⇤ ), (0, s) ⇥ (x, 1)). ⇤ Remark 2 When = 0, Theorem 1 implies that lim t!1 (s(t), x(t)) = (s ⇤ , s in s ⇤
) for all initial conditions that are valued in (0, 1) 2 . Our proof in the next section will show that (s(t), x(t)) 2 (0, 1) 2 for all t 0 for all initial states in (0, 1) 2 . The functions ¯ and ¯ in the ISS estimate will depend on d, x, and s. Our proof of Theorem 1 can be used to provide an algorithm for constructing ¯ and ¯ . ⇤ 4 Proof of Theorem 1 Fix any initial state (s(0), x(0)) 2 (0, s) ⇥ (x, 1) and any corresponding solution (s(t), x(t)) for the perturbed system (9), in closed loop with (12).

First

Step. We first prove that the solution (s(t), x(t)) is valued in (0, s) ⇥ (0, 1) for all t 0. To show that s and x stay positive, note that at any possible time t when s(t) = 0, we would have ṡ(t) > 0, since D(t)s in > 0. Hence, (s(t), x(t)) 2 (0, 1) 2 for all t 0. Also, ṡ(t) < 0 for all t such that s(t) s in , so s(t) stays in (0, s). We next compute useful bounds using the new variable

z(t) = s in s(t) x(t). ( 13
)
By summing the equations in (9), we get ṡ(t) + ẋ(t) = D(s in s(t) x(t)) = Dz(t), so our formula (12) for D and (13) combine to give

ż(t) = D(t)z(t) = µ1(s⇤) 1+ (s(t ⌧ (t))) z(t). (14)
Here and in the sequel, all equalities and inequalities are along all solutions of (9), unless otherwise noted. Hence, the subadditivity of the square root gives

|(x(t), s(t))|  s(t)+x(t) = z(t)
|z(t)|  |z(0)|exp ⇣ R t 0 µ1(s⇤)d1 + (s(` ⌧ (`))) ⌘  |z(0)|e tµ 1 (s⇤ ) 1+ (s) . (16)
We also use the error variable X that is defined by

X(t) = (s(t) s ⇤ , x(t) s in + s ⇤ ). (17)
Then for all t 0, the triangle inequality and (17) give

|z(t)| = |s ⇤ s(t) (x(t) s in + s ⇤ )|  |s ⇤ s(t)| + |x(t) s in + s ⇤ |  2|X(t)|.
(18)

Second

Step. We build a function T a 2 K 1 such that s(t) 2 (0, s in ) for all t T a (|X(0)|) for all possible values of X(0). Fix an unbounded function ⌧ ⇤ : (0, 1) ! [0, 1) such that for each function p : [0, 1) ! (0, 1) that satisfies the di↵erential inequality ṗ(t) (1+d)µ1(sin) µ1(s⇤) 1+ (s) p(t) for all t 0, (

we have p(t) 2(|X(0)| + s in ) for all t ⌧ ⇤ (|X(0)| + p(0)), and such that ⌧ ⇤ is strictly increasing on [(s in s ⇤ )/2, 1). Such a function ⌧ ⇤ exists because (10) implies that the numerator in ( 19) is positive. We next prove the following claim, by arguing by contradiction: Claim 1:

s(t) < s in for some t 2 [0, ⌧ ⇤ (|X(0)| + x(0))]. Suppose that s(t) s in for all t 2 [0, ⌧ ⇤ (|X(0)| + x(0))].
Then for all t in this interval, the first equation in (9) gives ṡ(t)  0, so

(1+ (t))µ1(s(t)) 1+ (s(t)) µ1(s⇤) 1+ (s(t ⌧ (t))) (1+ (t))µ1(sin) 1+ (s(t)) µ1(s⇤) 1+ (s(t ⌧ (t))) (1+ (t))µ1(sin) µ1(s⇤) 1+ (s(t ⌧ (t))) (1+d)µ1(sin) µ1(s⇤) 1+ (s) , (20)
since µ 1 is nondecreasing and (t) d, where we also used the fact that (s(t))  (s(t ⌧ (t)))  (s), which fol-lows because is nondecreasing and s is nonincreasing. Hence, by our choice (12) of D, the x(t) component of ( 9) is a positive valued solution of (19), so (18) gives x(t)

2(|X(0)| + s in ) |z(0)| + 2s in at t = ⌧ ⇤ (|X(0)| + x(0)), contradicting (15), since s in > 0. Also, if s(t) = s
in , then ṡ(t) < 0, so s(t) cannot increase to above s in , so s(t) < s in for all t ⌧ ⇤ (|X(0

)| + x(0)). If |X(0)|  1 2 (s in s ⇤ ), then |s(0) s ⇤ |  1 2 (s in s ⇤ ), which gives s(0) s in = s(0) s ⇤ (s in s ⇤ )  1 2 (s in s ⇤ ) < 0, so s(t) stays in (0, s in )
. Also, ( 15) and ( 18) give 0 < x(0 , where s 4 will be independent of X(0) and . First, notice that our choice (13) of z implies that

)  |z(0)| + s in  2|X(0)| + s in , which gives ⌧ ⇤ (3|X(0)| + s in ) ⌧ ⇤ (|X(0)| + x(0)) if |X(0)| (s in s ⇤ )/2.
ẋ(t)= h (1+ (t))µ1(sin x(t) z(t)) 1+ (sin x(t) z(t)) µ1(s⇤) 1+ (sin x(t) z(t)+s(t ⌧ (t)) s(t)) i x(t) . (21)
Also, since the initial functions for s are constant, we can use the Fundamental Theorem of Calculus to get

s(t ⌧ (t)) s(t) = R t t d ṡ(m)dm and (22) R t t d ṡ(m)dm = R t t d n⇣ µ1(s⇤) 1+ (s(m ⌧ (m))) (1+ (m))µ1(s(m)) 1+ (s(m)) ⌘ [s in s(m)] o dm + R t t d (1+ (m))µ1(s(m)) 1+ (s(m)) z(m)dm (23)
for all t 0, where t d = max{0, t ⌧ (t)}. Since s in s(t) 0 and 1 + (t) 1 + d > 0 hold for all t T a (|X(0)|), we get the following for all t T a (|X(0)|):

s(t ⌧ (t)) s(t) = R t t d ṡ(m)dm µ a ⌧ M n 1 + d µ 1 (s in ) R t t d |z(m)|dm o , (24)
by combining ( 22)-( 23), where µ a and ⌧ M are from ( 5) and ( 6), since the quantity in curly braces in ( 23) is bounded above by µ 1 (s ⇤ )(s in s(m)) and is nonnegative valued. Since is nondecreasing, we can use (24) to obtain

µ1(s⇤) 1+ (sin x(t) z(t)+s(t ⌧ (t)) s(t)) µ1(s⇤) 1+ (sin x(t) µa⌧ M z(t) (1+ d)µ1(sin) R t t d |z(m)|dm) . (25)
Using ( 25) to lower bound the second term inside the brackets in (21), and then the nonnegativity of x(t), we get ẋ(t)

[q + (x(t)) + (x(t), z t )] x(t) for all t T a (|X(0)|), where q denotes the left side of (10), (x(t), z t ) = (1+d)µ1(sin x(t) z(t)) 1+ (sin x(t) z(t))

(1+d)µ1(sin x(t)) andz t is defined by z t (`) = z(t + `) for all ` 0, and where we extend the domain of µ 1 so that µ 1 is zero on ( 1, 0). We can also use (15) and our exponential decay estimate (16) on z(t) to find a function T b 2 K 1 such that (x(t), z t ) q/2 and therefore also ẋ(t

1+ (sin x(t)) + µ1(s⇤) 1+ (sin x(t) µa⌧ M ) µ1(s⇤) 1+ ⇣ sin x(t) µa⌧ M z(t) (1+ d)µ1(sin) R t t d |z(m)|dm ⌘ and (x) = (1+d)µ1(sin x) 1+ (sin x) (1+d)µ1(sin) 1+ (sin) + µ1(s⇤) 1+ (sin µa⌧ M ) µ1(s⇤) 1+ (sin x µa⌧ M ) ,
) [0.5q + (x(t))] x(t) (26)
for all t T b (|X(0)|). Also, since (0) = 0, the continuity of  provides a constant x p 2 (0, (s in s ⇤ )/2) such that for all x 2 [0, x p ], we have (x) 0.25q. By enlarging T b , we can also assume that for all t T b (|X(0)|), we have

x(t) x p . ( 27
) This follows because if t T b (|X(0)|), then (26) gives ẋ(t) [0.5q+(x(t))] x(t) q 4 x(t) if x(t) 2 (0, x p ]. (28) To enlarge T b , notice that if |X(0)|  1 2 (s in s ⇤ ), then our formula (17) for X gives |x(0) (s in s ⇤ )|  0.5(s in s ⇤ ),
and then the triangle inequality gives x(0) 0.5(s in s ⇤ ) x p , which gives x(t) x p for all t 0, by (28). On the other hand, we can use (28) to find a positive valued function M such that x(t) x p for all t M(x(0)) and all choices of x(0) x and such that M is strictly increasing on [x, 1). Also, the triangle inequality gives where

|X(0)| + s in s ⇤ |x(0) (s in s ⇤ )| + s in s ⇤ x(0). Hence, M(|X(0)| + s in s ⇤ ) M(x(0)) if |X(0)| 0.5(s in s ⇤ ),
I(t) = R t t ⌧ (t)
0 (s(m)) ṡ(m)dm.

We next choose

U 1 (s) = R s s⇤ 0 m sin s⇤ m dm, (35)
which is C 1 over [0, s in ), and nonnegative valued at s(t) for all times t T c (|X(0)|), by the third step. It follows from (34) that its time derivative along all solutions of the closed loop system for all t T

c (|X(0)|) satisfies U1 (t) = s(t) s⇤ sin s (1 + (t))µ(s(t))z(t) (t)µ(s(t))(s(t) s ⇤ ) [µ1(s⇤) µ1(s(t))][1+ (s(t))]+µ1(s(t))I(t) [1+ (s(t ⌧ (t)))][1+ (s(t))] (s(t) s ⇤ ) , (36)
where we use U (t) to denote (d/dt)U (s(t)) to make our notation concise. From the third step and the fact that µ(s)  µ(s M ) for all s 0, we deduce that if t T

c (|X(0)|), then U1 (t)  |s(t) s ⇤ | µ1(s(t)) R t t ⌧ (t) 0 (s(m))| ṡ(m)|dm 1+ (s(t ⌧ (t))) + (s(t) s⇤)(µ1(s⇤) µ1(s(t))) 1+ (s(t ⌧ (t))) + c|s(t) s ⇤ |(|z(t)| + | (t)|)
holds, where c = µ(s M ) max{(1 + d)/(s in s 4 ), 1}. From the definition of ⇢ l in (7) and the fact that s(t) < s holds for all t T

c (|X(0)|), we obtain U1 (t)  h (s(t) s⇤)(µ1(s⇤) µ1(s(t))) 1+ (s(t ⌧ (t))) i + ĉ(t) + ⇢ l |s(t) s ⇤ | µ1(s(t)) R t t ⌧ (t) | ṡ(m)|dm 1+ (s(t ⌧ (t)))  (s(t) s⇤)(µ1(s⇤) µ1(s(t))) 2[1+ (s(t ⌧ (t)))] $ s (s(t) s⇤) 2 2[1+ (s(t ⌧ (t)))] + ĉ(t) + {|s(t) s ⇤ |} µ 1 (s(t ⌧ (t))+1.1µ a ⌧ M ) ⇥⇢ l R t t ⌧ (t) | ṡ(m)|dm o 1 1+ (s(t ⌧ (t))) (37) for all t T d (|X(0)|), where ĉ(t) = c|s(t) s ⇤ |(|z(t)| + | (t)|) and T d 2 K 1 is such that T d (r) T c (
r) for all r 0 and is such that the term in curly braces in (24) is bounded 5 Example Our theorem ensures asymptotic convergence for all componentwise positive initial states. To illustrate the theorem in an example, we use the growth rate and constant µ(s) = 0.5s 1+0.25s+2s 2 and s in = 1 .

