C Briat
email: cbriat@math.kth.se

O Sename

J F Lafay

Memory Resilient Gain-scheduled State-Feedback Control of Uncertain LTI/LPV Systems with Time-Varying Delays

Keywords: Time delay systems, Controller Delay-resilience, Linear parameter varying systems, Robust LMIs, Relaxation

The stabilization of uncertain LTI/LPV time delay systems with time varying delays by statefeedback controllers is addressed. At the difference of other works in the literature, the proposed approach allows for the synthesis of resilient controllers with respect to uncertainties on the implemented delay. It is emphasized that such controllers unify memoryless and exact-memory controllers usually considered in the literature. The solutions to the stability and stabilization problems are expressed in terms of LMIs which allow to check the stability of the closed-loop system for a given bound on the knowledge error and even optimize the uncertainty radius under some performance constraints; in this paper, the H ∞ performance measure is considered. The interest of the approach is finally illustrated through several examples.

Introduction

Since several years, time-delay systems [START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF][START_REF] Niculescu | Delay effects on stability[END_REF][START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Han | Absolute stability of time-delay systems with sector-bounded nonlinearity[END_REF][START_REF] Gouaisbaut | Delay dependent robust stability of time delay-systems[END_REF][START_REF] Fridman | Stability of systems with uncertain delays: a new 'complete' Lyapunov-Krasovskii Functional[END_REF][START_REF] Suplin | H ∞ control of linear uncertain time-delay systems -a projection approach[END_REF][START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF][START_REF] Ariba | Input-output framework for robust stability of time-varying delay systems[END_REF] have attracted more and more interest since they arise in various problems [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF][START_REF] Niculescu | Delay effects on stability[END_REF] such as chemical processes, biological systems, economic systems, etc. The presence of delays, in the equations describing the process, is often responsible of destabilizing effects and performance deterioration. Indeed, in fast systems, even a small time-delay may have a very harmful effect, and thus cannot be neglected. This has motivated the development of many types of stability tests and matched controller design techniques [START_REF] Zhong | Robust Control of Time-Delay Systems[END_REF][START_REF] Fridman | Descriptor discretized Lyapunov functional method: Analysis and design[END_REF][START_REF] Xu | New results on delay-dependent robust H ∞ control for systems with time-varying delays[END_REF][START_REF] Briat | A LFT/H ∞ state-feedback design for linear parameter varying time delay systems[END_REF][START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays[END_REF]. Nevertheless, while the theories for stability analysis and stabilization of LTI systems with constant delays are well established, the case of time-varying delays is still not well understood [START_REF] Louisell | New examples of quenching in delay differential equations having timevarying delay[END_REF][START_REF] Papachristodoulou | Stability analysis of linear systems with time-varying delays: Delay uncertainty and quenching[END_REF].

On the other hand, over the past recent years, LPV systems [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF][START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF][START_REF] Wu | A generalized LPV system analysis and control synthesis framework[END_REF][START_REF] Scherer | LPV control and full-block multipliers[END_REF][START_REF] Iwasaki | LPV system analysis via quadratic separator for uncertain implicit systems[END_REF] have been heavily studied since they offer a very general approach for the modeling and the control of complex systems such as nonlinear systems. This fresh upsurge of gain-scheduling based techniques is mainly due to the emergence of LMIs [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF], which provide a powerful formalism for the expression of solutions of many problems arising in systems and control theory. It is important to note that many problems in LPV framework remain open and major improvements are still expected for stability analysis as well as for control law synthesis.

The stability analysis of LPV time-delay systems is still an open problem and is discussed for instance in [START_REF] Zhang | Stability analysis of LPV time-delayed systems[END_REF][START_REF] Zhang | Delay-dependent stability analysis and H ∞ control for state-delayed LPV system[END_REF][START_REF] Briat | Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach[END_REF][START_REF] Briat | Control and observation of LPV time-delay systems[END_REF] while the control of LPV time-delay systems have also been studied in [35,[START_REF] Zhang | Stability analysis of LPV time-delayed systems[END_REF][START_REF] Zhang | Delay-dependent stability analysis and H ∞ control for state-delayed LPV system[END_REF][START_REF] Briat | Control and observation of LPV time-delay systems[END_REF][START_REF] Briat | H ∞ delay-scheduled control of of linear systems with time-varying delays[END_REF][START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays -a LPV approach[END_REF] but still remains sporadic. These systems belong to the intersection of two families and hence inherit from the difficulties of each one. Additionally, new difficulties arise, for instance several robust control tools which are used to deal with LPV systems (such as the projection lemma [START_REF] Gahinet | linear matrix inequality approach to H ∞ control[END_REF] or the dualization lemma [START_REF] Scherer | LPV control and full-block multipliers[END_REF]) are difficult to apply to LPV time-delay systems. Indeed, the stability analysis of such systems cannot be tackled using classical Lyapunov functions (as for finite dimensional LPV systems) but must be analyzed using adapted tools, namely Lyapunov-Razumikhin functions and Lyapunov-Krasovskii functionals, which increase the number of decision variables.

