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

Michael Malisoff
email: malisoff@lsu.edu

Stabilization and Robustness Analysis for Time-Varying Systems with Time-Varying Delays using a Sequential Subpredictors Approach *

Keywords: Delay, robustness, stability, time-varying

We provide a new sequential predictors approach for the exponential stabilization of linear time-varying systems. Our method circumvents the problem of constructing and estimating distributed terms in the control laws, and allows arbitrarily large input delay bounds, pointwise time-varying input delays, and uncertainties. Instead of using distributed terms, our approach to handling longer delays is to increase the number of predictors. We obtain explicit formulas to find lower bounds for the number of required predictors. The formulas involve bounds on the delays and on the derivatives of the delays. We illustrate our method in three examples.

Introduction

This work continues our search (which we began in Mazenc and [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF]) for predictive control methods for time-varying systems that can be applied without computing Lie derivatives and without computing distributed terms, and which can compensate for arbitrarily long input delays. Our work is motivated by the ubiquity of input delays across engineering, coupled with the challenges that one may encounter when building delay tolerant feedback controls, if one applies traditional emulation or prediction methods that can involve distributed terms. See, e.g., Bekiaris-Liberis and Krstic (2013), [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF], and [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] for overviews on delay compensating control, [START_REF] Sharma | Predictorbased compensation for electromechanical delay during neuromuscular electrical stimulation[END_REF] for constant electromechanical input delays in muscle response in neuromuscular electrical stimulation (or NMES), and [START_REF] Kamalapurkar | Time-varying input and state delay compensation for uncertain nonlinear systems[END_REF] and [START_REF] Merad | Isometric torque control for neuromuscular electrical stimulation with time-varying input delay[END_REF] for extensions to NMES under time-varying input delays.

For constant coefficient linear systems, it often suffices to use linear matrix inequalities (or LMIs) to build delay tolerant controls, but many important linear systems are time-varying. For instance, when tracking reference trajectories and linearizing around the reference trajectories, we obtain time-varying linear systems, even if the original system is time invariant. See Mazenc et al. (2014), and[START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF] for systems with delayed outputs, which lead to input delayed systems when the output for one system is an input for another system.

Traditional input delay compensation methods can roughly be grouped into three approaches. One approach is to solve a stabilization problem with the input delay set equal to 0, and to then look for upper bounds on the input delay that the resulting closed loop system can tolerate, without sacrificing the desired stability properties. Two advantages of this so-called emulation approach are that (a) it makes it possible to use relatively simple controls for undelayed systems (such as Lie derivative feedbacks and other approaches from [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]) and (b) the strict Lyapunov functions that one obtains from solving the feedback design problem for the corresponding undelayed system can often be transformed into Lyapunov-Krasovskii functionals, which can in turn be used to compute bounds on the input delays that the closed loop system can tolerate.

See [START_REF] Fridman | On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast-varying delays[END_REF] for background on Lyapunov-Krasovskii functionals, and [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] for ways to transform strict Lyapunov functions for undelayed systems into Lyapunov-Krasovskii functionals for the corresponding input delayed systems.

Another approach is the reduction model method, where the control is expressed implicitly as a solution of an integral equation, which can lead to challenging problems of numerically computing controls, especially when the system has uncertainty or the delay is time-varying. Prediction is another useful method, where the states in the controls are replaced by numerical predictions of the states. Standard prediction or reduction model methods can compensate for arbitrarily long input delays, and so may have an advantage over emulation for communication networks or multi-agent problems that are prone to long input delays, but the distributed terms in their controls (which use all values of the input or the state on some interval of times) may make them harder to implement; see [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], Bekiaris-Liberis and Krstic (2012), [START_REF] Mazenc | Lyapunov-Krasovskii functionals and application to input delay compensation for linear time-invariant systems[END_REF][START_REF] Witrant | Remote stabilization via communication networks with a distributed control law[END_REF].

This paper provides a new sequential predictors approach to exponential stabilization of time-varying linear systems with time-varying input delays, and therefore builds on recent notable works such as [START_REF] Najafi | Closed-loop control of dead time systems via sequential sub-predictors[END_REF] (which used LMI methods for time invariant linear systems to build sequential predictors) and [START_REF] Léchappé | Predictive Control and Estimation of Uncertain Systems with an Input Delay[END_REF] (which extended [START_REF] Najafi | Closed-loop control of dead time systems via sequential sub-predictors[END_REF] by studying constant coefficient linear systems with time-varying delays, which is a smaller class of systems than the time-varying systems that we consider here). It also extends [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF], which was confined to constant delays. Since we do not use any distributed terms or Lie derivatives in our control, our work is also very different from the classical reduction model or the more recent prediction approaches that have been used by M. Krstic and others (as in Bresch-Pietri and Petit (2014), Karafyllis andKrstic (2013), and[START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]).

Our approach uses several dynamic extensions. Each dynamic extension has the same dimension as the original system. This contrasts with the reduction model approach, where the integral equation that produces the control has the same dimension as the control. Since we do not use distributed terms, our work differs from [START_REF] Ahmed-Ali | Robust stabilization of nonlinear globally Lipschitz delay systems[END_REF][START_REF] Ahmed-Ali | Global exponential sampled-data observers for nonlinear systems with delayed measurements[END_REF], and other works that use several dynamic extensions and distributed terms. We obtain closed form control formulas and ways to compute lower bounds on the number of required extensions. Our work is mainly a theoretical and methodological development. However, we illustrate our work in three examples, including a pendulum dynamics that we studied in Mazenc et al. (2014), where we used a reduction model approach and distributed controls but did not cover time-varying delays.

There are several other notable works that use prediction without using distributed terms, but which do not co-ver the problems that we solve in the present work. The works [START_REF] Ahmed-Ali | Cascade high gain predictors for a class of nonlinear systems[END_REF], Cacace et al. (2016c), Cacace et al. (2016a), and[START_REF] Zhou | Truncated predictor feedback for linear systems with long timevarying input delays[END_REF] focused on time-invariant linear systems ẋ = Ax + Bu, and they use eigenvalue conditions on A and bounds on the delays or controllability conditions, without guaranteeing robustness under uncertainty; strict feedback systems were covered in Cacace et al. (2016b) by adding conditions on the coefficient matrices of a new system that is obtained by using a diffeomorphic transformation, but we do not require such conditions here; Cacace et al. (2014b) proved asymptotic stability results with prescribed decay rates for linear time-invariant systems by using partial spectrum assignment; Cacace et al. (2014c) was also limited to time-invariant linear systems; [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF] used a globally drift-observability condition (which we also do not require here) to cover nonlinear systems; and Zhou (2014a) and [START_REF] Zhou | Truncated Predictor Feedback for Time-Delay Systems[END_REF] gave sufficient conditions for stabilizability for time-varying linear systems under pseudo-predictor feedback with an integral delay system that is also not needed in the present work. Our work is also reminiscent of Cacace et al. (2014a), which is devoted to chain observers for time invariant nonlinear systems with time-varying measurement delays, and so does not cover the uniform global exponential stabilization results that we present here.

We use standard notation and definitions. Throughout the sequel, the dimensions are arbitrary, unless otherwise noted. We omit arguments of functions when they are clear, and we assume that the initial times t 0 for our solutions of our systems are t 0 = 0, but we can write analogs for general choices of t 0 ≥ 0. We use | • | to denote the usual Euclidean norm and the induced matrix norm, |φ| ∞ (resp., |φ| I ) is the essential supremum (resp., supremum over any interval I) for any bounded measurable function φ, and I n is the n × n identity matrix.

Our preliminary version [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] of this work only provides a sketch of the proof of its main result, while here we provide a complete and new proof, and a new example from identification theory that was not in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF]. Our new proof includes a new Lyapunov-Krasovskii functional approach that can allow smaller values for the number of required sequential subpredictors than what was required in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF]; see Remark 4.

Main Result

We study systems of the form

ẋ(t) = A(t)x(t) + B(t)u(t -h(t)) + δ(t), (1)
where the state x and the control u are valued in R n and R , respectively, h : R → [0, ∞) is a known time-varying delay, and δ : [0, ∞) → R n is an unknown measurable essentially bounded function representing unmodeled features or actuator errors. We make two assumptions:

Assumption 1 The function h is C 1 and bounded from above by a constant c h > 0, ḣ has a finite lower bound h ∈ R, ḣ is bounded from above by a constant l h ∈ (0, 1), and ḣ has a global Lipschitz constant n h > 0.

Assumption 2 The functions A and B are bounded and continuous, and there is a known bounded continuous function

K : [0, ∞) → R ×n such that ẋ(t) = [A(t) + B(t)K(t)]x(t) (2)
is uniformly globally exponentially stable on R n to 0.

Assumption 1 can model many delays, e.g., by using a standard denseness argument to arbitrarily closely approximate many non-C 1 delays, including discontinuous delays; see Remark 2. In terms of an integer m > 1 that we specify later, we use the functions

Ω i (t) = t -i m h(t) and θ j (t) = Ω -1 m-j+1 (Ω m-j (t)) (3)
for all i ∈ {0, ..., m} and j ∈ {1, . . . , m}, and define

R 1 = θ1 and R i (t) = θi (t)R i-1 (θ i (t)) for i > 1. (4)
The preceding functions are used to define the coefficients in our subpredictors and to produce the required exponential decay estimate of our transformed error vector in our theorem. Such functions exist because our upper bounds c h and l h ∈ (0, 1) from Assumption 1 imply that the Ω i 's have the range R and are strictly increasing. Hence, the θ i 's are also strictly increasing and C 1 . The inverses in (3) can be computed numerically using standard programs, e.g., the command

g = InverseFunction[Function[t, t -h[t]/m]] (5)
in Mathematica (2015) to compute Ω -1 1 (t) for a known choice of h(t). We also assume that the initial functions are constant on (-∞, 0]. In terms of the constants

u c = n h c h (1-l h ) 2 + l h 1-l h (6) and b 1 = 1 + 1 + uc m m |A| ∞ 1 + uc m m |A| ∞ , b 2 = 1 + 1 + uc m m |A| ∞ 2 , and b 3 = b 2 1 + uc m , (7)
we then prove:

Theorem 1 If Assumptions 1-2 hold, and if the constant integer m ≥ 2 satisfies

2 c h (1-l h )m 2 (b 2 1 + 4b 2 3 ) < 1, (8)
then we can find positive constants µ 1 and µ 2 such that for all solutions of (1) in closed loop with the control

u(t) = K(t + h(t))z m (t), (9)
where z m is the last n components of the system

ż1 (t) = R 1 (t)A(θ 1 (t))z 1 (t) + R 1 (t)B(θ 1 (t))u(Ω m-1 (t)) + L 1 (t)[z 1 (θ -1 1 (t)) -x(t)] żi (t) = R i (t)A(G i (t))z i (t) + R i (t)B(G i (t))u(Ω m-i (t)) + L i (t)[z i (θ -1 i (t)) -z i-1 (t)], 2 ≤ i ≤ m (10)
with the choices

L i (t) = -I n -R i (t)A(G i (t))
and

G i (t) = Ω -1 m (Ω m-i (t)), (11)
we have

|(x(t), E(t))| ≤ µ 1 |(x, E)| [-c h ,0] e -µ2t + |δ| ∞ (12)
for all t ≥ 0, where

E(t) = (z 1 (t) -x(θ 1 (t)), z 2 (t) - z 1 (θ 2 (t)), . . . , z m (t) -z m-1 (θ m (t))).
Remark 1 The z i dynamics in ( 10) are called the ith sequential subpredictor for each i. By the formula

e uc = lim m→∞ 1 + uc m m/uc uc , (13)
condition ( 8) holds if m is large enough. The estimate (12) says that the combined (x, E) dynamics are exponentially input-to-state stable (or ISS) with respect to δ; see [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] for ISS properties. Three key features of the preceding construction are (a) our allowing arbitrarily large delay bounds c h > 0, (b) the fact that our control (9) has no distributed terms and no Lie derivatives, and that (c) only the bottom n components of (10) are used to compute the control. This differs from standard predictive controls that contain distributed terms, and emulation controls that limit the allowable upper bounds on the delay. Longer delays and larger values of |A| ∞ lead to larger m values. See our examples below.