(47)

Then Assumption 1 and the requirements of Lemma 1 hold using s M = 1/ p 2 and µ 1 (s) = 0.5s and (s) = 0.25s + 2s 2 for all s 0. We take s ⇤ = 0.8 2 (s M , s in ). Then the constants from Section 3 are $ s = $ l = 0.5, ⇢ l = 0.25 + 4 = 17/4, µ a = 0.4, and

⇢ m = ✓ 17 4 ◆ 2 1 4 max `2[0,1] (`+ 0.44⌧ M ) 2 1 + 0.25`+ 2`2 . ( 48
for di↵erent choices of the perturbation (t), where bac = max{j 2 {0, 1, 2, . . .} : j  a} is the floor function; see Mathematica (2015). In Figs. 34, we plot the components of the state and the control for di↵erent initial states, with our control D(t) from Theorem 1. For Fig. 3, we chose = 0, so (s(t), x(t)) converges to (s ⇤ , s in s ⇤ ) = (0.8, 0.2). In Fig. 4, we simulated our closed loop system with our feedback with the choice (t) = 0.15(1 + sin(t)), and the states instead converge toward an oscillation around the equilibrium, which agrees with our ISS result. Hence, our simulations help validate our theory. Our results also apply if we allow nonzero ⌧ f 's and nonconstant sample rates.

Conclusions

We

Fig. 1 .

 1 Fig. 1. Uptake Function from (3), Showing Maximizer sM = 1/ p 2 as Blue Dot and sin = 1 as Red Dot.

Fig. 2 .

 2 Fig.2. Plots of (1 + )µ(s) with µ from (3), sM = 1/ p 2 as Blue Dot, and sin = 1 as Red Dot, in Special Cases where = 0.75 (Red), = 0 (Blue), = 1.5 (Purple), and = 4 (Olive).

  ⇤ )/2, which are requirements for K 1 functions. The validity of the formula for T

							Therefore,
		(	⇣		⌘
	T	a (p) =	2p sin s⇤ ⌧ ⇤ ⌧ ⇤ (3p + s	3(sin s⇤) 2 in ),	+ s	in	, p < sin s⇤ 2 p sin s⇤ 2
	satisfies our requirements. (The formula for the restriction
	of T T a is 0 at 0, strictly increasing on [0, (s a 2 K 1 to [0, (s in s ⇤ )/2) was chosen to ensure that in s ⇤ )/2), and continuous at (s

in s a on [(s in s ⇤ )/2, 1) follows from Claim 1.) Third Step. We build a function T c 2 K 1 and a constant s 4 2 (0, s in ) such that s(t) < s 4 for all t T c (|X(0)|)

  so it su ces to enlarge T , M ⇤ } 2 K 1 . Fourth Step. We construct a functional U 1 , which we later add to a double integral term to prove our ISS property. Using z as defined in (13), and the structure of µ in (4), we get this for all t 0:

								for all t T	c (|X(0)|), where T	c = max{T	a , T
								h	i
								ṡ(t) =	µ1(s⇤) 1+ (s(t ⌧ (t)))	µ1(s(t)) 1+ (s(t))	[s	in	s(t)]	(32)
								+(1 + (t))µ(s(t))z(t)	(t)µ(s(t))(s	in	s(t)) .
								Since the Fundamental Theorem of Calculus gives
								(s(t))	(s(t ⌧ )) =	R	t	0 (s(m)) ṡ(m)dm, (33)
								t ⌧ (t)
								we can use a common denominator in (32) to get
								ṡ(t) =
								n	o
								[µ1(s⇤) µ1(s(t))][1+ (s(t))]+µ1(s(t))I(t)	[s	in	s(t)]	(34)
								[1+ (s(t ⌧ (t)))][1+ (s(t))]
								+ {(1 + (t))µ(s(t))z(t)	(t)µ(s(t))(s	in	s(t))}
							b so that
				(
	T	b (r)	2r sin s⇤ M(1.5(s M(r + s in s ⇤ ), in	s ⇤ )), r  sin s⇤ 2 r> sin s⇤ 2	(29)
	b on [0, (s s ⇤ )/2] was chosen for the same reason that we chose the in where the formula for the lower bound for T corresponding part of the formula for T a in the second step.
	Also, our exponential decay condition (16) on z(t) and
	(18) provide a function M ⇤ 2 K 1 such that for all t M ⇤ (|X(0)|), the inequalities
				|z(t)|  0.25x	p  0.25s	in	(30)
	hold. Using the definition of z and (27)-(30), and setting
	s 4 = s	in	0.75x	p , we deduce that s 4 2 (0, s	in ) and
	s(t) = x(t) z(t) + s  x p + 0.25x p + s in = s 4 in  x(t) + 0.25x	p + s	in	(31)

b

   0.23 and d = 0. To illustrate our findings, we ran Mathematica simulations with ⌧ M = 0.23, t j = 0.23j for all j 0, and ⌧ f = 0, using the NDSolve command and the closed loop dynamics

					)
	If we set ⌧	M = 0.5 in (48), then we obtain ⇢ m = 2.371,
	so ⇢ from Assumption 2 hold if ⌧ m  2.371 for all ⌧ M 2 (0, 0.5]. The requirements (11) M 2 (0, 4/17). Also, our con-dition (10) from Assumption 2 reads
		0.5(1+d)	0.5(0.8)
		13/4 = 0.0264 > 0 1+0.25(1 0.4⌧ M )+2(1 0.4⌧ M ) 2	(49)
	when we choose any ⌧ of our assumptions hold if ⌧ M 2 (0, 0.24) and d = 0. Hence, all
	8
	> > >	ṡ(t) =	0.4(1 s(t)) 1+0.25s(0.23bt/0.23c)+2s 2	(0.23bt/0.23c)
	>
	> <			0.5(1+ (t))s(t) 1+0.25s(t)+2s 2 (t) x(t)
	> > >	ẋ(t) = 0.5(1+ (t))s(t)x(t) 1+0.25s(t)+2s 2 (t)
	>
	> :			0.4x(t) 1+0.25s(0.23bt/0.23c)+2s 2	(0.23bt/0.23c)

M

  used a new barrier Lyapunov function approach to prove input-to-state stability with respect to uncertainties in the growth functions in two state chemostats, in closed loop with output feedback controls. Our results are significant because only delayed and sampled measurements of the substrate level are available for use in the control, and because we allow a general class of growth functions that Fig.3. Top: Solution (s(t), x(t)) of (50) for Initial State (1, 1) Converging to (0.8, 0.2) with ⌧M = 0.23, (t) = 0, s(t) in Red, and x(t) in Blue. Bottom: Control D(t) in Green.

	1.0
	0.8
	0.6
	0.4
	0.2
	0		50		100	150	200
	0.24
	0.22
	0.20
	0.18
	0.16
	0.14
	0		50		100	150		200
	1.0
	0.8
	0.6
	0.4
	0.2
		10	20	30	40	50	60	70
	0.35
	0.30
	0.25
	0.20
	0	10	20	30	40	50	60	70

Fig. 4

. Top: Solution (s(t), x(t)) of (50) for Initial State (0.25, 1) with ⌧M = 0.23, (t) = 0.15(1 + sin(t)), s(t) in Red, and x(t) in Blue. Bottom: Control D(t) in Green.

version and this paper. Maliso↵ was supported by NSF-ECCS Grants 1102348 and 1408295 and Roy P. Daniels Professorship #3 in the LSU College of Science. The first author acknowledges financial support from the MATH-AmSud project "Stability and Dichotomies in Di↵erential and Delay Equations". Some of the third author's work was done while he visited the US National Science Foundation in Arlington, VA. He appreciated their hospitality during the period. The authors thank Paul Varnell for assisting with the simulations.

above by 0.1µ a ⌧ M for all t T d (|X(0)|); such a T d can be found using the exponential decay estimate (16) on z(t) and ( 18), and will depend on d. The second inequality in (37) follows from the definition of $ s in (7), which gives (µ 1 (s ⇤ ) µ 1 (s(t))(s ⇤ s(t)) $ s (s ⇤ s(t)) 2 (38)

when we use the Mean Value Theorem (to upper bound the first quantity in square brackets in (37) by the first two terms after the second inequality in (37)).

Hence, our choice of ⇢ m > 0 from (8) gives U1 (t)  (s(t) s⇤)(µ1(s⇤) µ1(s(t)))

where we used Young's inequality ab  $s 2 a 2 + 1 2$s b 2 with a and b chosen to be the first and second terms in curly braces in (37) respectively, and Jensen's inequality to get

(40)

Fifth

Step. We define

along all solutions of our dynamics. Then (39) gives U2 (t)  (s(t) s⇤)(µ1(s⇤) µ1(s(t)))

for all t T d (|X(0)|). Let a and b denote the first and second terms in curly braces in (34) respectively, and let p 0 2 (0, 1) be a constant that we will specify later. Then (34) gives ( ṡ(t)

, where we also used the relation (c

1 and c 2 to bound b 2 . Using our definition of $ l in ( 7), (40), and (ā + b) 2  2ā 2 + 2 b2 for suitable choices of ā 0 and b 0, it follows that for all t T

and so also U2 (t)  (s(t) s⇤)(µ1(s⇤) µ1(s(t)))

since µ 1 is nondecreasing. We conclude from our upper bounds from (11) on ⌧ M , the fact that µ

and the positivity of $ s in (38) that for a small enough constant p 0 2 (0, 1), there is a constant c e > 0 such that along all trajectories of ( 9 Remark 3 The constantness of the initial functions was used to obtain (23). Theorem 1 remains true if instead of assuming that both components s and x of the initial functions are constant, we only assume that the initial function for s is constant (by the same proof ). In fact, we can drop the constantness assumptions on the initial functions entirely, by viewing the calculations in the proof of Theorem 1 as holding for t ⌧ M , instead of all t 0. See [START_REF] Mazenc | Stabilization in a chemostat with sampled and delayed measurements[END_REF], where the case of nonconstant initial functions was handled in the special case where the 's are zero. ⇤ Appendix B: Finishing the Proof of Theorem 1

We convert the decay estimate (46) into the final ISS estimate. Since s(t) 2 (0, s) for all t 0, it follows from the structure of our z(t) dynamics in ( 14) and the fact that is nondecreasing that the time derivative of

g z 2 (t) along all trajectories of the z dynamics. Therefore, the time derivative of U


Frédéric Mazenc

Michael Malisoff
email: malisoff@lsu.edu.