The main contribution of the paper concerns the synthesis of non-fragile controllers with respect to an uncertain knowledge of the implemented delay. Several papers mention non-fragility of observers/controllers but the robust stability analysis is done only after the synthesis [START_REF] Verriest | Robust observer-controller for delay-differential systems[END_REF][START_REF] Sename | Observer-based H ∞ control for time-delay systems: a new LMI solution[END_REF]. In such a case, the non-fragility radius (maximal tolerable delay uncertainty) is difficult to guarantee or optimize. In the proposed approach, the non-fragility radius can be fixed by the designer or even maximized. Moreover, since the approach is based on LMIs, time-varying and uncertain systems can be also easily handled at the difference of latter results which were based on frequency domain methods, thus restricted to LTI systems and difficult to generalize to the uncertain case. A close result but notably different is also provided in [START_REF] Ivȃnescu | Dynamical compensation for timedelay systems: an LMI approach[END_REF] where the control of a time-delay system with constant delay is performed using a controller with a different delay. However, no relationship between the delays is considered and thus the maximal admissible error cannot be analyzed from this result. Results on resilience of controllers with delay uncertainties are also provided in [START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays -a LPV approach[END_REF][START_REF] Briat | H ∞ delay-scheduled control of of linear systems with time-varying delays[END_REF] in the framework on delay-scheduled controllers where the delay acts as a gain-scheduling parameter on the controller expression. In the current paper, the initial timedelay system system structure for both the system and the controller is kept, in order to develop a stabilization result in this domain. The resulting problem can be equivalently represented as a stabilization problem of a system with two delays where the delays are coupled through an algebraic inequality. Thus the problem reduces in a correct and efficient accounting of this inequality in the LMI-based stabilization result.

It is worth mentioning that almost all the literature addresses the stabilization problem with memoryless (conservative) or exact-memory (non-implementable) controllers. The approach of the paper is more pragmatic and aims at designing controller with delays different from the system one. Delay estimation techniques, as in [START_REF] Belkoura | A convolution approach for delay systems identification[END_REF], could be used in order to determine the controller delay. Such a controller class find applications in the control of physical systems with delay on states such as the ones in [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF].

The goal of this article is not to provide better results on the stability of time-delay systems by introducing new Lyapunov-Krasovskii functionals but is to show that it is possible to consider uncertainties of the delay knowledge and take it into account in an efficient way in the synthesis. The proposed approach is very general and can be extended to many types of (less conservative) different functionals.

For a real square matrix M we define M S := M + M T where M T is its transpose. The space of signals with finite energy is denoted by L 2 and the energy of

v ∈ L 2 is ||v|| L2 := +∞ 0 v * (t)v(t)dt 1/2 . S k
++ denotes the cone of real symmetric positive definite matrices of dimension k. R + (R ++ ) denotes the set of nonnegative (positive) real numbers. denotes symmetric terms in symmetric matrices and in quadratic forms. ⊗ and × denote the Kronecker and the cartesian product respectively. co{S} stands for the convex hull of the set S. col i (λ i ) is the column vector with components λ i .

The paper is structured as follows. In Section 2, definitions and objectives of the paper are defined. In Section 3 we provide several delay-dependent stability results for uncertain LPV timedelay systems with time-varying delays. Section 4 is devoted to the development of constructive sufficient conditions to the existence of three types of parameter dependent state-feedback controllers. Finally, in Section 5, examples and discussions on the proposed approach are provided.