Remark 2 A motivation for ( 9)-( 10) is that the maximum delay is divided among the predictors, and each predictor refers to a specific and constant time ahead. The C 1 property of h is needed to ensure the C 1 properties of the θ i 's in our stabilization analysis. Using Gaussian smoothing and interpolation, we can sometimes approximate discontinuous delays (which violate Assumption 1) by delays h(t) that satisfy Assumption 1, and then we can use h(t) in our analysis, so our Assumption 1 is widely applicable through delay approximations. However, one cannot extend our result to arbitrary sawtooth shaped delays (which arise from sampling), as illustrated by

ẋ1 (t) = x 2 (t) ẋ2 (t) = -x 1 (t) + u(t k ), t ∈ [t k , t k+1 ) (14)
with the sample times t k = 2πk for all integers k ≥ 0, where the control values u(t k ) are arbitrary, since ( 14) cannot be globally asymptotically stabilized to 0 by any feedback. To see why ( 14) cannot be globally asymptotically stabilized to 0, first notice that the function

γ(t) = cos(t)x 1 (t) -sin(t)x 2 (t) (15)
satisfies γ(t) = -sin(t)u(t k ) along all solutions of ( 14) for all t ∈ [t k , t k+1 ) and integers k ≥ 0, which gives

γ(2π(k + 1)) -γ(2πk) = 2π(k+1) 2πk
γ(s)ds = 0 ( 16)

so (15) gives x 1 (2π(k + 1)) = γ(2π(k + 1)) = γ(2πk) = . . . = γ(0) = x 1 (0) for all integers k ≥ 0, so x 1 (t) cannot converge to 0.
3 Proof of Theorem 1

We use this lemma, which we prove in the appendix:

Lemma 1 The θ i 's in (3) and the constant (6) satisfy

| θi (t) -1| ≤ uc m and |θ -1 i (t) -t| ≤ c h 1-l h 1 m (17)
for all t ≥ 0 and i ∈ {1, 2, . . . , m}.

By Lemma 1 and our definition of the R i 's in (4), we get

|R i (t)| ≤ 1 + uc m m , 1 ≤ i ≤ m, (18)
where throughout the proof, all equalities and inequalities are for all t ≥ 0, unless otherwise indicated. In terms of the states z i of (10), set ξ i (t) = z i-1 (θ i (t)) for i ≥ 1, where z 0 = x. The rest of the proof has four parts.

Part 1: Error Dynamics. We first derive a useful formula for the dynamics of the error E = (E 1 , . . . , E m ), where

E i (t) = z i (t) -ξ i (t) = z i (t) -z i-1 (θ i (t)) ( 19
)
for all i ∈ {1, ..., m}. Our formulas (3) give

ξ1 (t) = θ1 (t)A(θ 1 (t))ξ 1 (t) + θ1 (t)δ(θ 1 (t)) + θ1 (t)B(θ 1 (t))u(θ 1 (t) -h(θ 1 (t))) = R 1 (t)A(θ 1 (t))ξ 1 (t) + θ1 (t)δ(θ 1 (t)) +R 1 (t)B(θ 1 (t))u(Ω m (θ 1 (t))), ( 20
) because θ -1 i (t) ≤ t for all t ≥ 0, (21) since (21) is equivalent to Ω m-i+1 (t) ≤ Ω m-i (t), which is equivalent to m -i + 1 ≥ m -i, so θ 1 (t) ≥ t. Since Ω m (θ 1 (t)) = Ω m-1 (t), we then have ξ1 (t) = R 1 (t)A(θ 1 (t))ξ 1 (t) +R 1 (t)B(θ 1 (t))u (Ω m-1 (t)) + θ1 (t)δ(θ 1 (t)). ( 22
)
Hence, ( 19) and the z 1 dynamics in (10) give

Ė1 (t) = R 1 (t)A(θ 1 (t))E 1 (t) + L 1 (t)E 1 (θ -1 1 (t)) -θ1 (t)δ(θ 1 (t)) . ( 23
) If 2 ≤ i ≤ m, then we have G i-1 • θ i = Ω -1 m • Ω m-i+1 • Ω -1 m-i+1 • Ω m-i = G i , where • denotes composition, so ξi (t) = θi (t) {R i-1 (θ i (t)) [A(G i (t))z i-1 (θ i (t)) + B(G i (t))u(Ω m-i+1 (θ i (t)))] +L i-1 (θ i (t))[z i-1 (θ -1 i-1 (θ i (t))) -z i-2 (θ i (t))]} = R i (t)B(G i (t))u(Ω m-i+1 (θ i (t))) + θi (t)L i-1 (θ i (t))[z i-1 (θ -1 i-1 (θ i (t))) -z i-2 (θ i (t))] + R i (t)A(G i (t))ξ i (t), (24)
by our definition of the R i 's. Since Ω m-i+1 (θ i (t)) = Ω m-i (t), we deduce that for all i ∈ {2, . . . , m}, we get

Ėi (t) = L i (t)[z i (θ -1 i (t)) -z i-1 (t)] -θi (t)L i-1 (θ i (t))[z i-1 (θ -1 i-1 (θ i (t))) -z i-2 (θ i (t))] + R i (t)A(G i (t))E i (t) = R i (t)A(G i (t))E i (t) + L i (t)E i (θ -1 i (t)) -θi (t)L i-1 (θ i (t))E i-1 (θ -1 i-1 (θ i (t))) . (25)
Since θ 1 = G 1 , this gives these error dynamics:

               Ė1 (t) = R 1 (t)A(G 1 (t))E 1 (t) + L 1 (t)E 1 (θ -1 1 (t)) -θ1 (t)δ(θ 1 (t)) Ėi (t) = R i (t)A(G i (t))E i (t) + L i (t)E i (θ -1 i (t)) -θi (t)L i-1 (θ i (t)) ×E i-1 (θ -1 i-1 (θ i (t))), 2 ≤ i ≤ m (26)
We require the following preliminary analysis.

Part 2: Fundamental System. We first study the system

ṡi (t) = R i (t)A(G i (t))s i (t) + L i (t)s i (θ -1 i (t)) = -s i (t) -L i (t) s i (t) -s i (θ -1 i (t)) (27)
for a fixed i ∈ {1, 2, . . . , m}, where the second equality used our choices

L i (t) = -I n -R i (t)A(G i (t)). By (21), we obtain ṡi (t) = -s i (t) -L i (t) t θ -1 i (t) R i (l)A(G i (l))s i (l)dl + L i (t) t θ -1 i (t) (I n + R i (l)A(G i (l)))s i (θ -1 i (l))dl. ( 28
)
To study stability properties for (28), we use Q(s i ) = |s i | 2 /2, and the lower bound θ -1 i (t) ≥ t-c h /((1-l h )m), which follows from the second bound in (17).

By our choices (7) of b 1 and b 2 , our second bound in (17) from Lemma 1, and ( 18), it follows that the time derivative of Q along the trajectories of (28) satisfies

Q(t) ≤ -|s i (t)| 2 + b 1 |s i (t)| t t- c h 1-l h 1 m |s i (l)|dl + b 2 |s i (t)| θ -1 i (t) θ -1 i t- c h 1-l h 1 m |s i (r)|| θi (r)|dr, ( 29
)
by using the change the variables r = θ -1 i (l). Here and in what follows, all inequalities and equalities are for all t ≥ 2c h /(m(1 -l h )), unless otherwise indicated. Hence, since ( 21) is satisfied, the bounds (17) give

Q(t) ≤ -|s i (t)| 2 + b 1 |s i (t)| t t- c h 1-l h 1 m |s i (l)|dl + b 3 |s i (t)| t t- 2c h 1-l h 1 m |s i (r)|dr. ( 30
)
We can apply Young's inequality s i (t)r ≤ 1 4 |s i (t)| 2 + r 2 twice for suitable r ≥ 0, and then Jensen's inequality (applied to the convex integrand |s i | 2 ), to get

Q(t) ≤ -1 2 |s i (t)| 2 + b 2 1 c h (1-l h )m t t- c h 1-l h 1 m |s i (l)| 2 dl + 2b 2 3 c h (1-l h )m t t- 2c h 1-l h 1 m |s i (r)| 2 dr, ( 31
)
where the first application was

b 1 |s i (t)| t t- c h (1-l h )m |s i (l)|dl ≤ 1 4 |s i (t)| 2 + b 2 1 c h (1-l h )m t t- c h (1-l h )m |s i (l)| 2 dl, ( 32
)
and the second one was analogous, except with b 1 and c h in (32) replaced by b 3 and 2c h , respectively.