Silviu-Iulian Niculescu
email: silviu.niculescu@lss.supelec.fr.

Frederic Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

† F Mazenc

Epi Disco

Stability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory-Based Approach

Keywords: delay, reduction approach, time-varying, stability

I Introduction

Delays occur in many engineering applications [START_REF] Belem Saldivar Marquez | Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach[END_REF][START_REF] Downey | Compensating for fatigue-induced time-varying delayed muscle response in neuromuscular electrical stimulation control[END_REF][START_REF] Petit | Motion planning for two classes of nonlinear systems with delays depending on the control[END_REF][START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], where it may not be feasible to continuously measure the current state of the system. Stability analysis for systems with time-varying delays is significant from the applied and theoretical viewpoints, especially when the delays have discontinuities. Motivations for nonconstant delays occur in networked systems, and systems with nonconstant delays have been widely studied; see [START_REF] Bekiaris-Liberis | Nonlinear Control under Nonconstant Delays[END_REF][START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF][START_REF] Dieulot | Tracking control of a nonlinear system with input-dependent delay[END_REF][START_REF] Fridman | Delay-dependent stability and H ∞ control: constant and time-varying delays[END_REF][START_REF] Kharitonov | On the stability of linear systems with uncertain delay[END_REF][START_REF] Krstic | Lyapunov stability of linear predictor feedback for time-varying input delay[END_REF][START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF][START_REF] Sbarciog | Nonlinear predictive control of processes with variable time delay. A temperature control case study[END_REF][START_REF] Zhou | Truncated Predictor Feedback for Time-Delay Systems[END_REF] and [START_REF] Gu | Stability of Time-Delay Systems[END_REF]Chapt. 6]. Systems with nonconstant delays are challenging, since in most cases, standard frequency-domain and Lyapunov-Krasovskii methods are not generally applicable.

We can use time-varying delays h(t) to model many situations, such as sampling in controllers [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] (where ḣ(t) = 1 almost everywhere, so the delay has jump discontinuities at each sampling time) and delayed muscle response due to fatigue [START_REF] Downey | Compensating for fatigue-induced time-varying delayed muscle response in neuromuscular electrical stimulation control[END_REF] (where h may increase). Moreover, the upper bound on the time-varying delays that preserve stability may exceed the corresponding bound for constant delays; see [START_REF] Liu | Stability of linear systems with general sawtooth delay[END_REF], as well as the discussion [START_REF] Louisell | New examples of quenching in delay differential equations having time-varying delay[END_REF] of quenching. Also, even if a dynamics of the form ẋ(t) = F(x(t), x(t -τ )) is exponentially stable for all constant delays τ on a given interval [0, τ max ], it may not necessarily be true that ẋ(t) = F(x(t), x(t -h(t))) is exponentially stable for all nonconstant delays h : [0, ∞) → [0, τ max ]; see the Markus-Yanabe phenomenon [START_REF] Haidar | Further remarks on Markus-Yamabe instability for time-varying delay differential equations[END_REF], [START_REF] Hale | Ordinary Differential Equations. Second Edition Robert E[END_REF], [START_REF] Markus | Global stability criteria for differential systems[END_REF]. There may be other types of time-varying delays, such as distributed delays. This makes the study of systems with nonconstant delays an important problem in control theory.

The study of systems with time-varying delays is often complicated by the fact that the systems may be time-varying when the delay is zero [START_REF] Gu | Stability of Time-Delay Systems[END_REF]. Time-varying systems often arise in tracking problems, even if we use a linear approximation (since the coefficient matrices in the linearization will be time-varying) [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF]. It is common in the literature to require the largest possible delay value to be sufficiently small [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], or that certain matrices associated with the vector field are confined to certain compact sets [START_REF] Gu | Stability of Time-Delay Systems[END_REF]Chapt. 6]. Another standard assumption on the delay h(t) is that φ(t) = t -h(t) is invertible [START_REF] Krstic | Lyapunov stability of linear predictor feedback for time-varying input delay[END_REF]. The work [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] is an exception. In [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF], an averaging assumption on an integral of | ḣ| 2 is imposed, instead of assuming that φ(t) = t -h(t) is invertible.

The present paper has connections with [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF], since our assumptions also involve integrals that incorporate the delays h. However, our main results do not require the differentiability of the delay function h, and [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] is confined to constant coefficient linear systems with predictive controls. To cope with the time-varying character of the systems as well as the time-varying delay, we apply a trajectory-based approach from [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF]. In our opinion, two potential advantages of the approach from [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF] are that it does not require the Lyapunov or small gain conditions that are common in the literature, and that it does not restrict the upper bounds on h(t). We are unaware of any other method to prove the results in the present paper. The size restrictions on h(t) from [START_REF] Gao | A new delay system approach to network-based control[END_REF] and [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF] are not in general satisfied by our systems. Furthermore, we also do not need the time-varying matrices involving the vector fields to belong to specific compact sets, as was imposed in [START_REF] Gu | Stability of Time-Delay Systems[END_REF]Chapt. 6]. Unlike [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF], we cover linear time-varying systems with a time-varying piecewise continuous delay h(t). The assumptions from [START_REF] Krstic | Lyapunov stability of linear predictor feedback for time-varying input delay[END_REF] may be violated by our systems, and our results imply exponential stability.

Two of our theorems here are inspired by emulation, insofar that they place assumptions on corresponding undelayed systems, and then they study the effects of the delays. Emulation is generally a two step process. In the first step, one solves a stabilization problem with the input delays set to zero, by constructing a Lyapunov function for the closed-loop undelayed system and then computing decay estimates for the Lyapunov function. In the second step, one reintroduces the input delays and converts the Lyapunov function into a Lyapunov-Krasovskii functional for the corresponding input delayed systems, in order to find upper bounds on the input delays that the system can tolerate while maintaining the stability in closed loop. A potential advantage of emulation is that it makes it possible to use relatively simple controllers, but emulation cannot always cover communications and other engineering applications where the delays are long relative to the total response time of the system [START_REF] Gao | A new delay system approach to network-based control[END_REF].

In such cases reduction appears as a useful alternative, in which the controller involves a dynamic extension, and can often compensate for arbitrarily long input delays [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF][START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF][START_REF] Figueredo | Robust stability criteria for uncertain systems with delay and its derivative varying within intervals[END_REF][START_REF] Kwon | Feedback stabilization of linear systems with delayed control[END_REF][START_REF] Mazenc | Lyapunov-Krasovskii functionals and application to input delay compensation for linear time-invariant systems[END_REF]. Although it has its origins in the Smith predictor [START_REF] Smith | A controller to overcome dead time[END_REF] for linear systems (hence the term prediction [START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF][START_REF] Krstic | Lyapunov stability of linear predictor feedback for time-varying input delay[END_REF]), recent prediction results apply to many other systems, including perturbed dynamics [2-5, 8, 27]. The work [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] used a transport PDE, reduction, and a generalized Halanay inequality [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF][START_REF] Ivanov | Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima[END_REF] to prove stability of linear time invariant systems with known input delays h(t) that can satisfy ḣ(t) > 1 for some t's. This sets [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] apart from the usual results that require sup t ḣ(t) ≤ 1, so [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] is a significant advance in the use of reduction to prove stability of systems with time-varying delays. One of the key assumptions in [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] is that | ḣ| 2 is small in a suitable averaged sense, without requiring a bound on | ḣ|. This makes it possible to use [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] to cover chattering in delays that occur in many engineering systems.

In our third and fourth theorems, we pursue a related line of research involving dynamic extensions for time-varying systems with time-varying delays, but our results differ from [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] in several key ways. First, we allow the delays to contain discontinuities and uncertainties. Also, we use a very different set of assumptions from [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF], which allow us to prove stability using our trajectory approach from [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF] instead of using transport PDEs. It is important to point out that the trajectory based method from [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF] was not used in [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF]. We also provide an alternative approach to dynamic controls under time-varying delays, using a dynamic extension from [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF][START_REF] Mondié | Finite spectrum assignment of unstable time-delay system with a safe implementation[END_REF] from the theory of spectrum assignment. In all of our results in this paper, we allow time-varying linear systems with additive uncertainty in the coefficient matrices, which can capture important cases where the system arises from linearizing around a desired reference trajectory, but where the reference trajectory itself is subject to uncertainties.

Our conference papers [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF][START_REF] Mazenc | Stability analysis for systems with time-varying delay: trajectory based approach[END_REF] presented special cases of this work in which there is no uncertainty in the coefficient matrices, and they did not include the proofs detailed below. By covering linear time-varying systems with uncertain coefficients and providing proofs, this work adds considerable value relative to [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF][START_REF] Mazenc | Stability analysis for systems with time-varying delay: trajectory based approach[END_REF].

II Definitions, Notation, and Prerequisites

Throughout this paper, all dimensions are arbitrary, unless indicated otherwise. The usual Euclidean norm, and its induced matrix norm, are denoted by | • |, | • | I denotes the (essential) supremum over any interval I ⊆ R, and I r is the identity matrix in dimension r. Let λ max (Q) > 0 (resp., λ min (Q) > 0) be the biggest (resp., smallest) eigenvalue of any positive definite symmetric matrix Q, and C 1 be the set of all continuously differentiable functions, whose domains and ranges will be clear from the context. For any matrices P and Q in R n×n , we use P < Q (resp., P ≤ Q) to mean that Q -P is positive definite (resp., positive semidefinite). For each constant delay bound τ > 0, let C([-τ, 0], R n ) be the set of all continuous R n -valued functions defined on [-τ, 0]. We let C in be the set of all absolutely continuous functions φ ∈ C([-τ, 0], R n ), which we call the set of all initial functions. For each continuous function ϕ : [-τ, ∞) → R n and all t ≥ 0, we set ϕ t (m) = ϕ(t + m) for all m ∈ [-τ, 0]. A function defined on an interval I ⊆ R is called piecewise continuous provided it is continuous at all points of I except at finitely many points on each bounded subinterval of I (which includes continuous functions as a special case). When referring to a function φ that is defined on [0, ∞) as being differentiable, we view φ (0) as being the right derivative at 0. Let K denote the set of all strictly increasing continuous functions α : [0, ∞) → [0, ∞) such that α(0) = 0; if, in addition, α is unbounded, then we say that α is of class K ∞ . We say that a continuous function

β : [0, ∞) × [0, ∞) → [0, ∞) is of class KL provided (a)
for each s ≥ 0, the function β(•, s) is in K and (b) for each choice of r ≥ 0, the function β(r, •) is non-increasing and satisfies lim s→∞ β(r, s) = 0. For simplicity, we assume that all of our initial times for our initial value problems are 0, but more general cases can be covered by similar arguments. We say that a system of the form ẋ(t) = f (t, x(t), u(t -h(t)), ε(t)), having a time delay h(t) that admits a constant τ > 0 such that 0 ≤ h(t) ≤ τ for all t ≥ 0, is input-to-state stable (or ISS) [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] with respect to ε : [0, ∞) → R m provided the following condition holds: There exist functions β ∈ KL and γ ∈ K ∞ such that for all initial functions and all locally bounded piecewise continuous functions ε : [0, ∞) → R m , the corresponding solution σ f (t) = φ(t, x 0 , ε) of the system is defined on [-τ, ∞) and satisfies

|φ(t, x 0 , ε)| ≤ β(|x 0 | [-τ,0] , t) + γ(|ε| [0,t]
) for all t ≥ 0. This reduces to the standard global exponential stability (or GES) condition when the perturbations ε are all zero and β(s, t) = c 1 se -c2t for some constants c 1 > 0 and c 2 > 0. We say that a function P : R × C in → C in is locally Lipschitz in its second argument provided that for each φ ∈ C in , there exist constants r > 0 and K r > 0 such that for all φ 1 ∈ C in and φ 2 ∈ C in such that |φ 1 -φ| [-τ,0] ≤ r and |φ 2 -φ| [-τ,0] ≤ r, and for all t ∈ R, we have

|P(t, φ 1 ) -P(t, φ 2 )| [-τ,0] ≤ K r |φ 1 -φ 2 | [-τ,0] .