Definitions and Objectives

The following class of systems will be considered in the paper:

ẋ(t) = A(λ, ρ)x(t) + A h (λ, ρ)x h (t) + B(λ, ρ)u(t) +E(λ, ρ)w(t) z(t) = C(λ, ρ)x(t) + C h (λ, ρ)x h (t) + D(λ, ρ)u(t) +F (λ, ρ)w(t) x(η) = φ(η), η ∈ [-h M , 0] (1) where x ∈ R n , x h (t) = x(t -h(t)) ∈ R n , u ∈ R m , w ∈ R p , z ∈ R q
and φ(•) are respectively the system state, the delayed state, the control input, the exogenous input, the controlled output and the functional initial condition. The system matrices are defined by

A A h B E C C h D F (λ, ρ) = N i=1 λ i A i A hi B i E i C i C hi D i F i (ρ) (2)
where λ = col i (λ i ) is time-invariant and belongs to the unitary simplex Λ defined by Λ := λ i ≥ 0 :

N i=1 λ i = 1, i = 1, . . . , N (3)
The delay is assumed to belong to the set

H := {h : R + → [0, h M ], h ≤ µ < 1} (4)
with h M < +∞. The vector of parameters ρ belongs to

P := ρ : R + → U ρ ⊂ R Np , ρ ∈ co{U ν } (5)
where N p > 0 is the number of parameters, U ρ is a connected compact subset of R Np . U ν is the set of vertices of the convex set in which the derivative of the parameters evolves and is defined by

U ν := × Np i=1 {ν i , νi } (6)
where ν i and νi denote respectively the upper and lower bound of ρi ; hence we have ρ ∈ co{U ν }.

The aim of the current paper is to find a control law based on a parameter dependent statefeedback of the form

u(t) = K 0 (ρ)x(t) + K h (ρ)x(t -d(t)) (7)
where the gains K 0 (ρ) and K h (ρ) are sought such that the controller stabilizes the uncertain LPV system [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF]. Note that the delay d(t) involved in the control law is allowed to be different from the system delay h(t). First, the usual case d(t) = h(t) will be considered and then the more general case d(t) = h(t) + η(t) with |η(t)| ≤ δ will be solved in turn. To this aim, the following set is introduced

D δ := {d : R + → [0, h M ], |d(t) -h(t)| ≤ δ, h ∈ H } (8)
and defines the set of controller delays.

Definition 2.1. The following terminology is used for the controllers (7):

• If K h (•) = 0 the controller is referred to as a memoryless controller;

• If d(t) = h(t) for all t ≥ 0 (i.e. δ = 0 in (8)) then the controller is referred to as an exact memory controller;

• If |d(t)-h(t)| ≤ δ for some δ > 0 then the controller is referred to as a δ-memory resilient controller.

The set D δ is parameterized by the uncertainty radius δ ≥ 0. Note that when δ = 0 the equality d(t) = h(t) holds for all t ≥ 0 and hence the δ-memory resilient controller coincides with exact memory controller. Note also that if δ = h M then the implemented delay may take any value inside [0, h M ] independently of the value of h(t). In such a case, h(t) can be considered as unknown, resulting then in the particular case where the memoryless and the h M -memory resilient controllers are actually quite near, structurally speaking. It will be illustrated in the examples that δ-memory resilient controllers connect together the well-known memoryless and exact-memory controllers by providing a unique and generalized expression for all controllers. With this in mind, it is possible to state the main problem of the paper:

Problem 2.1. Find a parameter dependent δ-memory resilient state-feedback controller (7) which

1. Robustly asymptotically stabilizes system (1): x(t) → 0 as t → +∞ with w(t) = 0, for all parameter trajectories ρ ∈ P, for all delays (h, d) ∈ H × D δ and for all λ ∈ Λ. 2. Provides a guaranteed L 2 performance attenuation gain from w to z satisfying ||z|| L2 < γ||w|| L2 with x(η) = 0, η ∈ [-h M , 0], w(t) = 0 and for all parameter trajectories ρ ∈ P, for all delays (h, d) ∈ H × D δ and for all λ ∈ Λ.

A Control Oriented Delay-Dependent Stability Result

This section is devoted to the stability analysis of LPV time-delay systems of the form (1). The results are based on the extension of [START_REF] Han | Absolute stability of time-delay systems with sector-bounded nonlinearity[END_REF][START_REF] Gouaisbaut | Delay dependent robust stability of time delay-systems[END_REF] to uncertain LPV time-delay systems with timevarying delays. As we shall see later, the immediate LMI conditions derived from the Lyapunov-Krasovskii theorem are not well suited for stability and synthesis problems due to:

• multiple products between system matrices and decision variables; and

• quadratic terms on the polytope variable λ.