Since our condition (8) allows us to find a constant λ > 1 that is close enough to 1 so that

λ c h (1-l h )m 2 (b 2 1 + 4b 2 3 ) < 1 2 , (33)
it follows from (31) that we can also find a constant c 0 > 0 such that the time derivative of

Q (s i,t ) = Q(s i (t)) + c h λ (1-l h )m b 2 1 t t- c h 1-l h 1 m t w |s i (r)| 2 drdw +2b 2 3 t t- 2c h 1-l h 1 m t w |s i (r)| 2 drdw along all solutions of (28) satisfies Q ≤ -c 0 Q (s i,t ), (34)
where (34) was obtained using the relations

t t- jc h 1-l h 1 m t w |s i (r)| 2 drdw ≤ jc h (1-l h )m t t- jc h 1-l h 1 m |s i (r)| 2 dr and d dt t t- jc h 1-l h 1 m t w |s i (r)| 2 drdw = jc h (1-l h )m |s i (t)| 2 - t t- jc h (1-l h )m
|s i (r)| 2 dr for j = 1 and j = 2 and where s i,t ( ) = s i (t + ) for all ∈ [-2c h /((1 -l h )m), 0]. For instance, we can choose

c 0 = min 1 -2λ c h (1-l h )m 2 (b 2 1 + 4b 2 3 ), (λ-1)(1-l h )m 2λc h .
This concludes our analysis of (27).

Part 3: Input-to-State Stability of the Error Dynamics (26). We first study the dynamics for

E (t) = E 1 (θ -1 1 • θ m (t)), E 2 (θ -1 2 • θ m (t)), . . . , E m (t)
with the disturbance δ(θ m (t)). We can find positive constants θ and θ such that θ ≤ (θ -1 i • θ m ) (t) ≤ θ hold for all t ≥ 0 and i ∈ {1, 2, . . . , m}. To find θ and θ, first note that for each i ∈ {1, 2, . . . , m}, we have

θ -1 i • θ m = (Ω -1 m-i+1 • Ω m-i ) -1 • Ω -1 1 = Ω -1 m-i • Ω m-i+1 • Ω -1 1 ,
so the chain rule gives

(θ -1 i • θ m ) (t) = 1 -m-i m ḣ(θ -1 i • θ m (t)) -1 × 1 -m-i+1 m ḣ(Ω -1 1 (t)) × 1 -1 m ḣ(Ω -1 1 (t)) -1
so the existence of the constants θ and θ follows from the lower and upper bounds on ḣ from Assumption 1.

Hence, the decay estimate (34) along trajectories of ( 27), combined with Young's inequality and our choice of the quadratic part of Q , implies that along all solutions of the dynamics for

(E 1 (t), E 2 (t)) = (E 1 (θ -1 1 • θ m (t)), E 2 (θ -1 2 • θ m (t))),
we have

d dt Q (E 2,t ) ≤ - θc0 2 Q (E 2,t ) + 2 θ4 θc0 | θ2 | 2 ∞ |L 1 | 2 ∞ Q (E 1,t ) d dt Q (E 1,t ) ≤ - θc0 2 Q (E 1,t ) + θ4 θc0 | θ1 | 2 ∞ |δ| 2 ∞ , (35)
where the first inequality used Young's inequality to get

θ2 |E 2 (θ -1 2 • θ m (t))|| θ2 | ∞ |L 1 | ∞ |E 1 (θ -1 1 • θ m (t))| ≤ θc0 2 Q (E 2,t ) + 2 θ4 θc0 (| θ2 | ∞ |L 1 | ∞ ) 2 Q (E 1,t ),
and the second inequality in (35) was obtained in an analogous way. Hence, the function

Q 2 (E 1,t , E 2,t ) = Q (E 2,t ) + 1 + 2 θ2 | θ2|∞|L1|∞ θc0 2 Q (E 1,t ) (36) satisfies Q 2 ≤ - θc0 2 Q E 1,t + Q E 2,t + 1 + 2 θ2 | θ2|∞|L1|∞ θc0 2 θ4 θc0 | θ1 | 2 |δ| 2 ∞ ( 37
)
and so is an ISS Lyapunov-Krasovskii functional for the (E 1 (t), E 2 (t)) dynamics. Continuing in a similar way (by adding one component of E at a time) provides positive constants ω i and η i such that

Q m (E t ) = ω 1 Q (E 1,t ) + . . . + ω m-1 Q (E m-1,t ) + ω m Q (E m,t ) satisfies Q m (E t ) ≤ -η 1 Q m (E t ) + η 2 |δ| 2 ∞ ( 38
)
along all solutions of the E dynamics.

Part 4: ISS of (1) with Our Control (9). By ( 19), we have

z m (t) = ξ m (t) + E m (t) = z m-1 (θ m (t)) + E m (t) = z m-2 (θ m-1 (θ m (t))) + E m-1 (θ m (t)) + E m (t), (39)
and therefore we can argue inductively to get

z m (t) = x(θ 1 (...θ m-1 (θ m (t))...)) + E 1 (θ 2 (...θ m (t)...)) + ... + E m (t). ( 40
)
The formulas

θ i (t) = Ω -1 m-i+1 (Ω m-i (t)) then give θ m-1 (θ m (t)) = Ω -1 2 (Ω 1 (Ω -1 1 (Ω 0 (t)))) = Ω -1 2 (t). We deduce that θ 1 (...θ m-1 (θ m (t))...) = Ω -1 m (t), and so z m (t) = x(Ω -1 m (t)) + E 1 (θ 2 (...θ m (t)...)) + ... + E m (t)
. Hence, our control (9) can be written as

u(t -h(t)) = K(t)[x(Ω -1 m (t -h(t))) + E 1 (θ 2 (...θ m (t -h(t)...))) + ... + E m (t -h(t))]. (41)
Since Ω m (t) = t -h(t), the closed loop system is then

ẋ(t) = (A(t) + B(t)K(t))x(t) +{B(t)[K(t)E 1 (θ 2 (...θ m (t -h(t)...))) +... + K(t)E m (t -h(t))] + δ(t)}. (42)
Also, Assumption 2 provides a C 1 bounded function P such that the Lyapunov function V (t, x) = x P (t)x has a quadratic lower bound in |x| and satisfies V ≤ -|x(t)| 2 along all solutions of (2) (using (Khalil, 2002, Theorem 4.14)). Then the triangle inequality gives

V ≤ -1 2 |x(t)| 2 + 2|P | 2 ∞ |δ | 2 ∞ ≤ -1 2|P |∞ V (t, x(t)) + 2|P | 2 ∞ |δ | 2 ∞ ≤ -1 2|P |∞ V (t, x(t)) + 2(m + 1) (|P | ∞ |B| ∞ |K| ∞ ) 2 × |E 1 (θ 2 (...θ m (t -h(t)...)))| 2 + . . . + |E m (t -h(t))| 2 + |δ| 2 ∞ (43)
along all solutions of (42), where δ is the quantity in curly braces in (42) viewed as a sum of m + 1 terms.

Also, for all i ∈ {1, 2, . . . , m -1}, we have

|E i (θ i+1 • . . . • θ m (t -h(t)))| 2 = 2Q(E i (θ i+1 • . . . • θ m (t -h(t)))) = 2Q(E i (M i (t))) ≤ 2 ωi Q m E Mi(t) (44)
where

M i (t) = θ -1 m • θ i • θ i+1 • . . . • θ m (t -h(t)) = Ω 1 • Ω -1 m-i+1 • Ω m (t),
and

|E m (t-h(t))| 2 = 2Q(E m (t-h(t))) ≤ 2 ωm Q m E Mm(t)
where M m (t) = t -h(t) = Ω m (t). We can find positive constants M and M such that M ≤ Ṁi (t) ≤ M for all t ≥ 0 and i ∈ {1, 2, . . . , m}, by reasoning as we did to obtain θ and θ. This and ( 43) give the decay estimates

V ≤ -1 2|P |∞ V (t, x(t)) + 2(m + 1)(|P | ∞ |B| ∞ |K| ∞ ) 2 × m i=1 2 ωi Q m E Mi(t) + |δ| 2 ∞ and d dt Q m E Mi(t) ≤ -Mη 1 Q m E Mi(t) + Mη 2 |δ| 2 ∞ which imply that Q + E t = V (t, x(t)) + 1 + 2(m+1) Mη1 (|P | ∞ |B| ∞ |K| ∞ ) 2 × m i=1 2 ωi Q m E Mi(t) (45)
admits constants d 0 > 0 and d 1 > 0 such that the ISS Lyapunov-Krasovskii decay condition

Q + (E t ) ≤ -d 0 Q + (E t ) + d 1 |δ| 2 ∞ ( 46
)
holds along all solutions of the dynamics for

E (t) = x(t), E M1(t) , . . . , E Mm(t) . (47)
Also, for all t ≥ 0 and i ∈ {1, 2, . . . , m}, we can use ( 21)

to obtain θ -1 i • θ m • M i (t) = θ -1 i • Ω -1 m-i+1 • Ω m (t) ≤ Ω -1 m-i+1 • Ω m (t) ≤ t.
Hence, there are positive constants c 1 and c 2 such that along all solutions of the dynamics for E and for all i ∈ {1, 2, . . . , m}, we have

|E i (θ -1 i • θ m • M i (t))| 2 ≤ Q + E t ≤ c 1 |(x, E)| 2 [-c h ,0] e -c2t + |δ| 2
∞ along all solutions of the dynamics for ( 47) for all t such that min

i θ -1 i • θ m • M i (t) ≥ 0.
The final exponential ISS estimate for the (x, E) dynamics can now be deduced from the quadratic structure of V and Q, the invertibility of the functions

H i = θ -1 i •θ m •M i
for all i, the existence of positive constants h a and h b such that H -1 i (t) ≥ h a th b for all i and t ≥ 0 (which follows from our bounds on ḣ from Assumption 1 and the Mean Value Theorem), and the subadditivity of the square root. This proves Theorem 1.

Remark 3 Our proof of Theorem 1 shows that our control can be written as

u(t -h(t)) = K(t)x(t)+ K(t)[E 1 (θ 2 (...θ m (t -h(t)...)))+ ... + E m (t -h(t))], (48)
which may at first seem to contradict our assumption that the current value of the state x(t) is not available to use in the feedback control for (1). However, since

θ 2 • . . . • θ m (t -h(t)) = Ω -1 m-1 • Ω m-2 • Ω -1 m-2 • Ω m-3 • . . . • Ω -1 1 • Ω m (t) = Ω -1 m-1 • Ω m (t) = θ -1 1 (t)
holds for all t ≥ 0, we can use (19) to get

E 1 (θ 2 (...θ m (t -h(t)...))) = z 1 (θ -1 1 (t)) -ξ 1 (θ -1 1 (t)) = z 1 (θ -1 1 (t)) -x(t), (49)
so the two terms in ( 48) and ( 49) containing x(t) cancel. Hence, our control (9) does not require current values x(t).