III Key Preliminary Results

This section provides three lemmas and a theorem that are key for proving the stability results in the next sections; see [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF], (69), and (83) for the classes of systems that we will later cover by our theory. We first establish a slightly less restrictive version of the trajectory based result from [35, Lemma 1]; it differs from [35, Lemma 1] because the offset term in [35, Lemma 1] was |d| [0,t] /(1 -ρ) 2 while the offset in our new version is |d| [0,t] /(1 -ρ). Since ρ ∈ (0, 1), our new lemma therefore provides a tighter estimate.

Lemma 1. Let T * > 0 be a constant. Let w : [-T * , ∞) → [0, ∞) be a piecewise continuous locally bounded function and d : [0, ∞) → [0, ∞) be piecewise continuous. Assume that there exists a constant ρ ∈ (0, 1) such that

w(t) ≤ ρ|w| [t-T * ,t] + d(t) (1)
holds for all t ≥ 0. Then the inequality

w(t) ≤ |w| [-T * ,0] e ln(ρ) T * t + 1 1-ρ |d| [0,t] (2)
holds for all t ≥ 0.

Proof. Set d b (t) = |d| [0,t] for all t ≥ 0 and α(t) = d b (max{0, t})/(1 -ρ) for all t ≥ -T * . Then, (1) implies that

w(t) ≤ ρ|w| [t-T * ,t] + d b (t)
for all t ≥ 0. Since w is nonnegative valued and α is nondecreasing, this leads to

w(t) -α(t) ≤ ρ|w| [t-T * ,t] + d b (t) -α(t) = ρ |w| [t-T * ,t] -α(t) ≤ ρ sup ∈[t-T * ,t] (w( ) -α( )) (3)
for all t ≥ 0. Let t ≥ 0 and Ω(t) = max{w(t) -α(t), 0}. If w(t) ≤ α(t), then Ω(t) = 0, which implies that

Ω(t) ≤ ρ|Ω| [t-T * ,t] . (4)
T * t = | max{w -α, 0}| [-T * ,0] e ln(ρ) T * t (5)
holds for all t ≥ 0. We now consider two cases. If t ≥ 0 is such that w(t) > α(t), then (5) leads to

0 < w(t) -α(t) ≤ | max{w -α, 0}| [-T * ,0] e ln(ρ) T * t . (6)
It follows that, in this case, we have the following:

0 < | max{w -α, 0}| [-T * ,0] = sup{w(m) -α(m) : m ∈ [-T * , 0], w(m) > α(m)} ≤ |w| [-T * ,0] (7)
and therefore [START_REF] Belem Saldivar Marquez | Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach[END_REF] gives

w(t) ≤ |w| [-T * ,0] e ln(ρ) T * t + α(t). (8)
On the other hand, if t ≥ 0 is such that w(t) ≤ α(t), then (8) holds again, since the first right side term in ( 8) is nonnegative, so (8) holds in both cases. This proves the lemma.

The next lemma differs from a Lyapunov type argument because the function p is allowed to take negative values, which makes it very different from a contractiveness lemma.

Lemma 2. Let Z : R → [0, ∞), R : R → [0, ∞) and p : R → R be functions such that Z is C 1 , p is piecewise continuous and locally bounded, R is locally integrable, and

Ż(t) ≤ -p(t)Z(t) + Z(t)R(t) (9)
holds for all t ≥ 0. Then, for all nonnegative values s and t such that s ≤ t, the inequality

Z(t) ≤ e -1 2 t s p(m)dm Z(s) + 1 2 t s e -1 2 t p(m)dm R( )d (10) holds.
Proof. Let > 0 be a constant and Y ( ) = Z( ) + -√ . Since R and Z are nonnegative valued, (9) leads to

Ẏ ( ) ≤ -p( ) Z( ) 2 √ Z( )+ + √ Z( ) 2 √ Z( )+ R( ) ≤ -p( ) 2 Y ( ) + 1 2 R( ) + p( ) 2 Y ( ) + -(Y ( )+ √ ) 2 Y ( )+ √ = -p( ) 2 Y ( ) + 1 2 R( ) - √ p( ) 2(Y ( )+ √ ) Y ( ) (11)
for all ∈ [s, t]. By integrating (11) on [s, t], and then substituting in the definition of Y , we obtain:

Z(t) + - √ ≤ e -1 2 t s p(m)dm Z(s) + - √ + t s e -1 2 t p(m)dm 1 2 R( )d - √ t s p( )e -1 2 t p(m)dm 1 2 Z( ) + Z( ) + - √ d . (12)
Since ( 12) is valid for all > 0, and since the quantity in curly braces in ( 12) is bounded by 1, the inequality ( 12) is true for = 0. This proves the lemma.

We introduced > 0 in the previous proof because it makes it possible to establish [START_REF] Dieulot | Tracking control of a nonlinear system with input-dependent delay[END_REF] with functions that are of class C 1 everywhere. By contrast, if we had set = 0 throughout the proof, then we would have needed to study the non-differentiable function Y 0 . The next lemma involves a decay condition that is reminiscent of a Riccati inequality, but is different since the function p(t) in the lemma can again take positive and negative values.

Lemma 3. Consider the system Ẋ(t) = F(t)X(t) + M(t, X t ) , ( 13
)
where X is valued in R n , F : [0, ∞) → R n×n is piecewise continuous and locally bounded, and M : [0, ∞)×C in → R n is locally integrable in its first argument and locally Lipschitz in its other argument. Assume that there is a C 1 function P : [0, ∞) → R n×n such that P is symmetric and positive definite for all t ≥ 0, constants p s and p l such that p l ≥ p s > 0, and a continuous function p : [0, ∞) → R such that Ṗ(t) + F(t) P(t) + P(t)F(t) ≤ -p(t)P(t) and p s I n ≤ P(t) ≤ p l I n [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF] hold for all t ≥ 0. Then

|X(t)| ≤ p l ps e -1 2 t r p(m)dm |X(r)| + p l ps t r e -1 2 t p(m)dm |M( , X )|d (15)
holds for all t and r such that t ≥ r ≥ 0 and for all solutions of (13).

Proof. Let ν(t, X) = X P(t)X. For each t ≥ 0, let P(t) 1/2 be a symmetric positive definite matrix such that

P(t) = P(t) 1/2 P(t) 1/2 . Then ν(t, X) = |X P(t) 1/2 | and p s |X| 2 ≤ ν(t, X) ≤ p l |X| 2
hold for all t ≥ 0 and X ∈ R n . From ( 14), it follows that the time-derivative of ν along the trajectories of ( 13) satisfies:

ν(t) ≤ -p(t)ν(t, X(t)) + 2 ν(t, X(t)) P(t) 1/2 M(t, X t ) . (16)
We now apply Lemma 2, with the choices

Z(t) = ν(t, X(t)) and R(t) = 2|P(t) 1/2 M(t, X t )|. It follows that for all r ≥ 0 and t ≥ r, ν(t, X(t)) ≤ e -1 2 t r p(m)dm ν(r, X(r)) + t r e -1 2 t p(m)dm P( ) 1/2 M( , X ) d (17)
holds. By our bounds

p s |X| 2 ≤ ν(t, X) ≤ p l |X| 2 , ( 17
) leads to |X(t)| ≤ p l ps e -1 2 t r p(m)dm |X(r)| + 1 √ ps t r e -1 2 t p(m)dm P( ) 1/2 M( , X ) d . (18)
Since |P(t) 1/2 | ≤ √ p l holds for all t ≥ 0, we conclude that ( 15) is satisfied.

The next result is important for proving the results in the next sections on time-varying linear systems with perturbations and delays. See Remark 1 for a way to convert its conclusion into an ISS estimate.

Theorem 1. Let g > 0 be any constant. Consider any system of the form

Ẋ(t) = F(t)X(t) + U(t, X t ) (19)
with initial conditions in C([-g, 0], R n ), where F : [0, ∞) → R n×n is piecewise continuous and locally bounded, and U is locally integrable in its first argument t and locally Lipschitz in its second argument. Assume that there are locally bounded piecewise continuous functions λ

: [0, ∞) → [0, ∞) and µ : [0, ∞) → [0, ∞) such that |U(t, φ)| ≤ µ(t) sup m∈[-g,0] |φ(m)| + λ(t) (20)
holds for all t ≥ 0 and all φ ∈ C in . Finally, assume there exist a

C 1 function P : [0, ∞) → R n such that P(t)
is symmetric and positive definite for all t ≥ 0; constants p * > 0, T ≥ g, and δ ∈ (0, 1); a continuous function p : [0, ∞) → [-p * , ∞); and two constants p s and p l such that p l ≥ p s > 0 and such that

Ṗ(t) + F(t) P(t) + P(t)F(t) ≤ -p(t)P(t) and p s I n ≤ P(t) ≤ p l I n (21)
hold for all t ≥ 0 and such that

p l ps e -1 2 t t-T p(m)dm + t t-T e -1 2 t p(m)dm µ( )d ≤ δ (22)
holds for all t ≥ T . Then for all t ≥ T , the inequality

|X(t)| ≤ e ln(δ) T +g (t-T ) |X| [-g,T ] + 2 √ p l (e T p * /2 -1) p * (1-δ) √ ps |λ| [0,t] (23)
is satisfied along all solutions of [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF].