The second part of the proof is then devoted to the relaxation of such a result in order to both reduce the number of these products (to one) and make the dependence on the polytope variable λ affine. This makes the derivation of stabilization result easier and overall more efficient than approaches based on relaxations made after substitution of the closed-loop system matrices into the LMI.

Another approach based on the projection lemma [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers[END_REF] and on adjoint systems [START_REF] Bensoussan | Representation and Control of Infinite Dimensional Systems -2 nd Edition[END_REF] was considered in [START_REF] Briat | Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach[END_REF]. This approach led to an equivalent problem independent of the controller matrices involving a nonlinear matrix inequality which was then solved using an iterative LMI algorithm. The current approach avoids such a computational complexity by tolerating an increase of conservatism. So, following this new idea, the following stability analysis result is stated:

Theorem 3.1. System (1) with no control input (ie. u(t) ≡ 0) is robustly asymptotically stable for all (h, ρ, λ) ∈ H × P × Λ if there exist continuously differentiable matrix functions P i : U ρ → S n ++ for all i ∈ {1, . . . , N }, a matrix function X : U ρ → R n×n , N constant matrices Q i , R i ∈ S n
++ and a scalar γ > 0 such that the parameter dependent LMIs

Θ i =           -X(ρ) S Φ 12i Φ 13i Φ 14i 0 X(ρ) T h M R i Φ 22i R i 0 C i (ρ) T 0 0 Φ 33i 0 C hi (ρ) T 0 0 -γI p F i (ρ) T 0 0 -γI q 0 0 -P i (ρ) -h M R i -R i           ≺ 0 (9) hold for all (ρ, ν, i) ∈ U ρ × co{U ν } × {1, . . . , N } with Φ 22i = ∂P i (ρ) ∂ρ ν -P i (ρ) + Q i -R i       [A(λ, ρ) T P (λ, ρ)] S + Q(λ) -R(λ) P (λ, ρ)A(λ, ρ) + R(λ) P (λ, ρ)E(λ, ρ) C(λ, ρ) T h M A(λ, ρ) T R(λ) -(1 -µ)Q(λ) -R(λ) 0 C h (λ, ρ) T h M A h (λ, ρ) T R(λ) -γI p F (λ, ρ) T h M E(λ, ρ) T R(λ) -γI q 0 -R(λ)       ≺ 0 (14) Φ 12i = P i (ρ) + X(ρ) T A i (ρ) Φ 13i = X(ρ) T A hi (ρ) Φ 14i = X(ρ) T E i (ρ) Φ 33i = -(1 -µ)Q i -R i
In such a case, system (1) satisfies ||z|| L2 < γ||w|| L2 for all (ρ, h, λ) ∈ P × H × Λ.

Proof : The choice of the functional V is inspired from [START_REF] Gouaisbaut | Delay dependent robust stability of time delay-systems[END_REF][START_REF] Han | Absolute stability of time-delay systems with sector-bounded nonlinearity[END_REF] and extended to the case of LPV time-delay systems with time-varying delays as in [START_REF] Briat | Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach[END_REF] and the supply rate s(w, z) is considered:

V = x T (t)P (λ, ρ)x(t) + t t-h(t) x T (θ)Q(λ)x(θ)dθ +h M 0 -h M t t+θ x T (η)R(λ)x(η)dηdθ s(w(t), z(t)) = γw T (t)w(t) -γ -1 z(t) T z(t) (10)
with

P (λ, ρ) = N i=1 λ i P i (ρ), Q(λ) = N i=1 λ i Q i , R(λ) = N i=1 λ i R i .
The supply-rate s(w, z) characterizes the L 2 -gain from w to z. Define the function H to be

H = V - t 0 s(w(θ), z(θ))dθ (11)
The derivative of the function can be bounded from above by

