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

Michael Malisoff
email: malisoff@lsu.edu.

Stability Analysis for Time-Varying Systems with Delay using Linear Lyapunov Functionals and a Positive Systems Approach

We prove stability of time-varying systems with delays, using linear Lyapunov functionals and positive systems, and we provide robustness of the stability with respect to multiplicative uncertainty in the vector fields. We allow cases where the delay may be unknown, and where the vector fields defining the systems are not necessarily bounded. We illustrate our work using a chain of integrators and other examples.

I. INTRODUCTION

This paper continues our search for new ways to prove stability properties for time-varying nonlinear systems with delays. Our search is motivated by physical phenomena governing many applications that lead to nonlinear systems with delay [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF], [START_REF] Germani | Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability[END_REF]. However, these systems are beyond the scope of classical methods, e.g., frequency domain and linear matrix inequality techniques. Therefore, despite their importance, relatively few contributions are devoted to timevarying systems with delays (but see [START_REF] Gu | Stability of Time-Delay Systems[END_REF] for a polytopic approach to time-varying delays which does not cover the results we give here).

Moreover, the existing stability results are largely limited to systems where the delays only occur in the control. Prediction is very useful for systems with arbitrarily long input delays [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF], [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF]. However, prediction may not always apply when the delays occur in the vector fields defining the system, which is the situation we are concerned with here (but see [START_REF] Bekiaris-Liberis | Simultaneous compensation of input and state delays for nonlinear systems[END_REF] for predictive Lyapunov function based methods under delays in the input and in the vector fields of the system). Also, it may not always be easy to find Lyapunov functions that can be transformed into the Lyapunov-Krasovskii functionals that are often used to prove stability properties for delay systems.

The work [START_REF] Mazenc | Stability analysis of time-varying neutral time-delay systems[END_REF] uses a different approach to prove stability of delayed neutral systems, based on nonnegative and cooperative systems. Here, we use analogous tools for time-varying systems. We use three key ingredients. The first involves operators with integral terms that produce systems with distributed delays, interconnected with integral equations. Then, we prove that all solutions of the interconnections are components of solutions of nonnegative systems. Finally, we use linear Lyapunov-Krasovskii functionals for the higher dimensional systems, which ensure global asymptotic stability for the original systems. Our Lyapunov functional design owes a great deal to the functionals in [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF] for time invariant systems; see also [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF], [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF].

We use decompositions of functions involving their cooperative parts, as was done in [START_REF] Efimov | Interval observers for time-varying discrete-time systems[END_REF], [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF] for interval observers or time invariant systems. This technique shares similarities with the internal positive representation from [START_REF] Cacace | A new approach to the internal positive representation of linear MIMO systems[END_REF], [START_REF] Germani | Representation of a class of MIMO systems via internally positive realization[END_REF]. Two advantages of our new technique are that (a) its assumptions are relatively simple and (b) it establishes exponential stability for systems that do not seem to be covered by other techniques. We establish our first theorem in Section III, under a bound on the delay. However, the result requires that certain vector fields satisfy suitable bounds, which may not always hold in practice. Therefore, we also present an approach in Section V for systems with unbounded vector fields and arbitrary unknown constant delays, under a condition related to the signs of the components of the vector fields of the system. Since we establish global asymptotic stability using stabilizing time-varying terms with delay, our results contrast with [START_REF] Tchangani | Robust stability and stabilization of systems with aftereffect[END_REF], [START_REF] Tchangani | Stability of nonlinear differential equations with distributed delay[END_REF], which use comparison systems and non-smooth time invariant vector Lyapunov functions to prove ultimate boundedness and local results (using the stabilizing effect of time invariant terms). See also [START_REF] Ngoc | Stability of positive differential systems with delay[END_REF], which proves local stability under delays, and global stability under homogeneity conditions that we do not require here.

Compared with treatments of more elementary systems (such as undelayed linear time invariant systems, where the stability analysis often entails solving a linear matrix inequality), our assumptions and analysis are more complicated. However, we believe that this is the price to pay to obtain the very general results we obtain here for nonlinear delay systems with state dependent coefficient matrices that may contain uncertainties. Our examples in Section VI illustrate the value added by our results and the checkability of our assumptions, and include a key chain of integrators where our allowable delay bound is larger than bounds that were available in the literature.

II. DEFINITIONS AND NOTATION

In what follows, the dimensions are arbitrary. For any matrix M ∈ R p×q , let mij denote its entry in row i and column j for all i and j. The k × n matrix in which each entry is 0 is denoted by 0. The usual Euclidean norm of vectors, and the induced norm of matrices, are denoted by | • |. All inequalities must be understood to hold for each entry of the corresponding matrices, i.e., given any matrices A and B of the same size, we write A ≤ B to mean that aij ≤ bij for all i and j. A square matrix is said to be cooperative or Metzler provided all of its off-diagonal entries are nonnegative. A matrix M ∈ R r×s is said to be nonnegative (resp., positive) provided every entry mij of M satisfies mij ≥ 0 (resp., > 0). For simplicity, we always take the initial times for the trajectories of our systems to be t0 = 0. We let M + denote the matrix whose position (i, j) entry is max{0, mij} for all i and j, and M -= M + -M . We let M s = M + + M -, so M s is obtained by taking the absolute values of all entries of M .

Let C 1 denote the set of all continuously differentiable functions, whose domains and ranges will be clear from the context. Given any constant τ > 0, we let C([-τ, 0], R n ) denote the set of all continuous R n -valued functions defined on [-τ, 0]. We often abbreviate this set as Cin, and we call it the set of all initial functions. A system is said to be positive for a class of initial functions S0 provided for each positive valued initial function in S0, the unique solution stays positive for all t ≥ 0. For any continuous function ϕ : [-τ, ∞) → R n and all t ≥ 0, we define ϕt by ϕt(m) = ϕ(t + m) for all m ∈ [-τ, 0], i.e., ϕt ∈ Cin is the translation operator.

III. STATEMENT OF FIRST RESULT AND DISCUSSION

We first provide a stability analysis for time-varying systems

ẋ(t) = A1(t, xt)x(t) + A2(t, xt)x(t -τ ) , ( 1
)
where Setting c5 = c3c4/c2, we also assume the following: Assumption 2: The delay τ ≥ 0 satisfies

x is valued in R n , τ > 0 is a constant
c 6 c 1 + c5 τ < 1 (3)
where the ci's are as above.

We can then prove the following: Theorem 1: If (1) satisfies Assumptions 1-2, then (1) is uniformly globally exponentially stable to 0 for all initial functions in Cin.

Remark 1: If Ba 1 + B a 1 is bounded from above by a constant Hurwitz matrix, then there is a constant vector p > 0 such that the first inequality in (2) holds for all t ≥ 0 [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]. When B a 1 is bounded from above by a suitable matrix B m (t), we can sometimes change coordinates to transform Ẋ = B m 1 (t)X into an autonomous positive system [START_REF] Mazenc | Asymptotically stable interval observers for planar systems with complex poles[END_REF], which may facilitate checking Assumption 1. Without extra assumptions, it is unclear how to drop the requirements in the last two inequalities in [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], which are analogous to the requirements in [16, Assumption A2, Eq. ( 4)] for the case of neutral systems.

Remark 2: Assumption 2 bounds τ , but does not require that τ be known. One cannot expect stability without assuming that τ is small enough, even if A1 is independent of xt, since Assumptions 1-2 do not imply that Ẋ = A1(t)X is asymptotically stable. A key difference between [START_REF] Mazenc | Stability analysis of time-varying neutral time-delay systems[END_REF] and Theorem 1 is that the potential stabilizing effect of the delayed term A2(t, xt)x(t -τ ) is not taken into account in [START_REF] Mazenc | Stability analysis of time-varying neutral time-delay systems[END_REF], but it is in Theorem 1; see Section VI.

Remark 3:

Conditions (2) hold if ṗ(t) + p(t) Ba 1 (t, φ, ψ)+B a 1 (t, φ, ψ) ≤ -c1p(t) , c2(1 ... 1) ≤ p(t) ≤ c3(1 ... 1) , p (t)(B a 2 (t, ψ)) s ≤ c4p (t), and
p (t)(B a 3 (t, φ, m, ψ)) s ≤ c6p (t) (4)
hold for all t ≥ 0 and all φ and ψ in Cin, since then we can satisfy (2) by taking B b (t) = sup φ∈C in ,ψ∈C in Ba 1 (t, φ, ψ)+B a 1 (t, φ, ψ) . Any conservativeness of Assumptions 1-2 mostly stems from the fact that B a 1 can be Hurwitz while Ba 1 + B a 1 is not Hurwitz [START_REF] Mazenc | Asymptotically stable interval observers for planar systems with complex poles[END_REF]. To broaden the class of systems for which Assumptions 1-2 apply, one can use changes of coordinates (like those in [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF]) that transform a Hurwitz matrix into a Hurwitz and Metzler matrix.