Proof. By [START_REF] Ivanov | Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima[END_REF], we deduce that Lemma 3 applies, with M = U. Hence, for all t and r such that 0 ≤ r ≤ t,

|X(t)| ≤ p l ps e -1 2 t r p(m)dm |X(r)| + p l ps t r e -1 2 t p(m)dm |U( , X )|d (24)
is satisfied. Using [START_REF] Hale | Ordinary Differential Equations. Second Edition Robert E[END_REF], it follows that if 0 ≤ r ≤ t, then

|X(t)| ≤ p l ps e -1 2 t r p(m)dm |X(r)| + p l ps t r e -1 2 t p(m)dm µ( ) sup m∈[ -g, ] |X(m)| + λ( ) d ≤ p l ps e -1 2 t r p(m)dm + t r e -1 2 t p(m)dm µ( )d |X| [r-g,t] + p l ps t r e -1 2 t p(m)dm λ( )d . (25)
It follows from [START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF], and from choosing r = t -T in [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF], that for all t ≥ T , we have the following:

|X(t)| ≤ δ|X| [t-T -g,t] + p l ps t t-T e -1 2 t p(m)dm λ( )d . ( 26
)
Let S( ) = |X( + T )| for all ≥ 0. By setting t = + T in [START_REF] Kharitonov | On the stability of linear systems with uncertain delay[END_REF], it follows that for all ≥ 0, we have

S( ) ≤ δS [ -T -g, ] + p l ps +T e -1 2
+T q p(a)da λ(q)dq .

(

) 27
Since δ ∈ (0, 1), we deduce from Lemma 1 with T * = T + g that for all ≥ 0, we have

S( ) ≤ S [-T -g,0] e ln(δ) T +g + 1 1-δ sup u∈[0, ] p l ps u+T u e -1 2
u+T q p(a)da λ(q)dq .

Consequently, by setting = t -T in [START_REF] Krstic | Lyapunov stability of linear predictor feedback for time-varying input delay[END_REF], it follows that for all t ≥ T , we have

|X(t)| ≤ |X| [-g,T ] e ln(δ) T +g (t-T ) + 1 1-δ p l ps sup u∈[0,t-T ] u+T u e -1 2 u+T p(a)da λ( )d . (29)
Then our final estimate [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF] follows from our choice of p * > 0.

Remark 1. Although [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF] only holds when t ≥ T , we can use the following Gronwall inequality argument to extend [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF] to all of [0, ∞) so that it becomes an ISS estimate. The first inequality in [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] (with the choice r = 0) and our lower bound -p * on p(t) imply that for all t ∈ [0, T ], we have

|X(t)| ≤
+ p l ps t 0 e t- 2 p * µ( ) sup m∈[0, ] |X(m)|d ≤ θ 1 (T ) t 0 sup m∈[0, ] |X(m)|d + θ 2 (t, T ) , (30)
where

θ 1 (T ) = p l ps e T 2 p * |µ| [0,T ] and θ 2 (t, T ) = p l ps e T p * 2 1 + T |µ| [0,T ] |X| [-g,0] + p l ps e T p * 2 T |λ| [0,t] . (31)
Hence, the function

G( ) = sup m∈[0, ] |X(m)| is such that |X(t)| ≤ G(t) ≤ θ 1 (T ) t 0 G( )d + θ 2 (t, T ) (32)
holds for all t ∈ [0, T ]. It follows from Gronwall's inequality that for all t ∈ [0, T ], we have

t 0 G( )d = t 0 sup m∈[0, ] |X(m)|d ≤ T e T θ1(T ) θ 2 (t, T ). ( 33
)
Combining [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF] with [START_REF] Markus | Global stability criteria for differential systems[END_REF], and then using the fact that δ ∈ (0, 1) and the definition of θ 2 , we obtain

|X(t)| ≤ θ 1 (T )T e T θ1(T ) θ 2 (t, T ) + θ 2 (t, T ) ≤ θ 1 (T )T e T θ1(T ) + 1 p l ps e T p * /2 (1 + T |µ| [0,T ] )e ln(δ) T +g (t-T ) |X| [-g,0]
+ θ 1 (T )T e T θ1(T ) + 1

p l ps e T p * /2 T |λ| [0,t] (34)
for all t ∈ [0, T ]. Since θ 2 (T, T ) ≥ |X| [-g,0]
, it follows from [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF] and the first inequality in [START_REF] Mazenc | New technique for stability analysis for time-varying systems with delay[END_REF] that

|X(t)| ≤ max |X| [-g,0] , θ 1 (T )T e T θ1(T ) + 1 θ 2 (T, T ) e ln(δ) T +g (t-T ) + 2 √ p l( e T p * /2 -1) p * (1-δ) √ ps |λ| [0,t] ≤ θ 1 (T )T e T θ1(T ) + 1 p l ps e T p * 2 1 + T |µ| [0,T ] |X| [-g,0] e ln(δ)
T +g (t-T )

+ θ 1 (T )T e T θ1(T ) + 1 T √ p l e T p * 2 √ ps + 2 √ p l (e T p * /2 -1) p * (1-δ) √ ps |λ| [0,t] (35)
holds for all t ≥ T , where the second inequality used the fact that X(t) does not depend on values λ(t) for t ≥ T , so we can replace θ 2 (T, T ) by θ 2 (t, T ). Comparing ( 34)- [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], it follows that [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF] holds for all t ≥ 0.

IV Delayed Systems with Stabilizing Undelayed Parts IV.1 Statement and Proof of Result

We next consider systems of the form

ẋ(t) = [A(t) + δ A (t)]x(t) + [B(t) + δ B (t)]x(t -h(t)) + ε(t) (36)
whose stabilizing part A(t)x(t) consists of a term without delay; but see the next section for other cases. We assume that the state x is valued in R n , the initial conditions are in [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF] as being a sum of known and unknown matrices as in [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF], because of how our assumptions involve A.

C in , h : [0, ∞) → [0, ∞) is a piecewise continuous function, and A : [0, ∞) → R n×n , B : [0, ∞) → R n×n ,
One consequence of our second theorem will be that even when the term B(t)x(t -h(t)) does not help stabilize the system, we can prove key asymptotic stability properties in cases where classical stability analysis techniques (such as Razumikhin's theorem or Lyapunov-Krasovskii functional constructions) do not seem to apply; see Section VIII for examples. We introduce the undelayed system ż(t) = A(t)z(t) [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF] with z valued in R n and the following two assumptions:

Assumption 1. There exist a continuous function p 1 : [0, ∞) → R, a C 1 function P : [0, ∞) → R n×n such that P (t) is symmetric and positive definite for all t ≥ 0, and positive constants p 2 , p 3 , and p 4 such that

p 1 (t) ≥ -p 4 and p 2 |z| 2 ≤ V (t, z) ≤ p 3 |z| 2 (38)
hold for all t ≥ 0 and z ∈ R n , where

V (t, z) = z P (t)z, (39)
and such that along all trajectories z(t) of ( 37), we have

d dt V (t, z(t)) ≤ -p 1 (t)V (t, z(t)) (40)
for all t ≥ 0. Also, the function h(t) is locally bounded and piecewise continuous.

Assumption 2. There is a constant g ≥ 0 such that 0 ≤ h(t) ≤ g for all t ≥ 0 .

Also, there are constants T > g and δ ∈ (0, 1) such that

p 3 p 2 e -1 2 t t-T p1(m)dm + t t-T e -1 2 t p1(m)dm |δ A ( )| + |B( ) + δ B ( )| d ≤ δ for all t ≥ T, (42)
where the p i 's are from Assumption 1.

See Section IV.2 for ways to check the preceding assumptions. Our first theorem for (36) is as follows:

Theorem 2. If Assumptions 1-2 hold, then, with the choices

H 1 = θ 1 (T )T e T θ1(T ) + 1 p3 p2 e T p4/2 , ∆(t) = |δ A (t)| + |B(t) + δ B (t)|, and θ 1 (T ) = p3 p2 e T p4/2 |∆| [0,T ] , (43)
all solutions x(t) of (36) satisfy

|x(t)| ≤ 1 + T |∆| [0,T ] H 1 |x| [-g,0] e ln(δ) T +g (t-T ) + H 1 T + 2 (1-δ)p4 p3 p2 e T p4/2 -1 |ε| [0,t] (44)
for all t ≥ 0.

Proof. Set R(t, x t ) = δ A (t)x(t) + [B(t) + δ B (t)]x(t -h(t)) + ε(t). (45)
Then ( 36) can be rewritten as ẋ(t) = A(t)x(t) + R(t, x t ). Also, for all t ≥ 0, we have

|R(t, x t )| ≤ |δ A (t)||x(t)| + |B(t) + δ B (t)||x(t -h(t))| + |ε(t)| ≤ (|δ A (t)| + |B(t) + δ B (t)|) sup m∈[t-g,t] |x(m)| + |ε(t)| . ( 46
)
We now apply Theorem 1, with the choices

U = R, µ = ∆, F = A, p l = p 3 , p s = p 2 , p * = p

IV.2 Interesting Features of Theorem 2

Before turning to our next theorem, we discuss several notable features of Theorem 2. A striking feature of our final ISS estimate [START_REF] Petit | Motion planning for two classes of nonlinear systems with delays depending on the control[END_REF] is that the overshoot is zero when ε = 0, in which case we have exponential stability of the origin even when there are additive uncertainties δ A and δ B on A and B. Also, we do not require p 1 to be nonnegative valued. On the other hand, Assumptions 1-2 imply that (37) is globally exponentially stable to zero (GES) when δ A = δ B = 0, since we can take B = 0 and ε = 0. Since we do not require p 1 to be periodic, it does not seem possible to transform V into a strict Lyapunov function for [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF]. We can sometimes prove exponential stability by using Razumikhin's theorem, but not always, as we illustrate in our examples below. Theorem 2 can be used to provide a bound on B + δ B such that (36) is GES if [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF] is GES. Moreover, we can find a bound on B + δ B that is independent of the bound g on h(t). To check that this can be done, we first assume that (37) is GES and that A is bounded and continuous; see below for more general cases. For simplicity, we assume that the perturbation terms ε and δ A in (36) are zero, but analogous reasoning applies when ε and δ A are nonzero. Then standard results (such as [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]Theorem 4.14]) make it possible to satisfy Assumption 1 using a constant positive value of p 1 (t) = p1 . Also, (42) holds if there is a constants T > 0 such that

e -T p1/2 + |B + δ B | [0,∞) T 0 e -p1r/2 dr < p 2 p 3 , (47)
or, equivalently, if

|B + δ B | [0,∞) < (p 1 /2)( p 2 /p 3 -e -T p1/2 )(1 -e -T p1/2 ) -1 .
Here is an analog where p 1 can take both positive and negative values. Assume that

ż(t) = N z(t) (48)
is GES, where N ∈ R n×n is constant, and consider any bounded continuous function q : R → R for which there are constants q 0 > 0 and T > 0 such that

t t-T q( )d ≥ q 0 ( 49
)
for all t ∈ R. This allows cases where q is a persistency of excitation function such as q( ) = sin 2 ( ) (by choosing T = π and q 0 = π/2), or a function such as q( ) = sin(π /T ) + 3/π that takes both positive and negative values (by taking any constant T > 0 and q 0 = T /π). By solving a Riccati equation, we can build a symmetric positive definite matrix P and also find positive constants α, α, and α * such that the time derivative of V (z) = z P z along all trajectories z(t) of ( 48) satisfies

d dt V (z(t)) = z(t) N P + P N z(t) ≤ -α * V (z(t)) (50)
and α|z| 2 ≤ V (z) ≤ α|z| 2 for all z ∈ R n , e.g., by taking α = λ min (P ) and ᾱ = λ max (P ). Set A(t) = q(t)N .