Remark 4 Instead of ( 8), the main result in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] required that m satisfy

max 1, 4 b1 √ 2 + b 3 c h 1-l h < m, ( 50
)
where the b i are as defined in (7). The statement of the theorem in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] is the same as in our theorem above for the special case where δ = 0, except for the condition on m. The sketch of the proof of the main result in [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] had the same structure as the complete proof we gave above with the same choice of Q, except that instead of (31), it used the estimates

Q(t) ≤ -1 2 |s i (t)| 2 + (b1/ √ 2) 2 b1/ √ 2+b3 t t- c h 1-l h 1 m |s i (l)| 2 dl + b 2 3 b1/ √ 2+b3 t t- 2c h 1-l h 1 m |s i (r)| 2 dr ≤ -1 2 |s i (t)| 2 + b1 √ 2 + b 3 t t- 2c h 1-l h 1 m |s i (r)| 2 dr, (51)
which follow from Jensen's and Young's inequalities, using the fact that (50) implies that

2c h m(1-l h ) < 1 b1/ √ 2+b3 , (52)
as well as the estimate (r 2 + s 2 )/(r + s) ≤ r + s, with the choices r = b 1 / √ 2 and s = b 3 . Then (50) implies that there are constants c 0 > 0 and λ > 1 such that along all solutions of the fundamental system (27), the function

Q old (s i,t ) = Q(s i (t)) +λ b1 √ 2 + b 3 t t- 2c h 1-l h 1 m t w |s i (r)| 2 drdw
satisfies a decay estimate of the form (34), except with Q replaced by Q old . By replacing (50) by ( 8), we can allow smaller values m, which is useful in applications. In the next section, we compare the values of m that can be obtained from (50) with the ones that we can obtain using our improved formula (8), in two examples.

Examples

Assumptions 1-2 hold for broad classes of systems of the form ẋ(t) = A(t)x(t) + B(t)u(t -h(t)), including all systems with constant controllable pairs (A, B). We first illustrate Theorem 1 in a scalar case. Then we revisit the linearized pendulum from Mazenc et al. (2014). Our work Mazenc et al. (2014) covered the linearized pendulum under constant delays, using a predictive controller with distributed terms. Hence, this section improves on [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] and Mazenc et al. (2014) in the case of the pendulum because in what follows, we allow nonconstant delays and because our control has no distributed terms. We also study a dynamics from identification theory with a time-varying delay, which we studied under constant delays in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF].

Scalar Example

Consider the scalar case

ẋ(t) = rx(t) + u(t -h(t)) + δ(t) (53)
for any constant r > 0 with the oscillating delay

h(t) = 1 + α sin(t) (54)
for any constant α ∈ (0, 1). Then Assumptions 1-2 are satisfied using n = 1, A = r, B = 1, K = -2r, c h = 1 + α, and n h = l h = α, so our ISS conclusion from Theorem 1 is satisfied using the control u(t) = -2rz m (t), where z m is the mth component of the dynamic extension from Theorem 1, under our conditions on m. To compute lower bounds on the allowable values of m, notice that in terms of our notation ( 6)-( 7), we have

b 1 = r 1 + r 1 + uc m m 1 + uc m m , b 2 = 1 + r 1 + uc m m 2 , u c = 2α (1-α) 2 ,
and b 3 = 1 + r 1 + uc m m 2 1 + uc m . (55)
Hence, our requirement on m ≥ 2 from ( 50) is

m -4 1+α 1-α r √ 2 1 + r 1 + uc m m 1 + uc m m + 1 + r 1 + uc m m 2 1 + uc m > 0. (56)
For any constants α ∈ (0, 1) and r > 0, we can compute the left side of (56) as a function φ α,r (m) of m and choose m such that φ α,r (m) > 0, since lim m→+∞ φ α,r (m) = +∞. For instance, if α = 1/7 and r = 1/2, then φ 1/7,1/2 (22) = 0.847755, and 22 is the smallest integer m for which (56) is satisfied for these α and r values. On the other hand, our condition (8) holds with m = 12 with the preceding parameter values for α and r, so Theorem 1 applies with m = 12. Hence, our new condition (8) on m produced a 45% reduction in the required number m of sequential subpredictors, as compared with (50). If we change the α value to α = 1/5 and keep the other parameters the same, then the smallest values of m for which (56) and ( 8) are satisfied are m = 31 and m = 17, respectively, so our new method for computing the number of required subpredictors again produces significant reductions in m. In general, increasing α can require increasing m, with our new method in this work producing smaller minimum allowable m values. See Fig. 1 for a comparison between the method for finding m from [START_REF] Mazenc | New prediction approach for stabilizing time-varying systems under time-varying input delay[END_REF] with the method from Theorem 1 above for the preceding scalar dynamics.

Pendulum

Consider the model of the simple pendulum with a time-varying delay h(t), where g = 9.8 m/s is the gravitational constant, l is the pendulum length in meters, M is the pendulum mass, and v is the input. As in [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] and Mazenc et al. (2014), our control objective is to track a given C 1 reference trajectory (r 1,s (t), r 2,s (t)) such that ṙ1,s (t) = r 2,s (t). Using the error variables ri = r i -r i,s (t) for i = 1, 2 and the change of feedback

   ṙ1 (t) = r 2 (t) ṙ2 (t) = -g l sin(r 1 (t)) + 1 Ml 2 v(t -h(t)) (57
u(t -h(t)) = 1 Ml 2 v(t -h(t)) -ṙ2,s (t) - g l sin(r 1,s (t)),
we obtain the tracking system

     ṙ1 (t) = r2 (t) ṙ2 (t) = g l [sin(r 1,s (t)) -sin(r 1 (t) + r 1,s (t))] + u(t -h(t)). (58)
Our work in Mazenc et al. (2014) showed that when r 1,s (t) = ωt and ω > 0 is a large enough constant and the delay is the constant h = 1, the linearization

ẋ1 (t) = x 2 (t) ẋ2 (t) = -g l cos(ωt)x 1 (t) + u(t -h), (59)
of ( 58) at 0 has the globally exponentially stabilizing distributed control u defined by u(t) = -0.6x 1 (t) -0.4x 2 (t) -t t-1 0.6(t -r -1) + 0.4 u(r)dr.

(60)

Our work [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] provided globally asymptotically stabilizing sequential predictor controls for the nonlinear system (58) when h is constant. However, it is not obvious how to extend [START_REF] Mazenc | Stabilization of nonlinear time-varying systems through a new prediction based approach[END_REF] to time-varying delays. Therefore, we apply Theorem 1 to its linearization (59) using the delay (54), where α ∈ (0, 1) is a known constant, but analogous reasoning applies for any delay h(t) satisfying Assumption 1.

To this end, first note that for any choice of the constant ω > 0, Assumption 2 is satisfied using

A(t) = 0 1 -g l cos(ωt) 0 , B = 0 1 , and
K(t) = g l cos(ωt) -1 -1 (61)
and condition (50) becomes

m 4 -1+α 1-α max{g/l,1} √ 2 1+max g l , 1 u m c,m u m c,m + 1 + max g l , 1 u m c,m 2 u c,m
> 0, where u c,m = 1 + uc m .

(62) Condition (62) holds for any choices of α and l, by choosing m large enough. For instance, if we pick α = 1/7, any ω > 0, any l > g, and m = 47, then we get the value 0.376057 for the left side of (62), and 47 is the smallest integer m such that (62) holds with the preceding parameter choices. On the other hand, (8) holds with m = 25 with the preceding parameter choices, so Theorem 1 applies with m = 25. Hence, we get a 47% reduction in the value of m using our new condition (8). See Fig. 2 for a comparison between the two conditions ( 8) and ( 62) for choosing m for the pendulum example. If we change α to α = 1/5 and keep the other parameters the same, then the conditions ( 8) and ( 62) for choosing m give the minimum allowable values m = 38 and m = 73, respectively, so our new method again gives smaller values for the minimum required numbers m of sequential predictors, and in general, larger α's can require larger m values.

Identification Theory Example

Theorem 1 applies to general time-varying linear systems for which the pairs (A(t), B(t)) are uniformly controllable and in particular allows nonzero drift, but in the special case where the drift is 0, Theorem 1 gives an explicit lower bound on the allowable m values. To illustrate this point, consider the system

ẋ(t) = -N (t)N T (t)[u(t -h(t)) + δ(t)] (63)
in which N : R → R n is a continuous function such that (i) |N (t)| = 1 for all t ∈ R and (ii) there are constants α ∈ (0, 1), β > 0 and c > 0 such that for all t ∈ R, we have

α I n ≤ t+c t N (τ )N T (τ )dτ ≤ β I n , (64)
where M 1 ≤ M 2 for matrices M 1 and M 2 means that M 2 -M 1 is positive semi-definite, and where the dimension n of the state x is arbitrary.

The system ( 63) is a benchmark model from identification theory, which we studied in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], using emulation methods. See (Anderson, 1977, Appendix A.1) for the derivation of (63) in a special case where the uncertainty δ = 0, where the dynamics are shown to arise when one wishes to identify coefficients of a stable plant transfer function, and see [START_REF] Aeyels | On the convergence of a time-variant linear differential equation arising in identification[END_REF], [START_REF] Aeyels | Asymptotic stability for time-variant systems and observability: uniform and nonuniform criteria[END_REF], and [START_REF] Peuteman | Exponential stability of nonlinear time-varying dierential equations and partial averaging[END_REF] for more recent results on (63). However, the preceding papers do not cover time-varying input delays, so our sequential subpredictors work seems novel and significant, even in the special case of identification dynamics.

To apply our method to (63), first note that (Mazenc et al., 2008, Lemma 8) says that with the choices κ = 1 + c 2 + 1 4α c4 and

P (t) = κI n + t t-c t s N (l)N T (l) dl ds, (65) the function V (x, t) = x T P (t)x satisfies V ≤ -α |x| 2 /2 (66)
along all trajectories ẋ(t) = -N (t)N T (t)x(t), and |P (t)| ≤ κ + c2 for all t ≥ 0, which implies that Assumption 2 is satisfied with K = I n .

Hence, Theorem 1 applies for all choices of h(t) that satisfy Assumption 1. Since (63) has the drift A(t) = 0, our constant b 1 is zero and b 2 = 1, and our condition (8) on m becomes

2 c h (1-l h )m 2 4 1 + uc m 2 < 1. ( 67
)
By multiplying (67) through by m 4 and taking square roots of both sides and finally completing the square, we obtain the explicit requirement

√ 2c h 1 -l h 2 + 2 √ 2c h u c 1 -l h + √ 2c h 1 -l h < m (68)
on the number m of sequential subpredictors for our conclusions of Theorem 1 to hold for our system (63).