IV. PROOF OF THEOREM 1

Step 1: Obtaining a Comparison System Since Assumption 2 ensures that A1 and A2 are bounded, all trajectories of (1) are defined over [-τ, ∞) [START_REF] Hale | Introduction to Functional Differential Equations[END_REF]. Therefore, by specifying one trajectory x(t), we can define

B1(t) = B a 1 (t, xt, xt+τ ), B1(t) = Ba 1 (t, xt, xt+τ ), B 1 (t) = B a 1 (t, xt, xt+τ ), B2(t) = B a 2 (t, xt+τ ), and B3(t, m) = B a 3 (t, xt, m, xm+τ ). For each t ≥ 0, set ξ(t) = x(t) + t t-τ A2(m + τ, xm+τ )x(m)dm . (5)
Since B1(t)B2(m) = B3(t, m) for all t and m, we then have

ξ(t) = B1(t)ξ(t) + t t-τ B3(t, m)x(m)dm x(t) = ξ(t) + t t-τ B2(m)x(m)dm . (6)
Hence, all solutions of (1) converge to 0 if all solutions of (6) with initial conditions in (φ ξ , φx) ∈ Cin satisfying the matching condition

φx(0) = φ ξ (0) + 0 -τ B2(m)φx(m)dm (7)
are defined over [-τ, ∞) and converge exponentially to 0. Note that the trajectory x(t) enters explicitly in (6), as well as through our formulas for the functions Bi that appear in [START_REF] Efimov | Interval estimation for uncertain systems with time-varying delays[END_REF], and the functions Bi are defined along a fixed choice of x(t). Nevertheless, since the constants ci in Assumptions 1-2 are independent of the trajectory, and since the constants in our final exponential stability estimate will be independent of x(t), we will still be able to establish our exponential stability estimate for all solutions (ξ(t), x(t)) of ( 6) that satisfy the matching condition [START_REF] Germani | Representation of a class of MIMO systems via internally positive realization[END_REF], which in particular will give the exponential stability result for the trajectory x(t) we used to define the Bi's.

One can prove that all the solutions of ( 6) for all initial conditions (φ ξ , φx) ∈ Cin satisfying the matching condition ( 7) are continuous and uniquely defined over [-τ, ∞) by noticing that they satisfy

     ξ(t) = B1(t)ξ(t) + t t-τ B3(t, m)x(m)dm ẋ(t) = B1(t)ξ(t) + t t-τ B3(t, m)x(m)dm + B2(t)x(t) -B2(t -τ )x(t -τ ) (8)
for all t ≥ 0. It is an interconnection of a system with a distributed delay. To analyze the stability of (8), we first write

B1(t) = B1(t) - B 1 (t), B2(m) = B + 2 (m) -B - 2 (m), and
B3(t, m) = B + 3 (t, m) - B -
3 (t, m), and observe that ( 6) is equivalent to

     ξ(t) = B1(t)ξ(t) -B 1 (t)ξ(t) + t t-τ (B + 3 (t, m) -B - 3 (t, m))x(m)dm x(t) = ξ(t) + t t-τ (B + 2 (m) -B - 2 (m))x(m)dm. (9)
We analyze the trajectories of the equivalent system (9).

Step 2: Analyzing the Comparison System [START_REF] Gu | Stability of Time-Delay Systems[END_REF] To analyze the stability of (9), we combine (i) our novel linear Lyapunov-Krasovskii functional approach and (ii) an approach from [START_REF] Moisan | Interval observers for non monotone systems, application to bioprocess models[END_REF] that doubles the dimension of the system. Consider the system

                                   ξ(t) = B1(t)ξ(t) + B 1 (t)Ψ(t) + t t-τ B + 3 (t, m)x(m)dm + t t-τ B - 3 (t, m)Z(m)dm x(t) = ξ(t) + t t-τ B + 2 (m)x(m)dm + t t-τ B - 2 (m)Z(m)dm Ψ(t) = B1(t)Ψ(t) + B 1 (t)ξ(t) + t t-τ B + 3 (t, m)Z(m)dm + t t-τ B - 3 (t, m)x(m)dm Z(t) = Ψ(t) + t t-τ B + 2 (m)Z(m)dm + t t-τ B - 2 (m)x(m)dm. (10)
One can easily check that (ξ, x, -ξ, -x) is a solution of (10), if (ξ, x) is a solution of (9). Hence, if all solutions of (10) satisfying the matching condition

x(0) = ξ(0) + 0 -τ B + 2 (m)x(m)dm + 0 -τ B - 2 (m)Z(m)dm Z(0) = Ψ(0) + 0 -τ B + 2 (m)Z(m)dm + 0 -τ B - 2 (m)x(m)dm (11)
are continuous on [-τ, ∞) and converge exponentially to 0, then all solutions of (6) that satisfy (7) converge exponentially to 0. Note that we are not asserting that ( 9) and ( 10) are equivalent, since positivity of (10) will not imply positivity of (9). However, the way we embed solutions of (9) as components of solutions of (10) will ensure that our stability result for [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] implies the desired stability for [START_REF] Gu | Stability of Time-Delay Systems[END_REF]. Arguing as we did when we studied the existence of the solutions of ( 6), one can prove that all solutions of (10) satisfying [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF] are continuous and uniquely defined over [-τ, ∞). Also, we prove in the appendix that [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] is positive for the class S0 of all initial functions satisfying [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]. Moreover, it is linear. Hence, it is globally exponentially stable if it is globally exponentially stable on only the positive orthant. To see why, let X be any solution of [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] with any nonzero initial condition φX = (φ ξ , φx, φΨ, φZ ) satisfying [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]. Then we can find a positive valued solution Xa of (10) and a negative valued solution X b of (10) (both satisfying the matching condition) such that the corresponding initial functions φX a and φX b satisfy

φX b (t) < φX (t) < φX a (t) for all t ∈ [-τ, 0]. (12)
To see why such Xa and X b exist, it suffices to find a negative valued φX b : [-τ, 0] → R 4n and a positive valued φX a : [-τ, 0] → R 4n satisfying [START_REF] Hale | Introduction to Functional Differential Equations[END_REF], both satisfying the matching condition, since then the positivity of Xa and the negativity of X b follow from our proof of the positivity of [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] in the appendix. To find φX b and φX a , let Li,j be the corresponding entries of B s 2 , so

B s 2 (m) = [Li,j(m)] for all m. Since B s 2 is bounded, there is a constant H > 0 such that max 1≤i≤n 0 -τ n j=1 Li,j(m)e Hm dm ≤ 1 2 . ( 13
)
Let δi be the integral in [START_REF] Hennet | Stability conditions of constrained delay systems via positive invariance[END_REF], so δi Next, assume that [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] satisfies the exponential stability property on the positive orthant. Since positivity and linearity of [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] give positivity of Xa(t) -X (t) and X (t) -X b (t) for all t ≥ 0, we get limt→∞(Xa(t) -X (t)) = limt→∞(X (t) -X b (t)) = 0, so limt→∞(Xa(t) -X b (t)) = 0, where the limits are exponential convergence. (The fact that the coefficient matrices in [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] depend on the state values xt does not matter, since the positivity proof does not use any information about the specific dependencies of the coefficient matrices on xt.) Since X = (X -X b ) + (X b -Xa) + Xa is a sum of three terms that exponentially converge to 0, and

∈ [0, 1/2] for all i ∈ {1, 2
|φχ a | [-τ,0] = |φχ b | [-τ,0] = 2|φX | [-τ,0] |φ d | [-τ,0] /φ0 where |•| [-τ,0]
is the sup norm, it follows that X (t) converges to 0 exponentially. Hence, we next study positive solutions of ( 10) satisfying [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF].

Step 3: Exponential Stability of Positive Solutions (ξ, x, Ψ, Z) of [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] Set c = x + Z and γ = ξ + Ψ. Then (10) and the decompositions

B s i = B + i + B - i for i = 2, 3 and B * 1 = B1 + B 1 give γ(t) = B * 1 (t)γ(t) + t t-τ B s 3 (t, m)c(m)dm c(t) = γ(t) + t t-τ B s 2 (m)c(m)dm. (14)
We use the linear function V1 : R × R n → R defined by V1(t, γ) = p(t) γ, where p is from Assumption 1 for the choice of x(t) that we used to define the Bi's at the start of the proof. Along all positive trajectories of ( 14), we get the following for all t ≥ 0:

V1(t) = ṗ(t) + p(t) B * 1 (t) γ(t) + t t-τ p(t) B s 3 (t, m)c(m)dm. ( 15
) Assumption 1 gives p (t)B s 3 (t, m)c(m) ≤ c6p (t)c(m) and p (t)B s 2 (m)c(m) ≤ c4p (t)c(m) ≤ c5p (m)c(m), since c is positive valued. Setting v(t) = V1(t, γ(t)) gives v(t) ≤ -c1v(t) + c6 t t-τ p(t) c(m)dm and p(t) c(t) ≤ v(t) + c5 t t-τ p (m)c(m)dm (16)
hold for all t ≥ 0, where we combined ( 14) and ( 15) and used the positivity of the solution (ξ, x, Ψ, Z). We next prove:

Claim 1: There are constants g ∈ (τ, 1/c5) and h > 0 such that hg -c1 < 0 and c6 + h(gc5 -1) < 0

hold, where c1, c5, and c6 are from Assumptions 1-2.

Proof of Claim 1. From (3), we get τ < 1/c5, so we can choose a constant g ∈ (τ, 1/c5). We rewrite our objectives [START_REF] Mazenc | Asymptotically stable interval observers for planar systems with complex poles[END_REF] as

c6 1 -gc5 < h < c1 g . (18)
There exists h > 0 such that (18) holds if and only if there is g ∈ (τ, 1/c5) such that (1 -gc5)/c6 > g/c1, i.e., 1 > (c6/c1 + c5)g. By Assumption 2, 1 > (c6/c1 + c5) τ . Therefore there exists g ∈ (τ, 1/c5) such that 1 > (c6/c1 + c5) g, which proves the claim. We use g and h from Claim 1 to define V2 and V3 by

V2(t, ct) = t t-τ (g -t + )p ( )c( )d and (19)
V3(t, γ(t), ct) = V1(t, γ(t)) + hV2(t, ct). ( 20
)
By the second inequality in ( 16), the time derivative of V2(t, ct) along all componentwise positive solutions of ( 14) satisfies

V2 = - t t-τ p ( )c( )d + gp (t)c(t) -(g -τ )p (t -τ )c(t -τ ) ≤ (gc5 -1) t t-τ p ( )c( )d + gv(t) (21)
for all t ≥ 0, since the fact that g ≥ τ lets us drop the term -(gτ )p(t -τ ) c(t -τ ). Along all positive valued solutions of (10),

g t t-τ p ( )c( )d ≥ V2(t, ct) ≥ (g-τ ) t t-τ p ( )c( )d ≥ (g -τ )c2 t t-τ n i=1 ci( )d (22)
for all t ≥ 0, by our lower bound on p(t) from Assumption 1.