Then Assumption 1 holds with p 1 (t) = α * q(t), p 2 = α, p 3 = α, and p 4 = |p 1 | [0,∞) . Then for each constant g > 0, we can argue as in the preceding paragraph to conclude that

ẋ(t) = q(t)N x(t) + (B(t) + δ B (t))x(t -h(t)) (51)
is GES for all piecewise continuous functions h : [0, ∞) → [0, g], and all choices of B : R → R n×n such that

|B + δ B | [0,∞) < inf t≥T t t-T exp - α * 2 t m q( )d dm -1 α ᾱ -exp - q 0 α * 2 (52)
holds. In fact, the GES property still holds if we replace (52) by the weaker condition

sup t≥T 1 T t t-T exp - α * 2 t m q( )d |B(m) + δ B (m)|dm < 1 T α ᾱ -exp - q 0 α * 2 , (53)
which simply says that |B + δ B | is small in a suitable averaged sense. We can use similar reasoning if N is time-varying, provided (a) ż = N (t)z is a uniformly globally exponentially stable system that admits a time invariant Lyapunov function of the form V (z) = z P z and (b) N is bounded and piecewise continuous. Hence, Theorem 2 can be viewed as a robustness result for GES of (37) with respect to the potentially destabilizing terms [B(t) + δ B (t)]x(t -h(t)) + ε(t).

V Systems Whose Undelayed Part Need Not Be Stabilizing

V.1 Statement and Proof of Result

We next study systems [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF] in the fundamental case where (37) is not necessarily exponentially stable. Consider

ż(t) = H(t)z(t) (54)
where H(t) = A(t) + B(t). We make the following assumption:

Assumption 3. The matrices A, δ A , B, and δ B and h are locally bounded and piecewise continuous, and there is a constant g such that 0 ≤ h(t) ≤ g for all t ≥ 0. Also, there are a bounded piecewise continuous function p 1 : [0, ∞) → R, positive constants p 2 and p 3 , and a C 1 function P : [0, ∞) → R n×n such that P (t) is symmetric and positive definite for all t ≥ 0, for which: For all t ≥ 0 and z ∈ R n , we have

p 2 |z| 2 ≤ V (t, z) ≤ p 3 |z| 2 , (55)
where V (t, z) = z P (t)z and the time derivative of V along all trajectories z(t) of (54) satisfies

d dt V (t, z(t)) ≤ -p 1 (t)V (t, z(t)) (56)
for all t ≥ 0.

We also make the convention that A, δ A , B, and δ B are 0 on [-g, 0], to extend their domains to [-g, ∞). We then add the following assumption: Assumption 4. There exist two constants T > g and δ ∈ (0, 1) such that with the choice

µ(t) = |δ A (t) + δ B (t)| + |B(t) + δ B (t)| t t-h(t) (|A( ) + δ A ( )| + |B( ) + δ B ( )|)d , (57)
we have

p3 p2 e -1 2 t t-T p1(m)dm + t t-T e -1 2 t p1(m)dm µ( )d ≤ δ (58)
for all t ≥ T , where g is from Assumption 3. Also, ε is locally bounded and piecewise continuous.

See Remark 5 for ways to check Assumption 4. The following ISS result reduces to global exponential stability when the perturbations ε are zero (but also see Remark 2 for results for more general initial functions): Theorem 3. If the system (36) satisfies Assumptions 3-4, then we can construct a constant c > 0 such that

|x(t)| ≤ ce ln(δ)t T +g + c 1 + |h| [0,t] |B + δ B | [0,t] |ε| [0,t] (59)
holds along all trajectories of (36) for all t ≥ 0 and all constant initial functions.

Proof. For all t ≥ 0, we can rewrite [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF] as

ẋ(t) = H(t)x(t) + (δ A (t) + δ B (t))x(t) -(B(t) + δ B (t))[x(t) -x(t -h(t))] + ε(t) . (60)
We apply Theorem 1. For all t ≥ 0, we can use the constantness of the initial functions to get

|x(t) -x(t -h(t))| ≤ t t-h(t) |A( ) + δ A ( )| + |B( ) + δ B ( )| d |x| [t-g,t] + h(t)|ε| [max{0,t-g},t] . (61)
Let U(t, X t ) denote the quantity in curly braces in (60) for all t ≥ 0. Then for all t ≥ 0, we get

ẋ(t) = H(t)x(t) + U(t, x t ), (62)
where

|U(t, φ)| ≤ µ(t)|φ| [-g,0] + λ(t) (63)
for all t ≥ 0 and all φ ∈ C([-g, 0], R n ), and where µ is from (57), and

λ(t) = (1 + h(t)|B + δ B | [0,t] )|ε| [0,t] . (64)
Then all assumptions from Theorem 1 hold, with X = x, p(t) = p 1 (t), p l = p 3 , and p s = p 2 , so the theorem follows from Theorem 1 and our ISS estimate (35) from Remark 1.

Remark 2. The upper bound in (61) was used to obtain the bound (63) which was in turn used to apply the stability condition from Theorem 1. In some cases, the upper bound in (61) may be significantly larger than the left side of (61), which may introduce conservativeness in our stability conditions. However, Theorem 3 presents the advantage of being based on easily checkable conditions, while the determination of the necessary and sufficient conditions for stability for ( 36) is an open problem, which seems to be formidably difficult.

Although we used the constantness of the initial functions to prove (61), we can generalize the preceding theorem to cover nonconstant initial functions, as follows: If t ≥ 0 is such that t ≥ h(t), then (61) still holds, even if the initial functions are not constant. On the other hand, if t ≥ 0 ≥ t -h(t), then

|x(t) -x(t -h(t))| ≤ |x(t) -x(0)| + |x(0) -x(t -h(t))| ≤ |x(t) -x(0)| + h(t)| ẋ| [-g,0] , (65)
and we can upper bound |x(t) -x(0)| by the right side of (61). Therefore, we can argue as in the last part of the proof of Theorem 3 to obtain

|x(t)| ≤ ce ln(δ)t T +g + c 1 + |h| [0,t] |B + δ B | [0,t] |ε| [0,t] + c|h| [0,t] | ẋ| [-g,0] (66)
along all trajectories of (36) for all t ≥ 0 and all essentially bounded absolutely continuous initial functions.

V.2 Notable Features of Theorem 3

Remark 3. Our assumptions on A, δ A , B, and δ B ensure that all solutions of (36) are well-defined and defined over [0, ∞). As before, we do not require p 1 to be nonnegative valued. Also, Assumptions 3-4 imply that the system (54) is globally exponentially stable to 0 (by taking δ A , δ B , h, and ε to be zero), and we can use [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF] as before to construct explicit comparison functions β ∈ KL and γ ∈ K ∞ in the final ISS estimate.

Remark 4. Theorem 3 is beyond the scope of [START_REF] Gu | Stability of Time-Delay Systems[END_REF]Chapt. 6], and can be extended to systems with different time-varying delays in different components of the state. Through Assumption 3 in the special case where δ B = 0, Theorem 3 takes into account the case where the term B(t)x(t -h(t)) has a stabilizing effect.

Remark 5. To motivate Assumption 4, let us consider the special case where the initial functions, p 1 , P , A and B are all constant, and where the uncertainties δ A and δ B are zero. Then (58) holds if

p 3 p 2 e -T p1/2 + |B|(|A| + |B|) t t-T h( )e ( -t)p1/2 d ≤ δ. ( 67
)
When B = 0, the preceding inequality holds provided there are constants T > 0 and δ ∈ (0, 1) such that

1 T t t-T h(m)e (m;-t)p1/2 dm ≤ δ p 2 /p 3 -e -T p1/2 T |B|(|A| + |B|) (68)
is satisfied for all t ≥ T , which is related to an average value of h.

VI Systems with Reduction Controls and Uncertain Delays

Although there did not exist any explicit controls in Theorems 2-3, they can be viewed as results on timevarying linear systems with time-varying linear state feedbacks with input delays, such as those that arise from tracking problems with latencies. On the other hand, we noted in the introduction that dynamic controllers are an important alternative to state feedbacks that have the potential advantage that they can often compensate for arbitrarily long input delays. Therefore, it is important to develop analogs of Theorems 2-3 for dynamic controllers. In the next two sections, we provide two such analogs. This section provides an analog for reduction method controllers, and the next section covers a different class of dynamic controllers with several useful degrees of freedom.

To highlight the most significant ideas, we confine our analysis in this section to systems of the form

ẋ(t) = [A + δ A (t)]x(t) + [B + δ B (t)]u(t -h(t) -γ(t)) (69)
where A ∈ R n×n and B ∈ R n×m are known constant matrices, but analogous results can be shown for cases where A and B are time-varying, or where there is additive uncertainty on the right side. Here, u is the control, h is a delay, and δ A , δ B , and γ represent uncertainty. Our first assumption is:

Assumption 5.
The function h is of class C 1 and there is a constant g > 0 such that 0 ≤ h(t) ≤ g holds for all t ≥ 0. The unknown function γ is piecewise continuous, and there is a known constant γ c ≥ 0 such that the inequalities 0 ≤ γ(t) ≤ γ c hold for all t ≥ 0.

Assumption 6. The pair (A, B) is controllable, and δ A and δ B are locally bounded and piecewise continuous.

Assumption 6 provides a matrix K ∈ R m×n such that R = A + BK is Hurwitz. Hence, we can find a constant c > 0 and a positive definite symmetric matrix Q ∈ R n×n for which

QR + R Q ≤ -cQ . ( 70
)
We also introduce the functions

µ a (t) = e |A|h(t) e |A|h(t) + |BK|L(|A|, h(t)) |δ A | [0,t] + |δ B | [0,t] |K| and µ b (t) = | ḣ(t)| |A| + 2e |A|h(t) -1 |BK| + 2e |A|h(t) |BK|, (71)
where L(|A|, h(t)) = (e |A|h(t) -1)/|A| when |A| = 0 and L(0, h(t)) = h(t), so µ a is zero when δ A and δ B are zero. Finally, we introduce this assumption:

Assumption 7. There exists a constant δ ∈ (0, 1) such that

λmax(Q) λmin(Q) e -c(γc+g)/2 + t t-γc-g e -c(t-)/2 (µ a ( ) + µ b ( ))d ≤ δ (72)
holds for all t ≥ γ c + g.

We prove the following result:

Theorem 4. If Assumptions 5-7 hold, then for any initial function for u, the control law

u(t) = K e Ah(t) x(t) + t t-h(t) e A(t-) Bu( )d (73)
renders the origin of (69) globally exponentially stable to 0.