Ḣ ≤ x T (t) ∂P (λ, ρ) ∂ρ ρ(t)x(t) + x(t) T A T (λ, ρ)P (λ, ρ) S x(t) +2x T h (t)A h (λ, ρ) T P (λ, ρ)x(t) + 2w T (t)E(λ, ρ) T P (λ, ρ)x(t) +x(t) T Q(λ)x(t) -(1 -ḣ)x h (t) T Q(λ)x h (t) +h 2 M ẋ(t) T R(λ) ẋ(t) + I -γw(t) T w(t) +γ -1 z(t) T z(t) z(t) = C(λ, ρ)x(t) + C h (λ, ρ)x(t -h(t)) + F (λ, ρ)w(t) I = -h M t t-h(t) ẋ(θ) T R ẋ(θ)dθ (12)
Note that -(1 -ḣ) ≤ -(1 -µ) and using Jensen's inequality [START_REF] Gu | Stability of Time-Delay Systems[END_REF] on I we obtain

I ≤ - t t-h(t) ẋ(s)ds T R(λ)( ) T (13)
Then expanding the expression of s(w, z) we get LMI [START_REF] Fridman | Stability of systems with uncertain delays: a new 'complete' Lyapunov-Krasovskii Functional[END_REF] after Schur complements. Due to products between decision matrices and system matrices, it is clear that the LMI is not linear in λ (e.g. A(λ, ρ)P (λ, ρ)). Moreover, the structure of ( 14) is not adapted to the controller design due to the presence of multiple products terms P A, P A h , RA and RA h preventing to find a linearizing change of variable even after congruence transformations. A relaxed version of ( 14) is then expected to remove the multiple products preventing the change of variables and limit the increase of conservatism. A similar approach has been used in [START_REF] Tuan | Robust and reduced order filtering: new LMI-based characterizations and methods[END_REF][START_REF] Tuan | Robust filtering for uncertain nonlinearly parametrized plants[END_REF][START_REF] Briat | Control and observation of LPV time-delay systems[END_REF].

In view of relaxing the latter result, define Θ = N i=1 λ i Θ i where Θ i is given in [START_REF] Briat | Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach[END_REF]. Below, it is proved that Θ ≺ 0 implies the feasibility of [START_REF] Fridman | Stability of systems with uncertain delays: a new 'complete' Lyapunov-Krasovskii Functional[END_REF]. First note that Θ can be written as (where the dependency on λ, ρ and ρ are dropped for clarity):

Θ| X=0 + Z T 1 XZ 2 + Z T 2 X T Z 1 ≺ 0 ( 15
)
where Z 1 = -I A A h E 0 0 and Z 2 = I 0 0 0 0 0 . Then invoking the projection lemma [START_REF] Gahinet | linear matrix inequality approach to H ∞ control[END_REF], the feasibility of Θ ≺ 0 implies the feasibility of the underlying LMI problem

N T 1 Θ| X=0 N 1 ≺ 0 (16a) N T 2 Θ| X=0 N 2 ≺ 0 (16b)
where N 1 and N 2 are basis of the null-space of Z 1 and Z 2 respectively. Note that since X only depends on ρ (and neither on ρ nor λ) then equivalence equivalence between ( 15) and ( 16) is lost and reduces then to an implication from ( 15) to ( 16) only.

After some tedious manipulations, it is possible to show that LMI (16a) is equivalent to ( 14) and thus that Θ ≺ 0 implies [START_REF] Fridman | Stability of systems with uncertain delays: a new 'complete' Lyapunov-Krasovskii Functional[END_REF]. Thus the conservatism of the approach is characterized by LMI (16b) and by the sole dependence of X on ρ.

LMIs (9) do not involve any multiple products and hence can be easily used for design purpose. The removal of multiple products has been allowed through the introduction of a 'slack' variable X(ρ). The additional conservatism and the (slight) increase of the computational complexity are the price to pay to get easily tractable conditions for the stabilization problem.