Conclusions and Future Work

We designed exponentially ISS sequential predictor controllers for a large class of time-varying linear systems that satisfy a uniform controllability condition, under general conditions on the time-varying input delays. Our closed loop systems are ISS with respect to additive uncertainty, and our controls do not contain any distributed terms of the type that would otherwise arise when using reduction model controllers. By allowing arbitrarily large bounds on the input delays, we also overcame a possible challenge in applying emulation methods to systems that can have long input delays. Our strategy for compensating for large delay bounds involves increasing the numbers of sequential subpredictors, and our theorem provides a useful mechanism for computing the required number of subpredictors.

We hope to extend our work to adaptive systems, where additional dynamic extensions would make it possible to identify unknown parameters. We also hope to cover nonlinear systems with time-varying delays, by building sequential predictors for the linear parts, and then proving that the controls ensure local stabilization of the original nonlinear systems and using the Lyapunov decay conditions to estimate the region of attraction, which would be a sequential predictor analog of our work in Mazenc and Malisoff (2014) on local stabilization of nonlinear systems under reduction model controls. Finally, we plan to merge our methods with the methods in [START_REF] Malisoff | Robustness of adaptive control under time delays for three-dimensional curve tracking[END_REF] for state constrained curve tracking problems, using robust forward invariance.

Appendix: Proof of Lemma 1

Fix any i ∈ {1, 2, . . . , m} and t ≥ 0. By (3), we have θ i (t) = Ω -1 m-i+1 (Ω m-i (t)) and Ω m-i (t) = Ω m-i+1 (t) + 1 m h(t), so

θ i (t) = t + Ω -1 m-i+1 Ω m-i+1 (t) + 1 m h(t) -Ω -1
m-i+1 (Ω m-i+1 (t)) .

(A.1)

Hence, the Mean Value Theorem provides a w ∈ R (depending on t and i in general) such that

θ i (t) = t + 1 Ωm-i+1(w) 1 m h(t).
(A.2) Therefore, Assumption 1 gives

|θ i (t) -t| ≤ 1 1-l h 1 m h(t) ≤ c h 1-l h 1 m . (A.3)
Hence, the second inequality in (17) follows by replacing t by θ -1 i (t) in (A.3). To check the first inequality in (17), first note that θi (t) = Since Ω m-i (t) = Ω m-i+1 (t) + 1 m h(t), the function

G i (t) = Ω -1 m-i+1 Ω m-i+1 (t) + 1 m h(t) (A.5)

Fig. 1 .

 1 Fig. 1. Criteria for Choosing m in Scalar Example. Top: φ 1/7,1/2 (m) Showing Smallest Value m = 22 for Which (56) is satisfied. Bottom: Left Side of (8) with |A|∞ = r = 1/2 Showing Smallest Value m = 12 for which (8) Holds.

Fig. 2 .

 2 Fig. 2. Criteria for Choosing m in Pendulum Example. Top: φ 1/7,1 (m) Showing Smallest Value m = 47 for Which (62) is satisfied. Bottom: Left Side of (8) with |A|∞ = 1 Showing Smallest Value m = 25 for which (8) Holds.

  (Ωm-i(t))) + 1 , (A.6) by writing ḣ(t) in the first numerator in (A.6) as ḣ(t) = ( ḣ(t) -ḣ(G i (t)) + ḣ(G i (t)). It follows that | θi (t) -1| ≤ mconstant n h for ḣ. Hence, our choice of G i and the Mean Value Theorem applied to Ω -1

Preprint submitted to Automatica

Malisoff was supported by NSF-ECCS Grant 1408295, and by Roy Paul Daniels Professorship #3 in the LSU College of Science.


Jean-Pierre Demailly

EXTENSION OF HOLOMORPHIC FUNCTIONS AND COHOMOLOGY CLASSES FROM NON REDUCED ANALYTIC SUBVARIETIES

Keywords: . 2010 Mathematics Subject Classification. Primary 32L10; Secondary 32E05 Compact Kähler manifold, singular hermitian metric, coherent sheaf cohomology, Dolbeault cohomology, plurisubharmonic function, L 2 estimates, Ohsawa-Takegoshi extension theorem, multiplier ideal sheaf

The goal of this survey is to describe some recent results concerning the L 2 extension of holomorphic sections or cohomology classes with values in vector bundles satisfying weak semi-positivity properties. The results presented here are generalized versions of the Ohsawa-Takegoshi extension theorem, and borrow many techniques from the long series of papers by T. Ohsawa. The recent achievement that we want to point out is that the surjectivity property holds true for restriction morphisms to non necessarily reduced subvarieties, provided these are defined as zero varieties of multiplier ideal sheaves. The new idea involved to approach the existence problem is to make use of L 2 approximation in the Bochner-Kodaira technique. The extension results hold under curvature conditions that look pretty optimal. However, a major unsolved problem is to obtain natural (and hopefully best possible) L 2 estimates for the extension in the case of non reduced subvarieties -the case when Y has singularities or several irreducible components is also a substantial issue.

Introduction and main results

The problem considered in these notes is whether a holomorphic object f defined on a subvariety Y of a complex manifold X can be extended as a holomorphic object F of the same nature on the whole of X. Here, Y is a subvariety defined as the the zero zet of a non necessarily reduced ideal I of O X , the object to extend can be either a section f ∈ H 0 (Y, E |Y ) or a cohomology class f ∈ H q (Y, E Y ), and we look for an extension F ∈ H q (X, E), assuming suitable convexity properties of X and Y , suitable L 2 conditions for f on Y , and appropriate curvature positivity hypotheses for the bundle E. When Y is not connected, this can be also seen as an interpolation problem -the situation where Y is a discrete set is already very interesting.

The prototype of such results is the celebrated L 2 extension theorem of Ohsawa-Takegoshi [START_REF] Ohsawa | On the extension of L 2 holomorphic functions[END_REF], which deals with the important case when X = Ω ⊂ C n is a pseudoconvex open set, and Y = Ω ∩ L is the intersection of Ω with a complex affine linear subspace L ⊂ C n . The accompanying L 2 estimates play a very important role in applications, possibly even more than the qualitative extension theorems by themselves (cf. section 4 below). The related techniques have then been the subject of many works since 1987, proposing either greater generality ([Ohs88, Ohs94, Ohs95, Ohs01, Ohs03, Ohs05], [START_REF] Manivel | Un théorème de prolongement L 2 de sections holomorphes d'un fibré vectoriel[END_REF], [START_REF] Demailly | On the Ohsawa-Takegoshi-Manivel L 2 extension theorem[END_REF], [START_REF] Popovici | L 2 extension for jets of holomorphic sections of a Hermitian line bundle[END_REF]), alternative proofs ( [START_REF] Berndtsson | The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman[END_REF], [START_REF] Chen | A simple proof of the Ohsawa-Takegoshi extension theorem[END_REF]), improved estimates ( [START_REF] Mcneal | Analytic inversion of adjunction: L 2 extension theorems with gain[END_REF], [START_REF] Varolin | Three variations on a theme in complex analytic geometry, Analytic and Algebraic Geometry[END_REF]) or optimal ones ([Blo13], [START_REF] Guan | A solution of an L 2 extension problem with an optimal estimate and applications[END_REF], [START_REF] Berndtsson | A proof of the Ohsawa-Takegoshi theorem with sharp estimates[END_REF]).

In this survey, we mostly follow the lines of our previous papers [START_REF] Demailly | Extension of holomorphic functions defined on non reduced analytic subvarieties[END_REF] and [START_REF] Cao | A general extension theorem for cohomology classes on non reduced analytic subspaces[END_REF], whose goal is to pick the weakest possible curvature and convexity hypotheses, while allowing the subvariety Y to be non reduced. The ambient complex manifold X is assumed to be a Kähler and holomorphically convex (and thus not necessarily compact); by the Remmert reduction theorem, the holomorphic convexity is equivalent to the existence of a proper holomorphic map π : X → S onto a Stein complex space S, hence arbitrary relative situations over Stein bases are allowed. We consider a holomorphic line bundle E → X equipped with a singular hermitian metric h, namely a metric which can be expressed locally as h = e -ϕ where ϕ is a quasi-psh function, i.e. a function that is locally the sum ϕ = ϕ 0 + u of a plurisubharmonic function ϕ 0 and of a smooth function u. Such a bundle admits a curvature current (1.1) Θ E,h := i∂∂ϕ = i∂∂ϕ 0 + i∂∂u which is locally the sum of a positive (1, 1)-current i∂∂ϕ 0 and a smooth (1, 1)-form i∂∂u. Our goal is to extend sections that are defined on a non necessarily reduced complex subspace Y ⊂ X, when the structure sheaf O Y := O X /I(e -ψ ) is given by the multiplier ideal sheaf of a quasi-psh function ψ with neat analytic singularities, i.e. locally on a neighborhood V of an arbitrary point x 0 ∈ X we have

(1.2) ψ(z) = c log |g j (z)| 2 + v(z), g j ∈ O X (V ), v ∈ C ∞ (V ).
Let us recall that the multiplier ideal sheaf I(e -ϕ ) of a quasi-psh function ϕ is defined by

(1.3) I(e -ϕ ) x 0 = f ∈ O X,x 0 ; ∃U ∋ x 0 , U |f | 2 e -ϕ dλ < +∞
with respect to the Lebesgue measure λ in some local coordinates near x 0 . As is well known, I(e -ϕ ) ⊂ O X is a coherent ideal sheaf (see e.g. [Dem-book]). We also denote by K X = Λ n T * X the canonical bundle of an n-dimensional complex manifold X ; in the case of (semi)positive curvature, the Bochner-Kodaira identity yields positive curvature terms only for (n, q)-forms, so the best way to state results is to consider the adjoint bundle K X ⊗ E rather than the bundle E itself. The main qualitative statement is given by the following result of [START_REF] Cao | A general extension theorem for cohomology classes on non reduced analytic subspaces[END_REF].

Theorem 1.1. Let E be a holomorphic line bundle over a holomorphically convex Kähler manifold X. Let h be a possibly singular hermitian metric on E, ψ a quasi-psh function with neat analytic singularities on X. Assume that there exists a positive continuous function δ > 0 on X such that

(1.4) Θ E,h + (1 + αδ)i∂∂ψ ≥ 0 in the sense of currents, for α = 0, 1.

Then the morphism induced by the natural inclusion I(he -ψ ) → I(h)

(1.5) H q (X, K X ⊗ E ⊗ I(he -ψ )) → H q (X, K X ⊗ E ⊗ I(h))
is injective for every q ≥ 0. In other words, the morphism induced by the natural sheaf surjection

I(h) → I(h)/I(he -ψ ) (1.6) H q (X, K X ⊗ E ⊗ I(h)) → H q (X, K X ⊗ E ⊗ I(h)/I(he -ψ ))
is surjective for every q ≥ 0.