Along all trajectories of (14), we can combine ( 16) and ( 21) to get

V3 ≤ -c1v(t) + c6 t t-τ p (m)c(m)dm + h(gc5 -1) t t-τ p ( )c( )d + hgv(t) = (hg-c1) v(t) + [c6 + h(gc5 -1)] t t-τ p (m)c(m)dm (23)
for all t ≥ 0. Then we can use the upper bound on V2 from ( 22) and ( 17) to show that the constant c7 = min{c1 -hg, (h(1 -gc5) -c6)/(gh)} is such that V3(t) ≤ -c7V3(t, γ(t), ct) for all t ≥ 0. For all t1 ≥ t2 ≥ 0, we can integrate the preceding inequality to get

V1(t1, γ(t1)) + hV2(t1, ct 1 ) ≤ exp(-c7(t1 -t2)) [V1(t2, γ(t2)) + hV2(t2, ct 2 )] , (24)
by our formula for V3 in [START_REF] Ngoc | Stability of positive differential systems with delay[END_REF]. Since [START_REF] Tchangani | Robust stability and stabilization of systems with aftereffect[END_REF] holds for all t ≥ 0 along all positive trajectories of (10), our choice of c2 in Assumption 1 gives

c2 n i=1 γi(t1) + h(g -τ )c2 n i=1 t 1 t 1 -τ ci( )d ≤ e -c 7 (t 1 -t 2 ) V1(t2, γ(t2)) + hV2(t2, ct 2 ) . (25)
We conclude from (25) that

γ(t) + t t-τ c(m)dm → 0 exponentially. (26)
It follows that c(t) → 0 exponentially, since ( 14) and (26) provide a constant matrix c > 0 and constants c 8 > 0 and c9

> 0 such that |c(t)| ≤ |γ(t) + c t t-τ c(m)dm| ≤ c9exp(-c 8 t)(|γ(0)| + τ |c| [-τ,0] ) ≤ c9exp(-c 8 t)(|c(0)| + (1 + |c|)τ |c| [-τ,0]
). Since (ξ(t), x(t), Ψ(t), Z(t)) > 0, c(t) = x(t) + Z(t), and γ(t) = ξ(t) + Ψ(t) hold for all t ≥ 0, it follows from (26) that all positive solutions of ( 10)-( 11) converge exponentially to zero. Hence, our argument at the end of Step 2 implies that ( 6) is uniformly globally exponentially stable to 0. This proves Theorem 1. Remark 4: Often, the exact values of the entries of the matrices A1 and A2 in (1) are uncertain, but their signs are known. We represent this case using multiplicative uncertainties on their entries, i.e.,

ẋ(t) = (δA1)(t, xt)x(t) + (δA2)(t, xt)x(t -τ ) ( 27
)
where the (i, j) entry of (δAp

)(t, xt) is δpij(t,
δ δ c 6 c 1 + c5 τ < 1. ( 28
)
Under these new assumptions, we can show that ( 27) is uniformly globally exponentially stable to zero. The proof is similar to the proof of Theorem 1, except one replaces A1 and A2 by (δA1) and (δA2) respectively throughout the earlier proof. The c6 is multiplied by δ2 in (28), because both matrices B a 1 and B a 2 in the formula for B a 3 (t, φ, m, ψ) have their entries multiplied by the uncertainties.

V. ANOTHER RESULT

A. Statement of Result

We next present an alternative result for systems of the form

ẋ(t) = M (t, xt)x(t) + P (t -τ, xt-τ )x(t -τ ) (29)
under the following assumptions: Assumption 3: For all t ≥ 0 and all functions φ in Cin, the matrix M (t, φ) ∈ R n×n is Metzler and P (t, φ) ∈ R n×n is nonnegative. Also, M and P are locally Lipschitz in φ and continuous in t.

Assumption 4: There exist a constant positive vector v ∈ R n and a locally Lipschitz function c : Cin → (0, ∞) such that v (M (t, φ) + P (t, φ)) ≤ -c(φ)v holds for all t ≥ 0 and φ ∈ Cin.

We can then prove: Theorem 2: Fix any constant τ > 0. If (29) satisfies Assumptions 3-4, then (29) is uniformly globally asymptotically stable to 0.

Remark 5: Theorem 2 differs from Theorem 1 because P is nonnegative valued, so P (t, xt-τ )x(t -τ ) never has a stabilizing effect. We do not require bounds or growth conditions on M or P , or a uniform positive lower bound on c or a bound on τ . Since the off diagonal entries of M + P are nonnegative valued and v > 0, Assumption 4 requires that all main diagonal entries of M be negative valued. While Theorem 1 shows global exponential stability, Theorem 2 shows the weaker global asymptotic stability condition.

B. Proof of Theorem 2

Consider any solution x(t) of ( 27). Fix a value T > 0 such that x(t) is defined on [-τ, T ). We also set Mτ (t) = M (t, xt), Pτ (t) = P (t, xt), and cτ (t) = c(xt). Then for all t ∈ [0, T ), we have

ẋ(t) = Mτ (t)x(t) + Pτ (t -τ )x(t -τ ). ( 30
) Let xL : [-τ, 0] → (0, ∞) be defined componentwise by xL,i(t) = |xi(t)| + max ∈[-τ,0] |x( )| for i = 1, 2, . . . , n, assuming without loss of generality that x0 is nonzero. Then -xL(t) < x(t) < xL(t) for all t ∈ [-τ, 0] and max ∈[-τ,0] |x( )| ≤ max ∈[-τ,0] |xL( )| ≤ 2 √ n max ∈[-τ,0] |x( )|. We extend xL to [-τ, T ), by solving ẋL(t) = Mτ (t)xL(t) + Pτ (t -τ )xL(t -τ ) (31)
on [0, T ). For all t ∈ [0, T ), the matrices Mτ (t) and Pτ (t) are Metzler and nonnegative valued, respectively. It follows that

-xL(t) ≤ x(t) ≤ xL(t) (32)
and -xL(t) ≤ 0 ≤ xL(t) hold for all t ∈ [-τ, T ). The bounds (32) follow by showing that xL(t) -x(t) and x(t) + xL(t) have positive valued initial functions and are solutions of ẏ(t) = Mτ (t)y(t) + Pτ (t -τ )y(t -τ ) and so are nonnegative valued on [-τ, T ); see the appendix below for analogous arguments. (As in the proof of Theorem 1, the fact that the coefficient matrices depend on the state does not matter, since the analysis does not use any information about the specific dependencies of the coefficient matrices on xt.) Next note that for all t ∈ [0, T ), we have v ẋL(t) = v Mτ (t)xL(t) + v Pτ (t -τ )xL(t -τ ). Hence, the function

a(t) = v xL(t) + t t-τ v Pτ (m)xL(m)dm satisfies ȧ(t) = v (Mτ (t) + Pτ (t))xL(t) ≤ -cτ (t)v xL(t) (33)
for all t ∈ [0, T ), by Assumption 4. Since a is nonnegative valued and cτ (t)v xL(t) ≥ 0 for all t ∈ [0, T ), it follows from (33) that a(t) is bounded over [0, T ). Hence, our formula for a(t) and the nonnegative valuedness of v Pτ (m)xL(m) imply that v xL(t) and so also xL(t) are bounded on [0, T ). It follows from (32) that x(t) is bounded on [0, T ), so the finite escape phenomenon cannot occur, so each trajectory of (29) is defined over [-τ, ∞). Also, xL(t) is bounded, since a(t) ≤ a(0) for all t ≥ 0. Hence, (32) ensures that each trajectory x(t) is bounded. Also, (32) hold for all t ≥ -τ , and (30) and (31) hold for all t ≥ 0.

Since xL is bounded and positive valued and cτ is positive valued, we can find a constant c * > 0 such that c * v Pτ (t)xL(t) ≤ cτ (t)v xL(t) and inf t≥0 cτ (t) = inf t≥0 c(xt) > c * , for all t ≥ 0. Therefore, we can use (33) to find a constant c * * > 0 such that the time derivative of

a (t) = a(t) + c * 2τ t t-τ t s v xL( )d ds satisfies ȧ (t) ≤ -1 2 cτ (t)v xL(t) -c * 2τ t t-τ v xL( )d ≤ -c * * a (t) (34)
for all t ≥ 0, because

d dt t t-τ t s v xL( )d ds = τ v xL(t) - t t-τ v xL( )d
holds for all t ≥ 0. The exponential decay estimate (34) provides a uniform global asymptotic stability estimate for xL, i.e., a function β ∈ KL [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]

such that |x(t)| ≤ |xL(t)| ≤ β(max ∈[-τ,0] |xL( )|, t) for all t ≥ 0, which gives the conclusion, because max ∈[-τ,0] |xL( )| ≤ 2 √ n max ∈[-τ,0] |x( )|.