Proof. In what follows, all inequalities and equalities should be understood to hold for all t ≥ 0, unless otherwise indicated. Set q(t) = | ḣ(t)| 2 . Let us introduce the operator

ζ(t) = e Ah(t) x(t) + Γ(t), where Γ(t) = t t-h(t) e A(t-) Bu( )d . (74)
Then the control (73

) is u(t) = Kζ(t). Also, Γ(t) = AΓ(t) + Bu(t) -(1 -ḣ(t))e Ah(t) Bu(t -h(t)
). Since the system (69) can be written as

ẋ(t) = Ax(t) + Bu(t -h(t) -γ(t)) + δ A (t)x(t) + δ B (t)u(t -h(t) -γ(t)), (75)
the time derivative of ζ along all trajectories of (69) is

ζ(t) = e Ah(t) ẋ(t) + ḣ(t)Ae Ah(t) x(t) + AΓ(t) + Bu(t) -(1 -ḣ(t))e Ah(t) Bu(t -h(t)) = [1 + ḣ(t)]Aζ(t) + Bu(t) -ḣ(t)A t t-h(t) e A(t-) Bu( )d + ḣ(t)e Ah(t) Bu(t -h(t)) + e Ah(t) B[u(t -h(t) -γ(t)) -u(t -h(t))] + e Ah(t) δ A (t)x(t) + δ B (t)u(t -h(t) -γ(t)) , (76)
where the second equality used the relation e Ah(t) Ax(t) + AΓ(t) = Aζ(t) twice. Then the fact that our control

(73) is u(t) = Kζ(t), combined with our choice R = A + BK, give ζ(t) = Rζ(t) + U(t, ζ t ) (77)
for all t ≥ 0, where

U(t, ζ t ) = ḣ(t)Aζ(t) + ω(t, ζ t ) + κ(t, ζ t ) + e Ah(t) δ A (t)x(t) + δ B (t)Kζ(t -h(t) -γ(t) , (78)
ω(t, ζ t ) = ḣ(t) e Ah(t) BKζ(t -h(t)) -A t t-h(t) e A(t-) BKζ( )d and κ(t, ζ t ) = e Ah(t) BK[ζ(t -h(t) -γ(t)) -ζ(t -h(t))] . (79)
We now use (77) to apply Theorem 1. Notice that

|ω(t, ζ t )| ≤ | ḣ(t)| e A|h(t) + |A| t t-h(t) e |A|(t-) d |BK||ζ| [t-g,t] = | ḣ(t)||BK| 2e |A|h(t) -1 |ζ| [t-g,t] and |κ(t, ζ t )| ≤ 2e |A|h(t) |BK||ζ| [t-g-γc,t] (80)
hold for all t ≥ 0. Also,

|x(t)| = e -Ah(t) (ζ(t) -Γ(t)) ≤ e h(t)|A| |ζ(t)| + |BK| t t-h(t) e A(t--h(t)) d |ζ| [t-g,t] ≤ e h(t)|A| + |BK| t t-h(t) e |A|(h(t)+ -t) d |ζ| [t-g,t] ≤ e h(t)|A| + |BK|L(|A|, h(t)) |ζ| [t-g,t]
(81) for all t ≥ 0. This leads to

|U(t, φ)| ≤ | ḣ(t)| |A| + 2e |A|h(t) -1 |BK| + 2e |A|h(t) |BK| + e |A|h(t) e |A|h(t) + |BK|L(|A|, h(t)) |δ A | [0,t] + |δ B | [0,t] |K| |φ| [-g-γc,0] (82)

VII Perturbed Systems with Different Dynamic Controller VII.1 Statement and Proof of Theorem

We next provide an alternative to Theorem 4 based on a different control law. We study systems of the form

ẋ(t) = [A + δ A (t)]x(t) + [B + δ B (t)]u(t -h(t)) + ε(t) , (83)
whose state space is R n , where A ∈ R n×n \{0} and B ∈ R n×m are known constant matrices, the unknown locally bounded piecewise continuous functions δ A and δ B represent uncertainties, ε is an unknown locally bounded piecewise continuous disturbance, and the delay h(t) is known, but extensions that allow discontinuous or unknown delays are possible using ideas from the preceding section. We introduce the following assumption:

Assumption 8. The pair (A, B) is controllable, and there is a constant g > 0 such that 0 ≤ h(t) ≤ g holds for all t ≥ 0. Also, h is in C 1 .

Assumption 8 implies that we can construct matrices A f ∈ R m×m and B f ∈ R m×n such that

H = A B B f A f ∈ R (n+m)×(n+m) (84)
is Hurwitz. See Section VII.2 for ways to construct suitable pairs (A f , B f ). Therefore, we can find a symmetric and positive definite matrix Q ∈ R (n+m)×(n+m) and a constant c > 0 such that QH + H Q ≤ -cQ holds, e.g., by solving the Riccati equation QH + H Q = -I m+n and setting c = 1/λ max (Q). We now set

V (ζ) = ζ Qζ where ζ = (ρ, β) ∈ R n × R m
, and we redefine the function µ by

µ( ) = | ḣ( )| |A| + 2e h( )|A| -1 |B| + e h( )|A| |δ A ( )| 1 + e h( )|A| -1 |B| |A| e g|A| + |δ B ( )| , (85)
which reduces to the much simpler formula µ( ) = | ḣ( )|(|A| + (2e h( )|A| -1)|B|) in the important special case where the perturbations δ A and δ B are zero. Our final assumption is:

Assumption 9.
There exist constants T > g and δ ∈ (0, 1) such that the inequality

λmax(Q) λmin(Q) e -cT /2 + t t-T e -c(t-)/2 µ( )d ≤ δ (86)
holds for all t ≥ T .

We can then prove:

Theorem 5. If Assumptions 8-9 hold, then (83) in closed-loop with u(t) = β(t), where β is any solution of

β(t) = A f β(t)+B f e Ah(t) x(t)+ t t-h(t) e A(t-) Bβ( )d , (87)
is ISS with respect to the set of all locally bounded piecewise continuous functions ε : [0, ∞) → R n .

Proof. Fix any choices of ε : [0, ∞) → R n , δ A : [0, ∞) → R n×n , and δ B : [0, ∞) → R n×m and let Assumptions 8-9 hold. We define ρ(t) to be the quantity in squared brackets on the right side of (87). Then, β(t) = A f β(t) + B f ρ(t) holds for all t. Reorganizing terms gives the following:

ρ(t) = A ḣ(t)

VII.2 Coefficient Matrices in Dynamic Extension

Theorem 5 requires a pair (A f , B f ) ∈ R m×m × R m×n , a positive definite symmetric matrix Q, and a constant c > 0 such that (84) is Hurwitz and such that QH + H Q ≤ -cQ. Different choices of A f , B f , and Q lead to different controllers u in Theorem 5 and different conditions in Assumption 9, so it is of interest to find general ways to build A f , B f , and Q. (See [START_REF] Mondié | Finite spectrum assignment of unstable time-delay system with a safe implementation[END_REF] for a possible choice of the pair (A f , B f ).) On the other hand, [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF] builds a large class of pairs (A f , B f ) that satisfy the preceding requirements, and that also contain a useful scaling tuning parameter. For completeness, we state our construction from [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF] in the sequel.

Proposition 1. Let (A, B) ∈ R n×n ×R n×m be any controllable pair, and K ∈ R m×n be such that M = A+BK is Hurwitz. Let P ∈ R n×n be a positive definite symmetric matrix and c > 0 be a constant such that

P M + M P ≤ -cP, (92)
Q = P + ν 2 K K -ν 2 K -ν 2 K ν 2 I m (93)
is such that V (t) ≤ -cV (ζ) holds along all trajectories of ζ = Hζ.

Proof. We have V (ζ) = ρ P ρ + 0.5ν(β -Kρ) (β -Kρ) for all ζ = (ρ, β) ∈ R n × R m . Hence, by using the Riccati inequality (92), it follows that its time derivative along all trajectories of ζ = Hζ satisfies

V (t) = 2ρ P [M ρ + B(β -Kρ)] + ν(β -Kρ) [A f β + B f ρ -K(Aρ + Bβ)] ≤ -cρ P ρ + (β -Kρ) [ν(A f -KB)β + (νB f + 2B P -νKA)ρ]. (94)
We choose

A f = KB -1 ν G f and B f = G f K ν -2 ν B P + KA, where G f ∈ R m×m is to be chosen. Then V (t) ≤ -cρ P ρ + (β -Kρ) [-G f β + G f Kρ] = -c ρ P ρ + 1 c (β -Kρ) G f (β -Kρ) . (95)
The proposition now follows by choosing G f = ν 2 cI m .

VIII Further Illustrative Examples

In Section IV.2, we saw how to apply some of our work under persistency of excitation conditions. In this section, we provide additional examples that illustrate the potential of our work.

VIII.1 Illustration of Theorem 2

Let d > 0 and c ∈ (0, 1) be any constants such that:

(1 -c)d < 1 -e -1/2 . ( 96
)
Let σ : [0, ∞) → {0, d} be the periodic function of period 1 defined by (i) σ(t) = 0 when t ∈ [0, c) and (ii)

σ(t) = d when t ∈ [c, 1]. We consider the system ẋ(t) = -[1 + δ A (t)]x(t) + [σ(t) + δ B (t)]x(t -1), (97)
where x is valued in R, and where the piecewise continuous locally bounded functions δ A and δ B represent uncertainties as before. When d > 1, Razumikhin's theorem does not apply; see [START_REF] Gu | Stability of Time-Delay Systems[END_REF]Chapt. 6]. To the best of the authors' knowledge, no Lyapunov-Krasovskii functional construction technique applies to this system. Let us verify that (97) satisfies Assumptions 1-2 with V (x) = 1 2 x 2 , ε = 0, and h(t) = 1, under suitable bounds on δ A and δ B . With the notation from Assumptions 1-2, we choose p 1 (t) = 1 and P (t) = 1/2 for all t ≥ 0 and the constants p 2 = p 3 = 1 2 and T = p 4 = 1. Then Assumption 1 is satisfied with A = -1. Also, the requirement (42) from Assumption 2 is that there is a constant δ ∈ (0, 1) such that e -1/2 + sup

t≥1 t t-1 e -(t-)/2 (|δ A ( )| + |σ( ) + δ B ( )|) d ≤ δ (98) holds. Since t t-1 σ( )d = d(1 -c
) for all t ≥ 1, it follows that (98) holds for some constant δ ∈ (0, 1) if

sup t≥1 t t-1 e -(t-)/2 (|δ A ( )| + |δ B ( )|) d < 1 -e -1/2 -d(1 -c), (99)
and the right side of (99) is positive, by (96). Theorem 2 allows us to conclude that the corresponding perturbed systems are globally exponentially stable to 0, when (99) holds. More generally, we can use Theorem 2 to cover cases where the system has time-varying delays, or additive uncertainty ε(t) on the right side of (97) to conclude input-to-state stability. We leave these generalizations to the reader.