Remark 1.2. (A) When h is smooth, we have I(h) = O X and

I(h)/I(he -ψ ) = O X /I(e -ψ ) := O Y
where Y is the zero subvariety of the ideal sheaf I(e -ψ ). Hence, the surjectivity statement can be interpreted an extension theorem with respect to the restriction morphism

(1.7) H q (X, K X ⊗ E) → H q (Y, (K X ⊗ E) |Y ).
In general, the quotient sheaf I(h)/I(he -ψ ) is supported in an analytic subvariety Y ⊂ X, which is the zero set of the conductor ideal

(1.8) J Y := I(he -ψ ) : I(h) = f ∈ O X ; f • I(h) ⊂ I(he -ψ ) ,
and (1.6) can thus also be considered as a restriction morphism.

(B) A surjectivity statement similar to (1.7) holds true when (E, h) is a holomorphic vector bundle equipped with a smooth hermitian metric h. In that case, the required curvature condition (1.4) is a semipositivity assumption

(1.9) Θ E,h + (1 + αδ)i∂∂ψ ⊗ Id E ≥ 0 in the sense of Nakano, for α = 0, 1.
(This means that the corresponding hermitian form on T X ⊗ E takes nonnegative values on all tensors of T X ⊗ E, even those that are non decomposable.)

(C) The strength of our statements lies in the fact that no strict positivity assumption is made. This is a typical situation in algebraic geometry, e.g. in the study of the minimal model program (MMP) for varieties which are not of general type. Our joint work [START_REF] Demailly | Extension theorems, Non-vanishing and the existence of good minimal models[END_REF] contains some algebraic applications which we intend to reinvestigate in future work, by means of the present stronger qualitative statements.

(D) Notice that if one replaces (1.4) by a strict positivity hypothesis

(1.10) Θ E,h + i∂∂ψ ≥ εω in the sense of currents, for some ε > 0, then Nadel's vanishing theorem implies H q (X, O X (K X ⊗ E) ⊗ I(he -ψ )) = 0 for q ≥ 1, and the injectivity and surjectivity statements are just trivial consequences.

(E) By applying convex combinations, one sees that condition (1.4) takes an equivalent form if we assume the inequality to hold for α varying in the whole interval [0, 1].

We now turn ourselves to the problem of establishing L 2 estimates for the extension problem, along the lines of [START_REF] Ohsawa | On the extension of L 2 holomorphic functions[END_REF]. The reader will find all details in [START_REF] Demailly | Extension of holomorphic functions defined on non reduced analytic subvarieties[END_REF].

Definition 1.3. If ψ is a quasi-psh function on a complex manifold X, we say that the singularities of ψ are log canonical along the zero variety

Y = V (I(e -ψ )) if I(e -(1-ε)ψ ) |Y = O X↾Y for every ε > 0.
In case ψ has log canonical singularities, it is easy to see by the Hölder inequality and the result of Guan-Zhou [START_REF] Qi'an | A proof of Demailly's strong openness conjecture[END_REF] on the "strong openness conjecture" that I(ψ) is a reduced ideal, i.e. that Y = V (I(ψ)) is a reduced analytic subvariety of X. If ω is a Kähler metric on X, we let dV X,ω = 1 n! ω n be the corresponding Kähler volume element, n = dim X. In case ψ has log canonical singularities along Y = V (I(ψ)), one can also associate in a natural way a measure

dV Y • ,ω [ψ] on the set Y • = Y reg of regular points of Y as follows. If g ∈ C c (Y •
) is a compactly supported continuous function on Y • and g a compactly supported extension of g to X, we set (1.11)

Y • g dV Y • ,ω [ψ] = lim sup t→-∞ {x∈X , t<ψ(x)<t+1} ge -ψ dV X,ω .
By the Hironaka desingularization theorem, one can show that the limit does not depend on the continuous extension g, and that one gets in this way a measure with smooth positive density with respect to the Lebesgue measure, at least on an (analytic) Zariski open set in Y • . In case Y is a codimension r subvariety of X defined by an equation σ(x) = 0 associated with a section σ ∈ H 0 (X, S) of some hermitian vector bundle (S, h S ) on X, and assuming that σ is generically transverse to zero along Y , it is natural to take (1.12)

ψ(z) = r log |σ(z)| 2 h S . One can then easily check that dV Y • ,ω [ψ] is the measure supported on Y • = Y reg such that (1.13) dV Y • ,ω [ψ] = 2 r+1 π r (r -1)! 1 |Λ r (dσ)| 2 ω,h S dV Y,ω where dV Y,ω = 1 (n -r)! ω n-r |Y • .
For a quasi-psh function with log canonical singularities, dV Y • ,ω [ψ] should thus be seen as some sort of (inverse of) Jacobian determinant associated with the logarithmic singularities of ψ. In general, the measure dV Y • ,ω [ψ] blows up (i.e. has infinite volume) in a neighborhood of singular points of Y . Finally, the following positive real function will make an appearance in several of our estimates :

(1.14) γ(x) = exp(-x/2) if x ≥ 0, γ(x) = 1 1 + x 2 if x ≤ 0.
The first generalized L 2 estimate we are interested in is a variation of Theorem 4 in [START_REF] Ohsawa | On the extension of L 2 holomorphic functions, V : Effects of generalization[END_REF]. One difference is that we do not require any specific behavior of the quasi-psh function ψ defining the subvariety: any quasi-psh function with log canonical singularities will do; secondly, we do not want to make any assumption that there exist negligible sets in the ambient manifold whose complements are Stein, because such an hypothesis need not be true on a general compact Kähler manifold -one of the targets of our study.

Theorem 1.4 (L 2 estimate for the extension from reduced subvarieties). Let X be a holomorphically convex Kähler manifold, and ω a Kähler metric on X. Let (E, h) be a holomorphic vector bundle equipped with a smooth hermitian metric h on X, and let ψ : X → [-∞, +∞[ be a quasi-psh function on X with neat analytic singularities. Let Y be the analytic subvariety of X defined by Y = V (I(e -ψ )) and assume that ψ has log canonical singularities along Y , so that Y is reduced. Finally, assume that the Chern curvature tensor Θ E,h is such that the sum

Θ E,h + (1 + αδ) i∂∂ψ ⊗ Id E
is Nakano semipositive for some δ > 0 and α = 0, 1. Then for every holomorphic section

f ∈ H 0 (Y • , (K X ⊗ E) |Y • ) on Y • = Y reg such that Y • |f | 2 ω,h dV Y • ,ω [ψ] < +∞, there exists an extension F ∈ H 0 (X, K X ⊗ E) whose restriction to Y • is equal to f , such that X γ(δψ) |F | 2 ω,h e -ψ dV X,ω ≤ 34 δ Y • |f | 2 ω,h dV Y • ,ω [ψ].
Remark 1.5. Although |F | 2 ω,h and dV X,ω both depend on ω, it is easy to see that the product |F | 2 ω,h dV X,ω actually does not depend on ω when F is a (n, 0)-form. The same observation applies to the product

|f | 2 ω,h dV Y • ,ω [ψ],
hence the final L 2 estimate is in fact independent of ω. Nevertheless, the existence of a Kähler metric (and even of a complete Kähler metric) is crucial in the proof, thanks to the techniques developped in [START_REF] Andreotti | Carleman estimates for the Laplace-Beltrami equation in complex manifolds[END_REF] and [START_REF] Demailly | Estimations L 2 pour l'opérateur ∂ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète[END_REF]. The constant 34 is of course non optimal; the technique developed in [START_REF] Guan | A solution of an L 2 extension problem with an optimal estimate and applications[END_REF] provides optimal choices of the function γ and of the constant in the right hand side.

We now turn ourselves to the case where non reduced multiplier ideal sheaves and non reduced subvarieties are considered. This situation has already been considered by D. Popovici [START_REF] Popovici | L 2 extension for jets of holomorphic sections of a Hermitian line bundle[END_REF] in the case of powers of a reduced ideal, but we aim here at a much wider generality, which also yields more natural assumptions. For m ∈ R + , we consider the multiplier ideal sheaf I(e -mψ ) and the associated non necessarily reduced subvariety Y (m) = V (I(e -mψ )), together with the structure sheaf O Y (m) = O X /I(e -mψ ), the real number m being viewed as some sort of multiplicity -the support |Y (m) | may increase with m, but certainly stabilizes to the set of poles P = ψ -1 (-∞) for m large enough. We assume the existence of a discrete sequence of positive numbers 0 = m 0 < m 1 < m 2 < . . . < m p < . . .

such that I(e -mψ ) = I(e -mpψ ) for m ∈ [m p , m p+1 [ (with of course I(e -m 0 ψ ) = O X )
; they are called the jumping numbers of ψ. The existence of a discrete sequence of jumping numbers is automatic if X is compact. In general, this still holds on every relatively compact open subset

X c := {x ∈ X , ρ(x) < c} ⋐ X,
but requires some some of uniform behaviour of singularities at infinity in the non compact case. We are interested in extending a holomorphic section

f ∈ H 0 (Y (mp) , O Y (mp ) (K X ⊗ E |Y (mp ) ) := H 0 (Y (mp) , O X (K X ⊗ C E) ⊗ O X O X /I(e -mpψ )).
[Later on, we usually omit to specify the rings over which tensor products are taken, as they are implicit from the nature of objects under consideration]. The results are easier to state in case one takes a nilpotent section of the form

f ∈ H 0 (Y (mp) , O X (K X ⊗ E) ⊗ I(e -m p-1 ψ )/I(e -mpψ )).
Then I(e -m p-1 ψ )/I(e -mpψ )) is actually a coherent sheaf, and one can see that its support is a reduced subvariety Z p of Y (mp) . Therefore I(e -m p-1 ψ )/I(e -mpψ )) can be seen as a vector bundle over a Zariski open set Z • p ⊂ Z p . We can mimic formula (1.11) and define some sort of infinitesimal "m p -jet"

L 2 norm |J mp f | 2 ω,h dV Z • p ,ω [ψ]
(a purely formal notation), as the measure on Z • p defined by (1.15)

Z • p g |J mp f | 2 ω,h dV Z • p ,ω [ψ] = lim sup t→-∞ {x∈X , t<ψ(x)<t+1} g | f | 2 ω,h e -mpψ dV X,ω
for any g ∈ C c (Z • p ), where g ∈ C c (X) is a continuous extension of g and f a smooth extension of f on X such that f -f ∈ I(m p ψ) ⊗ O X C ∞ (this measure again has a smooth positive density on a Zariski open set in Z • p , and does not depend on the choices of f and g). We extend the measure as being 0 on Y (mp) red Z p , since I(e -m p-1 ψ )/I(e -mpψ )) has support in Z • p ⊂ Z p . In this context, we introduce the following natural definition.