VI. ILLUSTRATIONS

A. System with a stabilizing term without delay

We first consider the one-dimensional system

ẋ(t) = l1 cos 2 (t)x(t) + l2 sin(t)x(t -τ ) , (35)
where τ ≥ 0, l1 ∈ R, and l2 ∈ R are constants. We use Theorem 1 to find conditions on τ , l1 and l2 that ensure that (35) is exponentially stable. Using the above notation with the dependencies on φ and ψ omitted, we choose B a 1 (t) = Ba 1 (t) = l1 cos 2 (t) + l2 sin(t + τ ), B a 1 (t) = 0, B a 2 (t) = -l2 sin(t + τ ), and B a 3 (t, m) = -l2 sin(m + τ )(l1 cos 2 (t) + l2 sin(t + τ )). Let us determine conditions ensuring that there are a C 1 positive valued function p : R → R and a constant c1 > 0 such that ṗ(t) + p(t)B a 1 (t) = -c1p(t), for all t ≥ 0, which is equivalent to ṗ(t) = -[c1 + l1 cos 2 (t) + l2 sin(t + τ )]p(t). Assuming that l1 < 0 and picking c1 = -0.5l1 gives ṗ(t) = [-0.5l1 cos(2t) -l2 sin(t + τ )] p(t), by the double angle formula for cosine. Thus, we can choose p(t) = exp(-(l1/4) sin(2t) + l2 cos(t + τ )). We deduce that Assumptions 1 and 2 hold with

c2 = exp(-|l1|/4 -|l2|), c3 = 1/c2, c5 = |l2|exp(|l1|/2 + 2|l2|), c6 = (|l1| + |l2|)|l2|, (36)
and all τ > 0 such that (c6/c1 + c5) τ < 1, which is equivalent to

|l2| 2 |l 1 | (|l1| + |l2|) + e 0.5|l 1 |+2|l 2 | τ < 1 . (37)
By Theorem 1, we conclude that the system (35) is uniformly globally exponentially stable to 0, provided l1 < 0 and (37) hold.

B. System with a stabilizing term with delay

We next illustrate Theorem 1 using the chain of integrators

ξ1(t) = v1(t -τ ), ξ2(t) = v2(t -τ ), ξ3(t) = v1(t)ξ2(t) (38)
for any constant delay τ that satisfies

τ ∈ 0, 1 3+2 √ e . (39)
In [15, Section 6.2], we solved a tracking problem for (38) for the reference trajectory (-cos(t), 0, 0) when τ = 0, by building a strict Lyapunov function, but there is no clear analog of this earlier construction under our condition (39). Here we solve the problem of globally asymptotically tracking the trajectory (sin(t), 0, 0) . Fix a constant a0 such that

a0 ∈ 0, 1 4π (40)
and set γ1(t) = ξ1(t) -sin(t) and v1(t) = cos(t + τ ) -a0 arctan(γ1(t)). This gives

γ1(t) = -a0 arctan(γ1(t -τ )), ξ2(t) = v2(t -τ ), ξ3(t) = v1(t)ξ2(t). (41)
Since the origin of γ1(t) = -a0 arctan(γ1(t -τ )) is globally asymptotically stable to zero, the tracking dynamics (41) will be globally asymptotically stable to zero if the origin of

ξ2(t) = v2(t -τ ), ξ3(t) = v1(t)ξ2(t) (42)
is globally exponentially stable (GES) to 0 with a GES estimate that is independent of γ1.

To show this GES property for (42), fix any γ1 satisfying γ1(t) = -a0 arctan(γ1(t -τ )) for all t ≥ 0, and apply backstepping. For each t ≥ 0, we set G(t) = cos(t + τ ) -a0 arctan(γ1(t -τ )), so v1(t+τ )G(t+τ ) = -a0 cos(t+2τ ) arctan(γ1(t)) -a0 arctan(γ1(t+τ )) cos(t+2τ ) +a 2 0 arctan(γ1(t+τ )) arctan(γ1(t))+cos 2 (t+2τ ).

(43)

Using γ2(t) = ξ2(t) + G(t)ξ3(t -τ ), (42) becomes

γ2(t) = v2(t -τ ) + G(t)v1(t -τ )ξ2(t -τ ) -sin(t+τ )- a 2 0 arctan(γ 1 (t-2τ )) 1+γ 2 1 (t-τ ) ξ3(t-τ ) ξ3(t) = v1(t)(γ2(t) -G(t)ξ3(t -τ )). ( 44
) Choose v2(t -τ ) = -γ2(t -τ ) -R(t, γ1(t -τ ), γ1(t -2τ ), ξ2(t - τ ), ξ3(t -τ ))
, where R is the quantity in brackets in (44). This gives

ẋ1(t) = v1(t)[x2(t) -G(t)x1(t -τ )] ẋ2(t) = -x2(t -τ ) , (45)
where x1 = ξ3 and x2 = γ2. Then Theorem 1 can be used to study (45). Using the notation from the proof of Theorem 1, we choose

B1(t) = -v1(t + τ )G(t + τ ) v1(t) 0 -1 and
B * 1 (t) = -v1(t + τ )G(t + τ ) |v1(t)| 0 -1 , (46)
by defining B * 1 = B1 + B 1 as before, and taking the upper right entries of B1(t) and B 1 (t) to be max{0, v1(t)} and max{0, v1(t)}-v1(t) for all t ≥ 0, respectively. Choose p(t) = (exp(0.25 sin(2t + 4τ )), 2(1 + πa0/2)e 1/4 ) . Then since a0 ∈ (0, 1/(4π)), condition (43) gives -v1(t + τ )G(t + τ ) ≤ -cos 2 (t + 2τ ) + 17πa0/16 for all t ≥ 0. Therefore, the double angle formula for cosine gives ≤ -1 2 -17 16 πa0 p(t) . Recalling our delay bound (39) and the second inequality in (40), it is now easy to check that (42) satisfies our Assumptions 1-2 with c1 = (1/2) -17πa0/16, c2 = e -1/4 , c3 = 2(1 + πa0/2)e 1/4 , c5 = 2e 1/2 (1+πa0/2) 3 , and c6 = (3/2)(1+πa0/2) when (39) holds and a0 > 0 is small enough. Hence, Theorem 1 applies. Combined with the GAS property of the γ1 subsystem of (41), we conclude that the tracking dynamics (41) is UGAS to zero, as claimed. See Fig. 1 for a simulation. We are not aware of any other technique that makes it possible to prove GES of (45) to 0 under our delay bound (39).

ṗ(t) + p(t) B * 1 (t) = 1 2 cos(2t + 4τ ) -v1(t + τ )G(t + τ ) e 1

C. System that is nonlinear in the state

We next illustrate Theorem 2 using the one-dimensional system ẋ(t) = -x(t)e -x(t) + 0.5x(t -τ )e -x(t-τ ) .

(47)

Omitting the time dependence in the coefficients, we can apply Theorem 2 with M (xt) = -e -x(t) and P (xt-τ ) = 0.5e -x(t-τ ) , since then M (φ) + P (φ) = -0.5e -φ holds for all φ ∈ Cin and Assumption 4 holds with c(φ) = 1 2 e -φ and v = 1. This example does not seem to be covered by earlier results, such as the Razumikhin Theorem or results based on Lyapunov-Krasovskii functionals.

VII. CONCLUSIONS

Stabilizing time-varying nonlinear systems with delays is challenging and beyond the scope of standard frequency domain and linear matrix inequality methods. The state-of-the-art results were largely limited to systems with input delays or required Lyapunov-Krasovskii functionals that may not always be easy to find [START_REF] Pepe | On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equation[END_REF]. Here, we used a very different approach, by expressing the original system as a coupling of (a) an integral equation and (b) a differential equation with a distributed delay. Another novel feature is our viewing the system solutions as solutions of a higher order system, and using positive systems to reduce the stabilization problem to a study of positive solutions of the higher order system. This improved on the work [START_REF] Mazenc | Stability analysis of time-varying neutral time-delay systems[END_REF] for neutral systems, which did not take the potentially stabilizing effect of the delayed term into account. The positivity helped us prove exponential stability, using linear Lyapunov-Krasovskii functionals. We illustrated our work in a chain of integrators, where we provided a larger upper bound on the allowable delays than was available in existing results. We plan to generalize our work to hyperbolic PDEs and difference equations [START_REF] Hennet | Stability conditions of constrained delay systems via positive invariance[END_REF].

APPENDIX: POSITIVENESS OF THE SYSTEM [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF] Let (φ ξ , φx, φΨ, φZ ) ∈ Cin be any positive valued initial condition satisfying [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]. We prove that the solution of ( 10) with (φ ξ , φx, φΨ, φZ ) ∈ Cin as the initial function is positive for all t ∈ [-τ, ∞). Throughout the sequel, let B1ij denote the (i, j) entry of B1 for all i and j. We prove the positivity of (10) by contradiction.

Case 1: Suppose that there were tc > 0 and i ∈ {1, ..., n} such that (ξ(t), x(t), Ψ(t), Z(t)) > 0 for all t ∈ [-τ, tc) and ξi(tc) = 0. Since B1(t) is Metzler and B 1 (t), B + 3 (t, m), and B - 3 (t, m) are nonnegative valued, it follows from (10) that ξi(t) ≥ B1ii(t)ξi(t) for all t ∈ [0, tc]. By integrating this inequality, we get ξi(tc) ≥ exp( tc 0 B1ii(m)dm)ξi(0) > 0. This yields a contradiction. Case 2: Suppose that there is tc > 0 and i ∈ {1, ..., n} such that (ξ(t), x(t), Ψ(t), Z(t)) > 0 for all t ∈ [-τ, tc) and xi(tc) = 0.

Since B + 2 (m) and B - 2 (m) are nonnegative valued, it follows from the structure of the x subdynamics of (10) that xi(tc) ≥ ξi(tc) ≥ 0. Hence, ξi(tc) = 0. From Case 1, we again have a contradiction.

Case 3: Suppose that there is tc > 0 and i ∈ {1, ..., n} such that (ξ(t), x(t), Ψ(t), Z(t)) > 0 for all t ∈ [-τ, tc) and Ψi(tc) = 0. Arguing as in Case 1 with ξi replaced by Ψi, we can conclude.

Case 4: Suppose that there is tc > 0 and i ∈ {1, ..., n} such that (ξ(t), x(t), Ψ(t), Z(t)) > 0 for all t ∈ [-τ, tc) and Zi(tc) = 0. Arguing as in Case 2, we can conclude from Case 3.

This concludes the proof of the positiveness of the system [START_REF] Haddad | Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay[END_REF].