VIII.2 Illustration of Theorem 3

Let l > 0 be a constant, and k ∈ N be an odd integer. We consider the system

ẋ(t) = q(t)x(t) -(1 -δ B (t))x(t -h k (t)) , (100)
where x is valued in R, the bounded piecewise continuous nonnegative valued function q admits a constant T ∈ (0, 1) such that the constant q defined by q = sup t≥T t t-T q( )d (101) satisfies q < T , and the time-varying delay is h k (t) = max{0, l sin k (t)}. Notice that this allows cases where q( ) > 1 for some choices of , as long the averaged value q/T is below 1. To the best of the authors' knowledge, Razumikhin's theorem does not apply to (100), and no Lyapunov-Krasovskii functional construction technique in the literature applies to this system. With the notation of Section V.1, we choose P (t) = 1, p 1 (t) = 2(1 -q(t)), p 2 = p 3 = 1, H(t) = q(t) -1, A(t) = q(t), and B = -1. Even though p 1 (t) can take both positive and negative values, Assumption 3 is still satisfied. Also, the inequality (58) from Assumption 4 holds if there is a constant δ ∈ (0, 1) such that e -T +q + t t-T exp r -t + t r q(a)da µ(r)dr ≤ δ for all t ≥ T, where µ

(r) = |δ B (r)| + |δ B (r) -1| r r-h k (r) (|q( )| + |δ B ( ) -1|) d . (102)
Pick any constant δ > 0 such that e -T +q + T e q δ < 1. We now allow only those piecewise continuous perturbations δ B such that |δ B | [0,∞) ≤ δ. By the Lebesgue Dominated Convergence Theorem, there is a constant k > 0 (which is sufficiently large, and which depends on l, T , |q| [0,∞) , and δ) such that there exists some constant δ ∈ (0, 1) satisfying (102) for all t ≥ T , by the following argument. First, notice that the inequality from (102) holds if

t t-T exp r -t + t r q(a)da h k (r)dr < δ -e -T +q -T e q δ δ + |q| [0,∞) + |δ B -1| [0,∞) |δ B -1| [0,∞) , (103)
where the δ in the denominator on the right side of ( 103) is to allow the case where (|q|

[0,∞) + |δ B -1| [0,∞) )|δ B - 1| [0,∞) = 0. Also,
exp r -t + t r q(a)da h k (r)dr = 0. (104)
There is an odd integer k > 0 such that (103) holds for all t ≥ T with δ = 0.5(1 + e -T +q + T e q δ), provided the convergence in ( 104 Theorem 3 now implies that (100) is globally exponentially stable to 0 for all perturbations δ B that are bounded by δ and all constant initial functions. More generally, we have the estimate (59) from Theorem 3, which also allows additive uncertainties ε(t) on the right side. Since l > 0 is arbitrarily large, we have exponential stability without any constraint on sup m≥0 h k (m). To the best of the authors' knowledge, such a result cannot be shown by any other technique.

VIII.3 Illustration of Theorem 5

We revisit the unstable second-order linear dynamics from [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF], which is the special case of the following when the unknown perturbation terms ε(t), δ A , and δ B are all zero functions:

ẋ(t) = 0 1 -1 1 + δ A (t) x(t) + 0 1 + δ B (t) u(t -h(t)) + ε(t). (106)
We only apply Theorem 5 and therefore assume that h ∈ C 1 is bounded by some constant g > 0, but we can also apply Theorem 4 to cover cases where the delay can have discontinuities and be uncertain. Choosing

Also, using the fact that |B| = 1 and lengthy but relatively simple calculations, one checks that Assumptions 8-9 will be satisfied provided there are also constants δ ∈ (0, 1) and T > 0 such that

This provides a notable alternative to the corresponding example in [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF], since it allows us to conclude input-tostate stability under additive uncertainties ε (which were not considered in [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF]) using the alternative controller from our Theorem 5 (which is based on a dynamic extension that was also not considered in [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF]), because we allow additive uncertainty on the coefficient matrices, and because we include the degree of freedom ν > 0.

IX Concluding Remarks

Time-varying delays can model many important phenomena, such as increasing latencies as a muscle becomes fatigued. Also, sampling in controllers can be modeled using a piecewise C 1 time-varying delay h(t) where ḣ(t) = 1 almost everywhere. This paper provided new stability techniques for piecewise continuous timevarying linear systems with time-varying delays, and with uncertain coefficient matrices. They can be applied when stabilizing control laws must be designed under delays. Since we also do not require the standard condition ḣ(t) < 1, our results are much broader in scope than much of the existing literature. Two of our theorems use dynamical extensions, including novel results on the reduction method under uncertain delays. Our work uses a trajectory-based method from [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], which circumvents the search for Lyapunov-Krasovskii functionals or small-gain conditions. The results in this paper are beyond the scope of [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], e.g., since we allow sup t h(t) to be arbitrarily large, and since our new integral conditions make it possible to establish exponential stability. Also, there are many degrees of freedom in applying our methods (such as a parameter ν > 0 in our dynamic extension). In our future work, we hope to exploit the degrees of freedom to make our work applicable to the broadest possible class of systems and chattering delays. Many other extensions seem possible. For instance, we conjecture that our results can be extended to cover distributed delays and systems that are nonlinear in the state, including applications to networked systems with uncertainty. It would also be useful to seek generalizations to hyperbolic PDEs, such as the dynamics for age structured chemostats where the control enters into the PDE directly instead of at the boundary [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF][START_REF] Karafyllis | Sampled-data feedback stabilization of age-structured chemostat models[END_REF] and where there may also be state dependent or rapidly varying delays [START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and statedependent delay perturbations[END_REF][START_REF] Fridman | On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast-varying delays[END_REF].

  µ( ) sup m∈[ -g, ] |X(m)|d + p l µ( )d |X| [-g,0] + p l

  4 , and λ(t) = |ε(t)|. Then (44) follows from (35) in Remark 1.

  for all t ≥ 0 and φ ∈ C in . Next set V (ζ) = ζ Qζ. Then the requirements from[START_REF] Ivanov | Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima[END_REF] hold with F = R, p(t) = c, p s = λ min (Q), and p l = λ max (Q), and[START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF] agrees with (72) with T = γ c + g. Hence, the result follows from Theorem 1, combined with the final stability estimate from[START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF] in Remark 1.

  and let ν > 0 be any constant. Then (a) the matricesA f = KB -c 2 I m and B f = KA -2 ν B P + c 2 K are such that (84) isHurwitz and (b) the matrix V (ζ) = ζ Qζ with the choice

  ) is uniform in t ≥ T . The convergence (104) will be uniform in t ≥ T if the convergence lim k→∞ T 0 h k ( + b)d = 0 (105) is uniform in b ∈ [0, 2π], by the periodicity of the sin function. On the other hand, the functions f k (b) = T 0 h k ( + b)d are continuous in b and they satisfy lim k→∞ f k (b) = 0 for all b ∈ [0, 2π] (again by the Dominated Convergence Theorem), and f k (b) ≥ f k+1 (b) for all k ≥ 1 and b ∈ [0, 2π]. Hence, the uniformity of the convergence (104) in t ≥ T follows from standard results, e.g., [46, Theorem 7.13, p.150]. Thus, Assumptions 3-4 hold.

1 ,

 1 and K = -[ 2 3 ] ,(107)the corresponding matrix M = A + BK is Hurwitz, so Assumption 8 is satisfied with n = 2 and m = 1. In fact, with the choices P M + M P ≤ -cP .To check Assumption 9 and apply Theorem 5, we must construct the required matrices A f , B f , and Q. To this end, we use the construction from Proposition 1 for the choices (108) and the special case where the tuning parameter is ν = 1, but we can take any constant ν > 0 and other choices of P . With the preceding choices, we get A f = KB -c 2 I m = -3.3675, B f = KA -2 ν B P + c 2 K = (1.9316, -6.7693), and

  t t-T | ḣ( )|d ≤ δ √ λmin(Q)/λmax(Q)-e -T c/2 -M * (t) |A|+2e |A|g -1(110)holds for all t ≥ T , whereM * (t) = e |A|g 1 +1 |A| e |A|g -1 e |A|g t t-T |δ A (r)|dr + t t-T |δ B (r)|dr .

  AeAh(t) x(t) + e Ah(t) Bβ(t -h(t)). We next apply Theorem 1 with the choices X = (ρ, β), the constant matrix P(t) = Q, the constant function p(t) = c, p s = λ min (Q), p l = λ max (Q), and U(t, X t ) being the quantity in curly braces in (89). Since |A| + 2e |A|h(t) -1 |B| |ζ| [t-g,t]and|R(t)| ≤ e |A|h(t) |δ A (t)| 1 + e |A|h(t) -1 |B|/|A| e g|A| + |δ B (t)| |ζ| [t-g,t] ,and therefore also |U(t, φ)| ≤ µ(t)|φ| [-g,0] + λ(t) for all t ≥ 0 and φ ∈ C in , where λ(t) = e g|A| | (t)| for all t ≥ 0 and µ is defined in (85). The theorem now follows from Theorem 1, since the contractiveness requirement[START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF] from Theorem 1 follows from our condition (86). Remark 6. A useful feature of the preceding reasoning is that it can give explicit formulae for the comparison functions β ∈ KL and γ ∈ K ∞ in the final ISS estimate. This follows from Remark 1.

	Since ζ(t) = (ρ(t), β(t)) ∈ R n+m for all t ≥ 0, we get
	ζ(t) = Hζ(t) + ḣ(t)	ψ(t) 0	+	e Ah(t) ε(t) + R(t) 0	,	(89)
	where ψ(t) = e Ah(t) x(t) = ρ(t) -	t t-h(t) e A(t-) Bβ( )d		(90)
	holds for all t ≥ 0, we can argue as in (81) to get
	|e Ah(t) x(t)| ≤ |ρ(t)| + |B| |A| e |A|h(t) -1 |β| [t-g,t] and |x(t)| ≤ e g|A| |ρ(t)| + |B| |A| e |A|h(t) -1 |β| [t-g,t]	(91)
	for all t ≥ 0. This gives
	|ψ(t)| ≤
	e Ah(t) x(t) + e Ah(t) (A + δ A (t)) x(t) + e Ah(t) (B + δ B (t)) u(t -h(t)) + e Ah(t) ε(t)

+ A ρ(t) -e Ah(t) x(t) + Bβ(t) -e Ah(t) Bβ(t -h(t))(1 -ḣ(t)) = Aρ(t) + e Ah(t) ε(t) + ḣ(t)Ae Ah(t) x(t) + Bβ(t) + ḣ(t)e Ah(t) Bβ(t -h(t)) + R(t),

(88)

where R(t) = e h(t)A [δ A (t)x(t) + δ B (t)β(t -h(t))].

  for each constant t ≥ T , we can use the Dominated Convergence Theorem to get

		t
	lim k→∞	t-T

The work of F. Mazenc was partly supported by l'Institut pour le Supported by NSF-ECCS Grants 1102348 and 1408295 and Roy P. Daniels Professorship #3 at LSU.