Definition 1.6. We define the restricted multiplied ideal sheaf

I ′ (e -m p-1 ψ ) ⊂ I(e -m p-1 ψ )
to be the set of germs F ∈ I(e -m p-1 ψ ) x ⊂ O X,x such that there exists a neighborhood U of x satisfying

Y (mp ) ∩U |J mp F | 2 ω,h dV Y (mp ) ,ω [ψ] < +∞.
This is a coherent ideal sheaf that contains I(e -mpψ ). Both of the inclusions

I(e -mpψ ) ⊂ I ′ (e -m p-1 ψ ) ⊂ I(e -m p-1 ψ )
can be strict (even for p = 1).

One of the geometric consequences is the following "quantitative" surjectivity statement, which is the analogue of Theorem 1.4 for the case when the first non trivial jumping number m 1 = 1 is replaced by a higher jumping number m p .

Theorem 1.7. With the above notation and in the general setting of Theorem 1.4 (but without the hypothesis that the quasi-psh function ψ has log canonical singularities), let 0 = m 0 < m 1 < m 2 < . . . < m p < . . . be the jumping numbers of ψ. Assume that

Θ E,h + i(m p + αδ)∂∂ψ ⊗ Id E ≥ 0
is Nakano semipositive for α = 0, 1 and some δ > 0.

(a

) Let f ∈ H 0 (Y (mp) , O X (K X ⊗ E) ⊗ I ′ (e -m p-1 ψ )/I(e -mpψ
)) be a section such that

Y (mp ) |J mp f | 2 ω,h dV Y (mp) ,ω [ψ] < +∞.
Then there exists a global section

F ∈ H 0 (X, O X (K X ⊗ E) ⊗ I ′ (e -m p-1 ψ ))
which maps to f under the morphism I ′ (e -m p-1 ψ ) → I(e -m p-1 ψ )/I(e -mpψ ), such that

X γ(δψ) |F | 2 ω,h e -mpψ dV X,ω [ψ] ≤ 34 δ Y (mp ) |J mp f | 2 ω,h dV Y (mp) ,ω [ψ].
(b) The restriction morphism

H 0 (X, O X (K X ⊗ E) ⊗ I ′ (e -m p-1 ψ )) → H 0 (Y (mp) , O X (K X ⊗ E) ⊗ I ′ (e -m p-1 ψ )/I(e -mpψ )) is surjective.
If E is a line bundle and h a singular hermitian metric on E, a similar result can be obtained by approximating h. However, the L 2 estimates then require to incorporate h into the definition of the multiplier ideals, as in Theorem 1.1 (see [START_REF] Demailly | On the cohomology of pseudoeffective line bundles, Complex Geometry and Dynamics[END_REF]). Hosono [START_REF] Hosono | The optimal jet L 2 extension of Ohsawa-Takegoshi type[END_REF] has shown that one can obtain again an optimal L 2 estimate in the situation of Theorem 1.7, when I(e -mpψ ) is a power of the reduced ideal of Y .

Question 1.8. It would be interesting to know whether Theorem 1.1 can be strengthened by suitable L 2 estimates, without making undue additional hypotheses on the section f to extend. The main difficulty is already to define the norm of jets when there is more than one jump number involved. Some sort of "Cauchy inequality" for jets would be needed in order to derive the successive jet norms from a known global L 2 estimate for a holomorphic section defined on the whole of X. We do not know how to proceed further at this point.

Bochner-Kodaira estimate with approximation

The crucial idea of the proof is to prove the results (say, in the form of the surjectivity statement), only up to approximation. This is done by solving a ∂-equation

∂u ε + w ε = v
where the right hand side v is given and w ε is an error term such that w ε = O(ε a ) as ε → 0, for some constant a > 0. A twisted Bochner-Kodaira-Nakano identity introduced by Donnelly and Fefferman [START_REF] Donnelly | L 2 -cohomology and index theorem for the Bergman metric[END_REF], and Ohsawa and Takegoshi [START_REF] Ohsawa | On the extension of L 2 holomorphic functions[END_REF] is used for that purpose. The technology goes back to the fundamental work of Bochner ([Boc48]), Kodaira ([Kod53a, Kod53b, Kod54]), Akizuki-Nakano ([AN54], [START_REF] Nakano | On complex analytic vector bundles[END_REF]), Kohn ( [FK]), Andreotti-Vesentini ([AV65]), Hörmander [START_REF] Hörmander | L 2 estimates and existence theorems for the ∂ operator[END_REF][START_REF] Hörmander | An introduction to Complex Analysis in several variables, 1st edition[END_REF]). The version we need uses in an essential way an additional correction term, so as to allow a weak positivity hypothesis. It can be stated as follows.

Proposition 2.1. (see [START_REF] Demailly | Extension of holomorphic functions defined on non reduced analytic subvarieties[END_REF]Prop. 3.12]) Let X be a complete Kähler manifold equipped with a (non necessarily complete) Kähler metric ω, and let (E, h) be a Hermitian vector bundle over X. Assume that there are smooth and bounded functions η, λ > 0 on X such that the curvature operator

B = B n,q E,h,ω,η,λ = [η Θ E,h -i ∂∂η -iλ -1 d∂η ∧ ∂η, Λ ω ] ∈ C ∞ (X, Herm(Λ n,q T * X ⊗ E))
satisfies B + εI > 0 for some ε > 0 (so that B can be just semi-positive or even slightly negative; here I is the identity endomorphism). Given a section v ∈ L 2 (X, Λ n,q T * X ⊗ E) such that ∂v = 0 and

M (ε) := X (B + εI) -1 v, v dV X,ω < +∞,
there exists an approximate solution f ε ∈ L 2 (X, Λ n,q-1 T * X ⊗ E) and a correction term

w ε ∈ L 2 (X, Λ n,q T * X ⊗ E) such that ∂u ε = v -w ε and X (η + λ) -1 |u ε | 2 dV X,ω + 1 ε X |w ε | 2 dV X,ω ≤ M (ε).
Moreover, if v is smooth, then u ε and w ε can be taken smooth.

In our situation, the main part of the solution, namely u ε , may very well explode as ε → 0. In order to show that the equation ∂u = v can be solved, it is therefore needed to check that the space of coboundaries is closed in the space of cocycles in the Fréchet topology under consideration (here, the L 2 loc topology), in other words, that the related cohomology group H q (X, F) is Hausdorff. In this respect, the fact of considering ∂-cohomology of smooth forms equipped with the C ∞ topology on the one hand, or cohomology of forms u ∈ L 2 loc with ∂u ∈ L 2 loc on the other hand, yields the same topology on the resulting cohomology group H q (X, F). This comes from the fact that both complexes yield fine resolutions of the same coherent sheaf F, and the topology of H q (X, F) can also be obtained by using Čech cochains with respect to a Stein covering U of X. The required Hausdorff property then comes from the following well known fact.

Lemma 2.2. Let X be a holomorphically convex complex space and F a coherent analytic sheaf over X. Then all cohomology groups H q (X, F) are Hausdorff with respect to their natural topology (induced by the Fréchet topology of local uniform convergence of holomorphic cochains).

In fact, the Remmert reduction theorem implies that X admits a proper holomorphic map π : X → S onto a Stein space S, and Grauert's direct image theorem shows that all direct images R q π * F are coherent sheaves on S. Now, as S is Stein, Leray's theorem combined with Cartan's theorem B tells us that we have an isomorphism H q (X, F) ≃ H 0 (S, R q π * F). More generally, if U ⊂ S is a Stein open subset, we have (2.1)

H q (π -1 (U ), F) ≃ H 0 (U, R q π * F)
and when U ⋐ S is relatively compact, it is easily seen that this a topological isomorphism of Fréchet spaces since both sides are O S (U ) modules of finite type and can be seen as a Fréchet quotient of some direct sum O S (U ) ⊕N by looking at local generators and local relations of R q π * F. Therefore H q (X, F) ≃ H 0 (S, R q π * F) is a topological isomorphism and the space of sections in the right hand side is a Fréchet space. In particular, H q (X, F) is Hausdorff.

Sketch of proof of the extension theorem

The reader may consult [START_REF] Demailly | Extension of holomorphic functions defined on non reduced analytic subvarieties[END_REF] and [START_REF] Cao | A general extension theorem for cohomology classes on non reduced analytic subspaces[END_REF] for more details. After possibly shrinking X into a relatively compact holomorphically convex open subset X ′ = π -1 (S ′ ) ⋐ X, we can suppose that δ > 0 is a constant and that ψ ≤ 0 (otherwise subtract a large constant to ψ). As π : X → S is proper, we can also assume that X admits a finite Stein covering U = (U i ). Any cohomology class in

H q (Y, O X (K X ⊗ E) ⊗ I(h)/I(he -ψ ))
is represented by a holomorphic Čech q-cocycle with respect to the covering U

(c i 0 ...iq ), c i 0 ...iq ∈ H 0 U i 0 ∩ . . . ∩ U iq , O X (K X ⊗ E) ⊗ I(h)/I(he -ψ ) .
By the standard sheaf theoretic isomorphisms with Dolbeault cohomology (cf. e.g. [Dem-e-book]), this class is represented by a smooth (n, q)-form f = i 0 ,...,iq c i 0 ...iq ρ i 0 ∂ρ i 1 ∧ . . . ∂ρ iq by means of a partition of unity (ρ i ) subordinate to (U i ). This form is to be interpreted as a form on the (non reduced) analytic subvariety Y associated with the ideal sheaf J = I(he -ψ ) : I(h) and the structure sheaf O Y = O X /J . We get an extension as a smooth (no longer ∂-closed) (n, q)-form on X by taking

f = i 0 ,...,iq c i 0 ...iq ρ i 0 ∂ρ i 1 ∧ . . . ∂ρ iq where c i 0 ...iq is an extension of c i 0 ...iq from U i 0 ∩ . . . ∩ U iq ∩ Y to U i 0 ∩ . . . ∩ U iq .
Without loss of generality, we can assume that ψ admits a discrete sequence of "jumping numbers"

(3.1) 0 = m 0 < m 1 < • • • < m p < • • • such that I(mψ) = I(m p ψ) for m ∈ [m p , m p+1 [.
Since ψ is assumed to have analytic singularities, this follows from using a log resolution of singularities, thanks to the Hironaka desingularization theorem (by the much deeper result of [START_REF] Qi'an | A proof of Demailly's strong openness conjecture[END_REF] on the strong openness conjecture, one could even possibly eliminate the assumption that ψ has analytic singularities). We fix here p such that m p ≤ 1 < m p+1 , and in the notation of [START_REF] Demailly | Extension of holomorphic functions defined on non reduced analytic subvarieties[END_REF], we let Y = Y (mp) be defined by the non necessarily reduced structure sheaf