  , . . . , n}. Set E * (t) = (e Ht , e Ht , . . . , e Ht ) ∈ R n and ∆ * = (1-δ1, 1-δ2, . . . , 1-δn) ∈ R n . Then the positive valued function φ d (t) = (φ d,ξ , φ d,x , φ d,Ψ , φ d,Z ) (t) = ∆ * , E * (t), ∆ * , E * (t) satisfies (11). Set φ = 2 max t∈[-τ,0] |φX (t)| and φ0 = min{1maxi δi, e -Hτ }. Then φX a (t) = ( φ/φ0)φ d (t) is positive valued and φX b (t) = -( φ/φ0)φ d (t) is negative valued, and (12) holds.

Fig. 1 .

 1 Fig.1. Simulations of Chain of Integrators (38) Tracking (sin(t), 0, 0) with our Closed Loop Control, Initial Function ξ 0 = (1, 2, 3) , Delay τ = 1/6 and a 0 = 1/(4π). Top to Bottom: ξ 1 (t), ξ 2 (t), ξ 3 (t), v 1 (t), and v 2 (t).

  xt)apij(t, xt) for all i and j in {1, 2, . . . , n} and the matrices Ap = [apij] are known for p = 1, 2. To prove our robustness property, we assume that each δpij : [0, ∞)×Cin → [δ, δ] is continuous but unknown, but that their constant lower and upper bounds δ > 0 and δ > 0 are known. We maintain our Assumptions 1-2 exactly as before, except we replace the first inequality in[START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF] by the requirement that ṗ(t) + p(t) (L + L)(t, xt, xt+τ ) ≤ -c1p(t) , where L(t, xt, xt+τ ) = δA1(t, xt) +

δA2(t + τ, xt+τ ), and where L and L are any matrices such that L is Metzler, L ≥ 0, and L = L -L on the domain of L; we replace the last inequality in (2) by p (t)((A1(t, φ)) s + (A2(t + τ, ψ)) s )B a 2 (m, ψ)) s ≤ c6p (t); and we replace (3) by the inequality

Supported by NSF-ECCS Grant 1102348.


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

Michael Malisoff
email: malisoff@lsu.edu

New Control Design for Bounded Backstepping under Input Delays *

Keywords: Backstepping, delay, stabilization

We provide a new backstepping result for time-varying systems with input delays. The novelty of our work is in the bounds on the controls, and the facts that (i) one does not need to compute any Lie derivatives to apply our controls, (ii) the controls have no distributed terms, and (iii) we do not require any differentiability conditions on the available controls for the subsystems.

Introduction

Backstepping is useful for designing feedback controls for many classes of nonlinear systems. The basic idea of backstepping is to design globally asymptotically stabilizing controllers, by building up from known globally asymptotically stabilizing controllers for certain subsystems of the original system, often by constructing Lyapunov functions for the closed loop subsystems. Many contributions have been written on this subject; see [START_REF] Karafyllis | Non-uniform stabilization of control systems[END_REF]; [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]; [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]. However, only a few works (such as [START_REF] Choi | Asymptotic stabilization of an input-delayed chain of integrators with nonlinearity[END_REF]; [START_REF] Mazenc | Backstepping design for time-delay nonlinear systems[END_REF]; [START_REF] Mazenc | Backstepping for nonlinear systems with delay in the input revisited[END_REF]) apply backstepping to input delayed systems and none of those works provide bounded feedbacks. In [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF], a bounded backstepping feedback for time-varying systems is designed, but we are not aware of any extension of [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF] in the literature to cases with input delays.

There are several valuable techniques for designing controls under input delays; see [START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF]; [START_REF] Kharitonov | On the stability of linear systems with uncertain delay[END_REF]; [START_REF] Niculescu | Delay Effects on Stability: A Robust Control Approach[END_REF]. One involves prediction, where the dynamic controls have distributed terms that usually require values of the state along a continuum of past times (but see [START_REF] Karafyllis | Numerical schemes for nonlinear predictor feedback[END_REF]; [START_REF] Karafyllis | Predictor-based tracking for neuromuscular electrical stimulation[END_REF] for sampling based prediction). Prediction has its origins in the Smith predictor for linear systems in [START_REF] Smith | A controller to overcome dead time[END_REF]. However, recent advances in prediction make stabilization possible for many classes of nonlinear systems of ordinary and partial differential equations, including adaptive cases. See, e.g., the recent monographs [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]; [START_REF] Krstic | Nonlinear Control Under Nonconstant Delays[END_REF] and Bekiaris-Liberis and Krstic (2013) for some state-ofthe-art work related to prediction. While prediction allows arbitrarily long delays, the distributed terms may not always be easy to compute. An alternative approach involves (a) building a control that ensures global asymptotic stability when the delay is zero and then (b) introducing a delay h > 0 into the control and computing upper bounds on the allowable delays h such that the closed loop system with the input delay h is still globally asymptotically stable. This approach is sometimes called emulation, and it usually does not allow arbitrarily long delays (but see [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] for drift-free control affine systems where a scaling of the control can ensure uniform global asymptotic stability under arbitrarily long delays). For control affine systems, the nominal controller in step (a) is often computed using Lie derivatives of a Lyapunov function for the corresponding undelayed system.

In this note, we propose a new extension of backstepping for time-varying systems that produces bounded controllers under input delays, and that differs from both prediction and emulation. Three potential advantages of our new approach are that (1) the control formula does not involve Lie derivatives along any vector fields of the systems, (2) our new controllers do not have any distributed terms, and (3) we can allow cases where the nominal controller is not necessarily C 1 . Moreover, we can often allow arbitrarily long input delays, and all trajectories of the closed loop system converge to 0. Our main assumption is a stability condition on a closed loop dynamics that is obtained by putting a suitably averaged nominal control into a subsystem of the original system; see (3).

In the next section, we provide definitions, to help make our work self contained. In Section 3, we state our bounded backstepping theorem under input delays, which we prove in Section 4. In Section 5, we give Lyapunov-like sufficient conditions for the main assumption of our theorem to hold. Our sufficient conditions ask that the nominal control stabilizes the system when the delay is set to zero. We illustrate our theorem in two examples, in Sections 6-7. In our first example, no negative values of the first components of the state are available for measurement (so the control depends on time delayed values of the output y = (max{0, x}, z) instead of current values, where (x, z) is the state of the system) but there is an upper bound on the allowable delays. In our second example, our method applies without imposing any upper bound on the delay, using a control scaling argument. In both examples, the assumptions of our theorem are satisfied, so we can certify stability for the cascade under time delays. In Section 8, we provide a technical discussion that compares our new backstepping theorem with [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] and other related results. We close in Section 9 by summarizing our contribution and by suggesting future research topics. This paper generalizes our conference version [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF]. Our work [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF] was confined to the case where the z subsystem in Theorem 1 below is scalar valued, and [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF] did not include Section 8 below, nor did [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF] include simulations or Proposition 1 below on ways to check Assumption 2. By covering more general systems and ways to check Assumption 2, this version adds considerable value relative to [START_REF] Mazenc | Bounded backstepping approach under input delays[END_REF].

Definitions and Notation

Unless otherwise noted, the dimensions of our vectors are arbitrary. The Euclidean norm of vectors in R n , and the induced norm of matrices, are both denoted by |•|. For each constant r > 0, we use B p,r to denote the closed radius r ball in R p centered at the origin. For each integer k ≥ 0, we let C k denote the set of all continuous functions whose partial derivatives up through order k are all continuous, when the domains and ranges are clear from the context, so C 0 is the set of all continuous functions. We always assume that the initial times t 0 ≥ 0 for all trajectories are t 0 = 0, but analogous results can be written for all t 0 ≥ 0. Given any constant h > 0, we let C in denote the set of all continuous functions φ : [-h, 0] → R n , which we call the set of all initial functions. A function G : [0, ∞)×R n → R p is called uniformly bounded with respect to the first variable provided that there is a nondecreasing continuous function

α : [0, ∞) → [0, ∞) such that |G(t, x)| ≤ α(|x|)
holds for all t ≥ 0 and x ∈ R n . We call a function G : [0, ∞) × R n → R p uniformly locally Lipschitz in the second variable provided that for each constant K > 0, there is a constant L K > 0 such that |G(t, x) -G(t, y)| ≤ L K |x -y| holds for all t ≥ 0 and for all x and y in B n,K . For each constant r > 0, we define the (standard) saturation σ r : R → R by σ r (p) = p if p ∈ [-r, r] and σ r (p) = rsign(p) for all p ∈ R \ [-r, r]. Then σ n,r (x) = (σ r (x 1 ), σ r (x 2 ), . . . , σ r (x n )) denotes the corresponding vector saturation for each x ∈ R n . Let K ∞ be the set of all C 0 functions γ : [0, ∞) → [0, ∞) such that γ(0) = 0, γ is strictly increasing, and lim s→∞ γ(s) = ∞. For any subset S of a Euclidean space such that 0 ∈ S, a function α : S → [0, ∞) is called positive definite provided α(0) = 0 and α(r) > 0 for all r ∈ S \ {0}. ) is called uniformly proper and positive definite provided there exist class K ∞ functions α and ᾱ such that α(|x|) ≤ V (t, x) ≤ ᾱ(|x|) hold for all t ≥ 0 and x ∈ R n . For each function x : [-h, ∞) → R n and each t ≥ 0, we define x t ∈ C in by x t (θ) = x(t+θ), so x t is the translation operator. Let L 2 (p, [0, ∞)) denote the set of all Lebesgue measurable functions d

A function V : [0, ∞) × R n → [0, ∞
: [0, ∞) → R p such that ∞ 0 |d(t)| 2 dt < ∞.

Statement of Main Result and Remarks

We consider the nonlinear time-varying system

ẋ(t) = f (t, x(t)) + g(t, x(t), z(t)), ż(t) = u(t -h) (1)
where x and z are valued in R n and R p , respectively, u is the input, and h > 0 is a constant delay (but see Remark 1 for generalizations). The initial functions (x 0 , z 0 ) ∈ C in are arbitrary. We assume:

Assumption 1 The functions f and g in (1) are uniformly locally Lipschitz in the state vectors x and z, f (t, 0) = 0 for all t, and f and g are uniformly bounded with respect to the first variable. Also, for each continuous bounded function z a : R → R p , all solutions of

ẋ(t) = f (t, x(t)) + g(t, x(t), z a (t)) (2)
are defined for all t ≥ 0.

Assumption 2 There are positive constants q, τ , and ū and a function u nom : [0, ∞) × R n → B p,ū such that u nom is uniformly locally Lipschitz in the second variable and continuous, u nom (t, 0) = 0 for all t ≥ 0, qτ < 1, and for each continuous function d : [0, ∞) → R p that exponentially converges to 0, all solutions of

ξ(t) = f (t, ξ(t))+ g t, ξ(t), t-h t-τ -h qe q(ℓ-t+h) 1-e -qτ u nom (ℓ, ξ(ℓ))dℓ + d(t) (3) converge to 0 as t → ∞.
Notice that we do not require u nom to be C 1 . See Section 5 for ways to check that Assumption 2 is satisfied. In terms of the nominal control u nom from Assumption 2, we now consider the augmented system

       ẋ(t) = f (t, x(t)) + g(t, x(t), z(t)) ż(t) = u(t -h) ȧ(t) = -qa(t) + q 1-e -qτ u nom (t, x(t)) -qe -qτ 1-e -qτ u nom (t -τ, x(t -τ )) (4)
with state space R n+2p , where the control u will be specified by our theorem. In terms of the variable s(t) = z(t) -a(th), we prove:

Theorem 1 Let Assumptions 1-2 hold. Let h, ǫ, and m be any constants such that

h > 0, ǫ ∈ 0, 1 2h
, and m > q + 3.2 τ ū .

(5)

Then, with the control

u(t -h) = -ǫσ p,1 (s(t-h))- σ p,m qa(t-h) - q 1-e -qτ u nom (t-h, x(t-h)) + qe -qτ 1-e -qτ u nom t-h -τ, x(t-h-τ ) , (6)
all solutions of (1) converge to 0 as t → ∞.

Before turning to the proof of Theorem 1, we point out several ways in which our control (6) differs from existing controls in the literature. First, our control ( 6) is bounded by √ p(m+ǫ), and therefore provides a time delayed analog of the bounded backstepping controllers from [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF], which only applied to undelayed systems. Second, there is no bound on the allowable delay h > 0. This contrasts with the emulation approach in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], which usually produces a hard bound on the delay. On the other hand, as we show below, it is sometimes useful to introduce a bound on the delay h to ensure that our Assumption 2 is satisfied. Also, while the usual delay compensation results (based on emulation) conclude uniform global asymptotic stability (UGAS), Theorem 1 does not ensure the uniformness of the asymptotic stability. On the other hand, if we strengthen Assumption 2 to requiring input-to-state stability (as defined in [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]) of (3) with respect to d, then a slight variant of our proof of Theorem 1 shows that (1) in closed loop with (6) satisfies UGAS to 0. Finally, notice that the lower bound on m in (5) can be made arbitrarily small by enlarging τ > 0 and reducing q, while maintaining our requirement that qτ < 1. However, τ must also be such that all trajectories of (3) converge to 0. See Section 5 below for ways to check the convergence properties for solutions of (3).

Proof of Theorem 1

Using Assumption 1, we can easily prove that the closedloop system given by ( 4) and ( 6) is forward complete. Next, we extend

u nom to [-h, ∞) × R n by setting u nom (ℓ, x) = u nom (0, x) for all ℓ ∈ [-h, 0] and all x ∈ R n . We also define Γ : [0, ∞) × C in → R p by Γ(t, φ) = 0 -τ qe qv 1-e -qτ u nom (t + v, φ(v))dv . (7) Then Γ(t, x t ) = t t-τ qe q(ℓ-t)
1-e -qτ u nom (ℓ, x(ℓ))dℓ (8) for all t ≥ 0. Also,

d dt Γ(t, x t )) = -qΓ(t, x t ) + q 1-e -qτ u nom (t, x(t)) -qe -qτ 1-e -qτ u nom (t -τ, x(t -τ )) (9)
holds for all t ≥ 0. Comparing the formula for ȧ(t) in ( 4) with ( 9), it follows that d dt (a(t) -Γ(t, x t )) = -q(a(t) -Γ(t, x t )) for all t ≥ 0, so since q is a positive constant, a(t) -Γ(t, x t ) converges exponentially to zero. Also, our choice

s(t) = z(t) -a(t -h) gives              ẋ(t) = f (t, x(t)) + g(t, x(t), s(t) + a(t -h)) ṡ(t) = u(t -h) + qa(t -h) -q 1-e -qτ u nom (t -h, x(t -h)) + qe -qτ 1-e -qτ u nom (t -h -τ, x(t -h -τ )) (10)
for all t ≥ τ . Since u nom is bounded by ū, and since

t t-τ qe q(ℓ-t)
1-e -qτ dℓ = 1 (11) holds for all t ≥ 0, we deduce that Γ is also bounded by ū. Also,

ℓ 1-e -ℓ ≤ 8 5 for all ℓ ∈ (0, 1] (12)
because the left side of ( 12) is increasing in ℓ on (0, 1].

Since a(t) -Γ(t, x t ) exponentially converges to zero and qτ < 1, it therefore follows from ( 6), (12) (with the choice ℓ = qτ ), and our lower bound on m from (5) that for each initial function, we have

u(t -h) = -ǫσ p,1 (s(t -h)) -qa(t -h) - q 1-e -qτ u nom (t -h, x(t -h)) + qe -qτ 1-e -qτ u nom (t -h -τ, x(t -h -τ )) (13)
when t > 0 is sufficiently large, since each component of the quantity in curly braces in (13) will be valued in [-m, m] when t is large enough, so the saturation σ p,m in our control (6) can be omitted. Hence,

ṡ(t) = -ǫσ p,1 (s(t -h)) (14)
also holds when t > 0 is sufficiently large. Also, our bound on ǫ > 0 from (5) implies that ( 14) is globally asymptotically and uniformly locally exponentially stable to 0; see the appendix below. Since a(t) -Γ(t, x t ) converges exponentially to zero, the x subsystem of (10

) is ẋ(t) = f (t, x(t)) + g (t, x(t), Γ(t -h, x t-h ) + d(t)), where d(t) = s(t) + a(t -h) -Γ(t -h, x t-h
) exponentially converges to 0. Hence, the result follows from Assumption 2, since

z(t) = s(t) + (a(t -h) -Γ(t -h, x t-h )) + Γ(t -h, x t-h )
is a sum of three terms that converge to 0.

Remark 1 The variable a(t) in (4) makes it possible to avoid having the distributed term Γ(t, x t ) in the control (6). We can easily extend Theorem 1 to cover

ẋ(t) = f (t, x(t)) + g(t, x(t), z(t)) ż(t) = -bz(t) + u(t -h) (15)
for any constant b > 0. Indeed, our choice of a(t) from ( 4) and the choice s(t) = z(t) -a(t -h) give

ṡ(t) = -bs(t) + u(t -h) + (q -b)a(t -h) -q 1-e -qτ u nom (t -h, x(t -h)) + qe -qτ 1-e -qτ u nom (t -τ -h, x(t -h -τ )).
( 16)

Hence, we can simply choose

u(t-h) = -σ p,m (q-b)a(t-h) -q 1-e -qτ u nom (t-h, x(t-h)) + qe -qτ 1-e -qτ u nom (t-τ -h, x(t-h-τ )) (17)
to realize our goal for any constant m > (|q -b| + 3.2 τ )ū, by a slight variant of our proof of Theorem 1.

Sufficient Conditions for Assumption 2 to Hold

A key feature of our work is that we do not require the nominal control u nom to render ξ

(t) = f (t, ξ(t)) + g(t, ξ(t), u nom (t, ξ(t))) (18)
UGAS to 0. On the other hand, if ( 18) is control affine (meaning, there is a function G such that g(t, ξ, u) = G(t, ξ)u for all t ≥ 0, ξ ∈ R n , and u ∈ R p ) and u nom is independent of time, then we can give sufficient Lyapunovlike conditions for Assumption 2 to hold, in terms of (18). One way to do so is by using the following, where V t and V ξ denote the partial derivative with respect to t and the gradient with respect to ξ, respectively:

Assumption 3 The functions f : [0, ∞) × R n → R n , G : [0, ∞) × R n → R n×p ,
V : [0, ∞) × R n → [0, ∞); a uniformly continuous positive definite function W : R n → [0, ∞);
positive constants κ, r 1 , and r 3 ; and a constant r 2 ≥ 0 such that

V t (t, ξ)+V ξ (t, ξ)(f (t, ξ)+G(t, ξ)u nom (ξ)) ≤ -W (ξ), |V ξ (t, ξ)G(t, ξ)| ≤ κ W (ξ), |u nom (ξ)| ≤ r 1 W (ξ), |f (t, ξ)| ≤ r 2 W (ξ), and |G(t, ξ)| ≤ r 3 ( 19
)
hold for all ξ ∈ R n and t ≥ 0. Also, u nom (0) = 0, and u nom admits a global Lipschitz constant C > 0.