O Y = O X /I(e -ψ ) = O X /I(e -mpψ ).
We now explain the choice of metrics and auxiliary functions η, λ for the application of Proposition 2.1, following the arguments of [Dem15b, proof of th. 2.14, p. 217]. Let t ∈ R - and let χ t be the negative convex increasing function defined in [Dem15b, (5.8 * ), p. 211]. Put η t := 1 -δ • χ t (ψ) and λ t := 2δ

(χ 2 t (ψ)) 2 χ ′′ t (ψ) . We set R t := η t (Θ E,h + i∂∂ψ) -i∂∂η t -λ -1 t i∂η t ∧ ∂η t = η t (Θ E,h + (1 + δη -1 t χ ′ t (ψ))i∂∂ψ) + δ • χ ′′ t (ψ) 2 i∂ψ ∧ ∂ψ. Note that χ ′′ t (ψ) ≥ 1 8 on W t = {t < ψ < t + 1}. The curvature assumption (1.4) implies Θ E,h + (1 + δη -1 t χ ′ t (ψ)) i∂∂ψ ≥ 0 on X.
As in [START_REF] Demailly | Extension of holomorphic functions defined on non reduced analytic subvarieties[END_REF], we find

(3.2) R t ≥ 0 on X and (3.3) R t ≥ δ 16 i∂ψ ∧ ∂ψ on W t = {t < ψ < t + 1}.
Let θ : [-∞, +∞[ → [0, 1] be a smooth non increasing real function satisfying θ(x) = 1 for x ≤ 0, θ(x) = 0 for x ≥ 1 and |θ ′ | ≤ 2. By using a blowing up process, one can reduce the situation to the case where ψ has divisorial singularities. Then we still have

Θ E,h + (1 + δη -1 t χ ′ t (ψ))(i∂∂ψ) ac ≥ 0 on X,
where (i∂∂ψ) ac is the absolutely continuous part of i∂∂ψ. The regularization techniques of [START_REF] Demailly | Pseudo-effective line bundles on compact Kähler manifolds[END_REF] and [Dem15a, Th. 1.7, Remark 1.11] produce a family of singular metrics {h t,ε } +∞ k=1 which are smooth in the complement X Z t,ε of an analytic set, such that I(h t,ε ) = I(h), I(h t,ε e -ψ ) = I(he -ψ ) and

Θ E,ht,ε + (1 + δη -1 t χ ′ t (ψ)) i∂∂ψ ≥ - 1 2 εω on X.
The additional error term -1 2 εω is irrelevant when we use Proposition 2.1, as it is absorbed by taking the hermitian operator B + εI. Therefore for every t ∈ R -, with the adjustment ε = e αt , α ∈ ]0, m p+1 -1[, we can find a singular metric h t = h t,ε which is smooth in the complement X \ Z t of an analytic set, such that I(h t ) = I(h), I(h t e -ψ ) = I(he -ψ ) and h t ↑ h as t → -∞. We now apply the L 2 estimate of Proposition 2.1 and observe that X Z t is complete Kähler (at least after we shrink X a little bit as X ′ = π -1 (S ′ ), cf. [START_REF] Demailly | Estimations L 2 pour l'opérateur ∂ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète[END_REF]). As a consequence, one can find sections u t , w t satisfying (3.4)

∂u t + w t = v t := ∂ θ(ψ -t) • f and (3.5) X (η t + λ t ) -1 |u t | 2 ω,ht e -ψ dV X,ω + 1 ε X |w t | 2 ω,ht e -ψ dV X,ω ≤ X (R t + εI) -1 v t , v t ω,ht e -ψ dV X,ω .
One of the main consequence of (3.3) and (3.5) is that, for ε = e αt and α well chosen, one can infer that the error term satisfies

lim t→-∞ X |w t | 2 ω,ht e -ψ dV X,ω = 0.
One difficulty, however, is that L 2 sections cannot be restricted in a continuous way to a subvariety. In order to overcome this problem, we play again the game of returning to Čech cohomology by solving inductively ∂-equations for w t on U i 0 ∩ . . . ∩ U i k , until we reach an equality

(3.6) ∂ θ(ψ -t) • f -u t = w t := - i 0 ,...,i q-1 s t,i 0 ...iq ∂ρ i 0 ∧ ∂ρ i 1 ∧ . . . ∂ρ iq with holomorphic sections s t,I = s t,i 0 ...iq on U I = U i 0 ∩ . . . ∩ U iq , such that lim t→-∞ U I |s t,I | 2 ω,ht e -ψ dV X,ω = 0.
Then the right hand side of (3.6) is smooth, and more precisely has coefficients in the sheaf C ∞ ⊗ O I(he -ψ ), and w t → 0 in C ∞ topology. A priori, u t is an L 2 (n, q)-form equal to u t plus a combination ρ i s t,i of the local solutions of ∂s t,i = w t , plus ρ i s t,i,j ∧ ∂ρ j where ∂s t,i,j = s t,j -s t,i , plus etc . . . , and is such that

X | u t | 2
ω,ht e -ψ dV X,ω < +∞.

Since H q (X, O X (K X ⊗ E) ⊗ I(he -ψ )) can be computed with the L 2 loc resolution of the coherent sheaf, or alternatively with the ∂-complex of (n, •)-forms with coefficients in C ∞ ⊗ O I(he -ψ ), we may assume that u t ∈ C ∞ ⊗ O I(he -ψ ), after playing again with Čech cohomology. Lemma 2.2 yields a sequence of smooth (n, q)-forms σ t with coefficients in C ∞ ⊗ O I(h), such that ∂σ t = w t and σ t → 0 in C ∞ -topology. Then

f t = θ(ψ -t) • f -u t -σ t is a ∂-closed (n, q)-form on X with values in C ∞ ⊗ O I(h) ⊗ O X (E), whose image in H q (X, O X (K X ⊗ E) ⊗ I(h)/I(he -ψ )) converges to {f } in C ∞ Fréchet topology.
We conclude by a density argument on the Stein space S, by looking at the coherent sheaf morphism

R q π * O X (K X ⊗ E) ⊗ I(h) → R q π * O X (K X ⊗ E) ⊗ I(h)/I(he -ψ ) .
Proof of the quantitative estimates. We refer again to [START_REF] Demailly | Extension of holomorphic functions defined on non reduced analytic subvarieties[END_REF] for details. One of the main features of the above qualitative proof is that we have not tried to control the solution u t of our ∂-equation, in fact we only needed to prove that the error term w t converges to zero. However, to get quantitative L 2 estimates, we have to pay attention to the L 2 norm of u t . It is under control as t → -∞ only when f satisfies the more restrictive condition of being L 2 with respect to the residue measure dV Y • ,ω [ψ]. This is the reason why we lose track of the solution when the volume of the measure explodes on Y sing , or when there are several jumps involved in the multiplier ideal sheaves.

Applications of the Ohsawa-Takegoshi extension theorem

The Ohsawa-Takegoshi extension theorem is a very powerful tool that has many important applications to complex analysis and geometry. We will content ourselves by mentioning only a few statements and references. 4.1. Approximation of plurisubharmonic functions and of closed (1, 1)-currents. By considering the extension from points (i.e. a 0-dimensional connected subvariety Y ⊂ X), even just locally on coordinates balls, one gets a precise Bergman kernel estimate for Hilbert spaces attached to multiples of any plurisubharmonic function. This leads to regularization theorems ( [START_REF] Demailly | Regularization of closed positive currents and Intersection Theory[END_REF]) that have many applications, such as the Hard Lefschetz theorem with multiplier ideal sheaves ([DPS01]), or extended vanishing theorems for pseudoeffective line bundles [START_REF] Cao | Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact Kähler manifolds[END_REF]). The result may consult [START_REF] Demailly | On the cohomology of pseudoeffective line bundles, Complex Geometry and Dynamics[END_REF] for a survey of these questions. Another consequence is a very simple and direct proof of Siu's result [START_REF] Siu | Analyticity of sets associated to Lelong numbers and the extension of closed positive currents[END_REF] on the analyticity of sublevel sets of Lelong numbers of closed positive currents. 4.2. Invariance of plurigenera. Around 2000, Siu [START_REF] Siu | Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type[END_REF] proved that for every smooth projective deformation π : X → S over an irreducible base S, the plurigenera p m (t) = h 0 (X t , K ⊗m Xt ) of the fibers X t = π -1 (t) are constant. The proof relies in an essential way on the Ohsawa-Takegoshi extension theorem, and was later simplified and generalized by Pȃun [START_REF] Pȃun | Siu's invariance of plurigenera: a one-tower proof[END_REF]. It is remarkable that no algebraic proof of this purely algebraic result is known! 4.3. Semicontinuity of log singularity exponents. In [START_REF] Demailly | Semicontinuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds[END_REF], we proved that the log singularity exponent (or log canonical threshold) c x (ϕ), defined as the supremum of constants c > 0 such that e -cϕ is integrable in a neighborhood of a point x, is a lower semicontinuous function with respect to the topology of weak convergence on plurisubharmonic functions. Guan and Zhou [START_REF] Qi'an | A proof of Demailly's strong openness conjecture[END_REF] recently proved our "strong openness conjecture", namely that the integrability of e -ϕ implies the integrability of e -(1+ε)ϕ for ε > 0 small; later alternative proofs have been exposed in [START_REF] Pham | The weighted log canonical threshold[END_REF] and [START_REF] Lempert | Modules of square integrable holomorphic germs[END_REF]. 4.4. Proof of the Suita conjecture. In [START_REF]Suita conjecture and the Ohsawa-Takegoshi extension theorem[END_REF] B locki determined the value of the optimal constant in the Ohsawa-Takegoshi extension theorem, a result that was subsequently generalized by Guan and Zhou [START_REF] Guan | A solution of an L 2 extension problem with an optimal estimate and applications[END_REF]. In complex dimension 1, this result implies in its turn a conjecture of N. Suita, stating that for any bounded domain D in C, one has c 2 D ≤ πK D , where c D (z) is the logarithmic capacity of C D with respect to z ∈ D and K D is the Bergman kernel on the diagonal. Guan and Zhou [START_REF] Guan | A solution of an L 2 extension problem with an optimal estimate and applications[END_REF] proved that the equality occurs if ond only if D is conformally equivalent to the disc minus a closed set of inner capacity zero.

Supported by the European Research Council project "Algebraic and Kähler Geometry" (ERC-ALKAGE, grant No. 670846 from September 2015).