While the control affine system ξ = f (t, ξ)+G(t, ξ)u nom (t, ξ) in Assumption 3 is not the most general class of nonlinear systems, it is a standard form that occurs in numerous applications, e.g., when one linearizes in the input. Assumption 3 is satisfied by a large class of linear time-varying systems in closed loop with linear feedback controls. More generally, if we are able to choose a control u nom such that ξ = f (t, ξ) + G(t, ξ)u nom (t, ξ) is globally exponentially stable to 0 and if G is bounded, then (under appropriate Lipschitzness conditions on f , G, and u nom ) (Khalil, 2002, Theorem 4.14) constructs a function V such that Assumption 3 holds with W (ξ) = |ξ| 2 . We prove the following, which is of independent interest (but see Remarks 2-3 for extensions, including a scaling that allows h to be arbitrarily large when the drift f in ( 18) is the zero function):

Proposition 1 If Assumption 3 holds, and if h > 0 and τ > 0 are any constants such that

h + τ < 1 2Cκ r 2 2 + 2r 2 1 r 2 3 , ( 20
)
then there is a constant q ∈ (0, 1/τ ) such that for each continuous function d ∈ L 2 (p, [0, ∞)), the following holds: All solutions of

ξ(t) = f (t, ξ(t))+ G(t, ξ(t)) t-h t-τ -h qe q(ℓ-t+h) 1-e -qτ u nom (ξ(ℓ))dℓ + d(t) (21)
converge to 0 as t → ∞. If, in addition, u nom is bounded, then Assumption 2 holds with g(t, ξ, u) = G(t, ξ)u.

Proof: Choose any function d as in the statement of the proposition and any solution ξ(t) of ( 21). In the rest of the proof, we only consider values of t ≥ τ + h. Then our bounds in (19), our Lipschitz constant C for u nom , and (11) imply that the time derivative V of V from ( 19), along all trajectories of (21

), satisfies V ≤ -W (ξ(t)) + V ξ (t, ξ(t))G(t, ξ(t)) (d(t)+ t t-τ qe q(ℓ-t) 1-e -qτ u nom (ξ(ℓ -h)) -u nom (ξ(t)) dℓ ≤ -W (ξ(t)) + κ W (ξ(t)) (d(t) C t t-τ qe q(ℓ-t) 1-e -qτ ξ(ℓ -h) -ξ(t) dℓ ≤ -W (ξ(t)) +κ W (ξ(t)) C t t-τ -h | ξ(ℓ)|dℓ + |d(t)| , (22)
where q will be specified and the last inequality uses (11) and the fact that

|ξ(ℓ -h) -ξ(t)| ≤ t t-h-τ | ξ(p)|dp for all ℓ ∈ [t -τ, t]. Hence, Jensen's inequality gives V ≤ -1 2 W (ξ(t)) + (τ + h)κ 2 C 2 t t-τ -h | ξ(ℓ)| 2 dℓ +κ 2 |d(t)| 2 , ( 23
)
where we also used Young's inequality to get b W (ξ(t)) ≤ 0.25W (ξ(t)) + b 2 twice for suitable b ≥ 0. Also, Assumption 3, (21), two applications of the triangle inequality, and Jensen's inequality give

| ξ(t)| 2 ≤ 2|f (t, ξ(t))| 2 + 2|G(t, ξ(t))| 2 5|d(t)| 2 + 5 4 t t-τ qe q(ℓ-t) 1-e -qτ u nom (ξ(ℓ -h))dℓ 2 ≤ 2r 2 2 W (ξ(t)) + 10r 2 3 |d(t)| 2 + 5 2 r 2 3 r 2 1 τ t t-τ q 2 e 2q(ℓ-t) (1-e -qτ ) 2 W (ξ(ℓ -h))dℓ, (24)
since (a + b) 2 ≤ (5/4)a 2 + 5b 2 holds for all a ≥ 0 and b ≥ 0. Set

V 1 (t) = V (ξ(t)) + (τ + h)κ 2 C 2 t t-τ -h t m | ξ(ℓ)| 2 dℓdm (25)
and ∆ = κ 2 (10(τ + h) 2 (Cr 3 ) 2 + 1). Combining ( 23)-( 24) and setting ℓ = qτ ∈ (0, 1) in ( 12), we get

V1 ≤ -1 2 W (ξ(t)) + (τ +h) 2 κ 2 C 2 | ξ(t)| 2 + κ 2 |d(t)| 2 ≤ -1 2 + 2(τ + h) 2 κ 2 C 2 r 2 2 W (ξ(t)) + ∆|d(t)| 2 + 4((τ +h)κCr1r3) 2 q 1-e -qτ t t-τ W (ξ(ℓ -h))dℓ. (26)
Hence, the time derivative of

V 2 (t) = V 1 (t) + 4((τ +h)κCr1r3) 2 q 1-e -qτ τ t t-h W (ξ(ℓ))dℓ + t-h t-τ -h t-h m W (ξ(ℓ))dℓdm (27)
along all trajectories of ( 21) satisfies

V2 ≤ -1 2 + 2((τ + h)κCr 2 ) 2 + 4((τ +h)κCr1r3) 2 qτ 1-e -qτ W (ξ(t)) + ∆|d(t)| 2 . ( 28
)
Since lim q→0 + qτ /(1 -e -qτ ) = 1 for each τ > 0, it follows from our bound (20) that for small enough q ∈ (0, 1/τ ), the quantity in curly braces in ( 28) is a negative constant. Remark 2 While stated as an existence result for a constant q ∈ (0, 1/τ ), the proof of Proposition 1 can be used to find an explicit bound on the allowable values of q, namely, q must be chosen such that the quantity in curly braces in (28) is negative. Reducing q > 0 may reduce the convergence rate for the closed loop system given by ( 4) and ( 6), since d dt (a(t) -Γ(t, x t )) = -q(a(t) -Γ(t, x t )) for all t. On the other hand, we can use ( 12) with the choice ℓ = qτ ∈ (0, 1) to show that for all q ∈ (0, 1/τ ), the quantity in curly braces in ( 28) is bounded above by -0.5+2((τ +h)κCr 2 ) 2 +6.4((τ + h)κCr 1 r 3 ) 2 , and this upper bound is negative if

h + τ < 1 2Cκ √ r 2 2 +3.2r 2 1 r 2 3 . ( 29
)
Hence, our proof of Proposition 1 leads to the following variant of Proposition 1 that applies for any q ∈ (0, 1/τ ): If Assumption 3 holds, and if h > 0 and τ > 0 are any constants such that (29) holds, then for all q ∈ (0, 1/τ ) and each continuous function d ∈ L 2 ([0, ∞)), the following holds: All solutions of (21) converge to 0 as t → ∞. If, in addition, u nom is bounded, then Assumption 2 holds with g(t, ξ, u) = G(t, ξ)u.

Remark 3 Consider the special case of Assumption 3 where V is independent of t, and where the drift term f (t, ξ) in ( 18) is the zero function. Then, we can choose r 2 = 0 in Assumption 3. Also, for any constant η ∈ (0, 1), Assumption 3 still holds if we replace u nom , W , C, r 1 , and κ by ηu nom , ηW , ηC, √ ηr 1 , and κ/ √ η, respectively in ( 19). With these replacements, our upper bound (20) changes to

h + τ < 1 2ηCκ √ 2r 2 1 r 2 3 → ∞ (30)
as η → 0. Hence, Proposition 1 applies for any h > 0, by scaling the nominal control u nom by a small enough constant η > 0. Then, Theorem 1 implies asymptotic convergence of all trajectories of (1) to 0 (with the control affine choice g(t, ξ, u) = G(t, ξ)u) for all h > 0 when we use the control (6) and the nominal control ηu nom . See Section 7 for an extension of this scaling argument that applies when V is not time invariant.

First Example: Output Feedback

Consider the two-dimensional system

ẋ(t) = σ 1 (|x(t)|) + z(t), ż(t) = u(t -h) (31)
for any delay h > 0, with the choice

u nom (x) = -2σ 1 (max{0, x}). (32)
Then σ 1 (|ξ|) + u nom (ξ) = -σ 1 (ξ) for all ξ ∈ R, so Assumption 3 holds with the choices

V (ξ) = ξ 0 σ 1 (r)dr, f (t, ξ) = σ 1 (|ξ|), G(t, ξ) = 1, W (ξ) = σ 2 1 (ξ), C = r 1 = 2
, and r 2 = r 3 = κ = 1. Therefore, since τ in Proposition 1 can be taken as small as desired, this leads to the bound

h < 1 2Cκ r 2 2 + 2r 2 1 r 2 3 ≈ 0.1. ( 33
)
This restriction on h is not surprising, because the local exponential instability of ẋ(t) = σ 1 (|x(t)|) strongly suggests that (31) cannot be stabilized by a feedback that does not incorporate past values of itself (as feedbacks provided by the reduction model approach do) when h is larger than a certain value. In the next section, we give an example that allows arbitrarily long delays.

One advantage of the feedback from Theorem 1 is that it does not depend on negative values of x. Hence, if only the output y = (max{0, x}, z) is measured, then our result still applies. To the best of our knowledge, no other result in the literature can be applied to this system when only y = (max{0, x}, z) is measured. One cannot apply classical backstepping if no negative values of x can be measured. Classical backstepping would entail finding a C 2 fictitious feedback z f (x) such that ẋ = σ 1 (|x|) + z f (x) is globally asymptotically stable to 0. Since z f is C 2 and z f (0) = 0, it admits a decomposition of the form z f (x) = ax + b(x)x 2 in some interval around 0, for some constant a ∈ R. Therefore, when x > 0 is small enough, we get ẋ = σ 1 (|x|) + z f (x) = x+ax+b(x)x 2 . We deduce that a ≤ -1 must hold, because we need ẋ ≤ 0 when x > 0 is close to 0. If z f (x) only depends on nonnegative values of x, then ax + b(x)x 2 is constant over (-∞, 0], which is impossible because a ≤ -1. This is perhaps one of the most important advantages of our new approach, namely, if u nom only depends on an output y, then we only need that output.

Although our approach has important advantages, Theorem 1 is an asymptotic convergence result that does not provide information about transient behavior of the system with our control (6), and so differs from results in the literature that prove uniform global asymptotic stability. Therefore, we next provide simulations of the closed loop dynamics, to show the transient behavior. In our simulations, we took the delay h = 0.09, τ = 1, q = 0.5, ǫ = 1/(4h) = 25/9, and m = 2.1(q + 3.2/τ ) = 7.77, which satisfy our requirements. We took the constant initial function (1, 1) for the dynamics. In Figure 1 below, we show the simulations we obtained using Mathematica; see Mathematica (2015). While our simulations show transient movement of the state away from the equilibrium, they show asymptotic convergence to the equilibrium and therefore help illustrate the value of Theorem 1.

7 Second Example: Arbitrarily Long Delays

In the preceding example, we imposed an upper bound on the allowable delay h. In this section, we provide an application of Theorem 1 that allows the delay to be arbitrarily large. Consider the system

ẋ(t) = cos 2 (t)z(t), ż(t) = u(t -h). (34)
We verify Assumption 3 with u nom (ξ) = -ησ 1 (ξ), where the constant η ∈ (0, 1] will be chosen later. We show that Assumption 3 is satisfied with the choices

V (t, ξ) = V 0 (ξ) + η π V0(ξ) 1+V0(ξ) t t-π t s cos 2 (r)drds where V 0 (ξ) = ξ 0 σ 1 (p)dp, f (t, ξ) = 0, r 3 = 1, G(t, ξ) = cos 2 (t), W (ξ) = η 2 V0(ξ) 1+V0(ξ) , r 2 = 0 κ = (1 + ηπ/2) 6/η, r 1 = √ 6η, and C = η . (35)
Note that V 0 is C 1 , even, and (uniformly) proper and positive definite. Also, ξ = f (t, ξ) + G(t, ξ)u nom (ξ) = -η cos 2 (t)σ 1 (ξ).

Hence, along all trajectories of ξ = f (t, ξ) + G(t, ξ)u nom (ξ), we have V0 (ξ) = -η cos 2 (t)σ 2 1 (ξ), and so also for all t ≥ 0 and ξ ∈ R. If |ξ| ≥ 1, then since V 0 is nondecreasing on [0, ∞) and even, we get

V t (t, ξ) + V ξ (t, ξ) [f (t, ξ) + G(t, ξ)u nom (ξ)] = -ησ 2 1 (ξ) cos 2 (t) 1 + η π 1 (1+V0(ξ)) 2 t t-π t s cos 2 (r)drds + η π V0(ξ) 1+V0(ξ) π cos 2 (t) - t t-π cos 2 (r)dr ≤ -η π V0(ξ) 1+V0(ξ) t t-π cos 2 (r)dr = -W (ξ), since V 0 (ξ) = 1 2 ξ 2 for all ξ ∈ [-1, 1], which gives V 0 (ξ)/(1+ V 0 (ξ)) ≤ σ 2 1 (ξ) for all ξ ∈ R. Also, |V ξ (t, ξ)| = σ 1 (ξ) + η π σ1(ξ) (1+V0(ξ)) 2 t t-π t s cos 2 (r)drds ≤ (1 + ηπ/2)|σ 1 (ξ)| (36)
κ W (ξ) ≥ κ ηV0(1) 2(1+V0(1)) = κ 0.5η 2(1+0.5) = κ √ η √ 6 = (1 + ηπ/2)|σ 1 (ξ)| ≥ |V ξ (t, ξ)G(t, ξ)|. (37) On the other hand, if |ξ| < 1, then since V 0 (ξ) = 1 2 ξ 2 ≤ 1 2 , we get κ W (ξ) ≥ κ √ 2 ηξ 2 (t)/2 3/2 = κ √ η √ 6 |σ 1 (ξ)| = (1 + ηπ/2)|σ 1 (ξ)| ≥ |V ξ (t, ξ)G(t, ξ)|. (38)
This also gives |u nom (ξ)| ≤ r 1 W (ξ), and the remaining requirements from Assumption 3 are satisfied because f ≡ 0 and G is bounded by 1. Also, with the choices (35), our upper bound ( 20) is τ + h < 1/(16.97η(1 + ηπ/2)), so we can allow any constant delay h > 0 as long as η > 0 satisfies

η(1 + ηπ/2) < 1 16.97(h + τ ) . (39)
Notice that this example used a scaling of the control to compensate arbitrarily long delays, even though V is timevarying. Therefore, this example provides an extension of the scaling argument from Remark 3. Figure 2 illustrates the performance of (34) with our bounded delay compensating controller (6). The figure shows plots from a Mathematica simulation of the con-trol loop system with the same constant initial function (1, 1) that we used to simulate (31). We also used the delay h = 1, q = 0.5, τ = 1, ǫ = 1/(4h) = 0.25, the scaling parameter η = 0.025, and the saturation bound m = (q + 3.2/τ )η = 0.0925 from our requirements. With the longer delay of h = 1, the convergence towards the zero equilibrium is slower than in the Section 6. However, our simulation illustrates the transient behavior in the x variable and convergence toward the equilibrium, and therefore helps illustrate our theorem.

Comparison with Earlier Backstepping Results

As we noted above, some potential benefits of our approach are that we can ensure boundedness of the control for the overall cascade (4), that the nominal control does not need to be differentiable, and that no Lie derivatives of Lyapunov functions are needed in the control design. This contrasts with our earlier work [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], which produced globally asymptotically stabilizing controllers for systems of the form

ẋ(t) = f (x(t)) + G(x(t))z(t) ż(t) = u(x(t -2h), x(t -h), z(t -h)), (40)
with state space R n × R m , under the following assumption: hold for all x ∈ R n and q ∈ R n .

Then [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]

satisfies UGAS to zero. Also, [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] shows input-to-state stability with respect to additive uncertainty on the control u in (40). However, (43) will not be bounded in general, even if the nominal control u s is bounded, and [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] requires the C 1 property for u s . Therefore, our current work is a potential improvement over the backstepping results (from [START_REF] Mazenc | Backstepping design for time-delay nonlinear systems[END_REF]; [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]) and other results that do not ensure boundedness of the control, or which require more regularity (such as C 1 ) for the nominal control.

Conclusions

Time delays arise in many applications, and often as input delays. Two approaches to stabilization under input delays involve (a) solving the stabilization problem with the delay set to zero and then finding upper bounds on the constant delays h > 0 that can be introduced into the controller such that the input delayed system with delay h is still asymptotically stable and (b) predictive methods that use dynamic controllers to compensate arbitrarily long delays.

In both approaches, backstepping is important. Backstepping involves recursively building controllers, using nominal controllers for subsystems. When input constraints and delays are both present, backstepping can be challenging. While [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF] provided a bounded backstepping approach for systems without delays, here we extended [START_REF] Mazenc | Backstepping with bounded feedbacks for time-varying systems[END_REF] to systems with bounded controls and input delays.

Other potential advantages of our new method are that (a) there are no distributed terms in our control of the type that would arise from prediction and (b) we allow nominal controllers that are not necessarily C 1 . Moreover, there are interesting cases where our controllers apply without imposing any upper bound on the allowable delays. We hope to extend our work to cover cases where there are both input constraints and state constraints or where the system is governed by a hyperbolic partial differential equation, either with a boundary control or with a control in the PDE.

∞τ

  +h W (ξ(ℓ))dℓ < ∞. It follows from Barbalat's Lemma that lim t→∞ ξ(t) = 0, as needed.

Fig. 1 .

 1 Fig. 1. Closed Loop System Performance of System (31) from First Example with Stabilizing Delay Compensating Control (6). Trajectory for x (Top Panel), z (Middle Panel), and Control u (Bottom Panel).

Fig. 2 .

 2 Fig. 2. Closed Loop System Performance of System (34) from Second Example with Stabilizing Delay Compensating Control (6). Trajectory for x (Top Panel), z (Middle Panel), and Control u (Bottom Panel).

Assumption 4

 4 The functions f : R n → R n and G : R n → R n×m are locally Lipschitz. Also, there are a C 1 function u s : R n → R m such that u s (0) = 0; a function γ ∈ K ∞ for which γ(r) ≤ r for all r ≥ 0; a C 1 uniformly proper and positive definite function V : R n → [0, ∞); positive constants L and K 1 ; and constants K i ≥ 0 (i = 2, 3, 4) such that∇V (x)[f (x) + G(x)u s (x)] ≤ -γ 2 ( √ n|x|), |∇V (x)G(x)| ≤ K 1 γ(|x|), |∇u s (x)| ≤ L, |f (x)| 2 ≤ K 2 γ 2 (|x|), |G(x)| 2 ≤ K 3 (γ(|x|) + 1), and [|G(x)||u s (q)|] 2 ≤ K 4 [γ 2 (|x|) + γ 2 (|q|)](41)

A preliminary version of some of this work (excluding Proposition 1 and Section 8) appeared in the proceedings of the 2015 European Control Conference (ECCFor a comparison between our ECC article and this journal version, see the end Supported by l'Institut pour le Contrôle et la Décision de l'Idex Paris-Saclay (iCODE) (Mazenc) and NSF Division

Appendix

We prove the global asymptotic and uniform local exponential stability of the system (14) for all constants ǫ ∈ 0, 1 2h , (A.1) which was needed in our proof of Theorem 1. Fix any i ∈ {1, 2, . . . , p}. Next, note that

5ǫ. This provides a function T such that the following conditions hold for all i ∈ {1, 2, . . . , p} and all initial functions φ for the dynamics: (i) |s i (t)| ≤ 1 for all t ≥ T (|φ| ∞ ) and (ii) T (p) = 0 for all p ∈ [0, 1]. This implies that (14) becomes ṡ(t) = -ǫs(t-h) for all t ≥ T (|φ| ∞ )+h, as well as for all s 0 such that |s 0 | ∞ ≤ 1. Therefore, if t ≥ T (|φ| ∞ ) + h, then we can use Jensen's inequality and the triangle inequality to check that V 0 (s) = 0.5|s| 2 satisfies

along the trajectories of ( 14). Hence, since