Delay-Dependent Stabilization by State-Feedback

Robust Stabilization using Memoryless and Exact-Memory State-Feedback Controllers

In this part, stabilizing control laws of the form

u(t) = K 0 (ρ)x(t) + K h (ρ)x(t -h(t)) (17)
are sought. The closed loop system obtained from the interconnection of system (1) and controller [START_REF] Gu | Stability of Time-Delay Systems[END_REF] is given by ẋ

(t) = A cl (λ, ρ)x(t) + A hcl (λ, ρ)x h (t) + E(λ, ρ)w(t) z(t) = C cl (λ, ρ)x(t) + C hcl (λ, ρ)x h (t) + F (λ, ρ)w(t) (18)
with

A cl (λ, ρ) = A(λ, ρ)+B(λ, ρ)K 0 (ρ), C cl (λ, ρ) = C(λ, ρ)+D(λ, ρ)K 0 (ρ), A hcl (λ, ρ) = A h (λ, ρ)+ B(λ, ρ)K h (ρ) and C hcl (λ, ρ) = C h (λ, ρ) + D(λ, ρ)K h (ρ).
The following theorem on robust stabilization is obtained:

Theorem 4.1. There exists a parameter dependent state-feedback control of the form (17) which robustly asymptotically stabilizes system (1) for all (h, ρ, λ) ∈ H ×P ×Λ if there exist continuously differentiable matrix functions

P i : U ρ → S n ++ , constant matrices Q i , R i ∈ S n ++ for i = 1, . . . , N , X ∈ R n×n , matrix functions Y 0 , Y h : U ρ → R m×n and a scalar γ > 0 such that the parameter dependent LMIs           Ξ 11i Ξ 12i Ξ 13i E i (ρ) 0 X h M R i Ξ 22i R 0 Ξ 24i 0 0 Ξ 33i 0 Ξ 34i 0 0 -γI p F i (ρ) T 0 0 -γI q 0 0 -P i (ρ) -h M R i -R i           ≺ 0 ( 19
)
hold for all (ρ, ν, i) ∈ U ρ × co{U ν } × {1, . . . , N } where

Ξ 11 = -X S Ξ 12i = P i (ρ) + A i (ρ)X + B i (ρ)Y 0 (ρ) Ξ 23i = A hi (ρ)X + B i (ρ)Y h (ρ) Ξ 22i = ∂P i (ρ) ∂ρ ν -P i (ρ) + Q i -R i Ξ 33i = -(1 -µ)Q i -R i Ξ 24i = [C i (ρ)X + D i (ρ)Y 0 (ρ)] T Ξ 34i = [C hi (ρ)X + D i (ρ)Y h (ρ)] T
In such a case, a suitable law is given by ( 17) with gains K 0 (ρ) = Y 0 (ρ)X -1 and K h (ρ) = Y h (ρ)X -1 . Moreover, the closed-loop system satisfies ||z|| L2 < γ||w|| L2 for all (h, ρ, λ) ∈ H × P × Λ.

Proof : Substitute the closed-loop system (18) into inequality [START_REF] Briat | Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach[END_REF] and set X to be a constant matrix. X is enforced to be constant in order to allow for the use of congruence transformations, otherwise, nonlinear terms would appear, making the solution to the problem difficult to solve (i.e. non LMI). Then performing a congruence transformation with respect to matrix

diag(I 3 ⊗ X -1 , I p+q , I 2 ⊗ X -1 )
and applying the following linearizing change of variable

X ← X -1 P i ← X -T P i X -1 Q i ← X -T Q i X -1 R i ← X -T R i X -1 Y 0 ← K 0 X -1 Y h ← K h X -1
yields LMIs [START_REF] Ivȃnescu | Dynamical compensation for timedelay systems: an LMI approach[END_REF].

Remark 4.1. The latter result can be used to both synthesize exact-memory and memoryless control laws. Memoryless structures can be obtained setting Y h (•) = 0.

Robust Stabilization using δ-Memory Resilient State-Feedback Controllers

This part establishes a new result on the stabilization of time-delay systems where the strong constraint on exact delay knowledge is relaxed. In the following, we will show that it is also possible to give stabilization conditions even in presence on time-varying uncertainties on the delay knowledge. We will see, as a direct consequence of the method, that Theorem 4.1 can be retrieved from Theorem 4.2, the main result of this section.

In the sequel, the following control law will be considered:

u(t) = K 0 (ρ)x(t) + K h (ρ)x(t -d(t)) (20)
where the approximate value of the delay d(t) ∈ D δ is used. To the authors' knowledge, the only work on such control laws using LMI techniques is [START_REF] Ivȃnescu | Dynamical compensation for timedelay systems: an LMI approach[END_REF]. However, the provided approach only considers constant time-delays and does not consider any relationship between the delays. In the current approach, time-varying delays are allowed and the delay knowledge maximal error is explicitly taken into account in the stabilization conditions. Moreover, with the provided approach, it is easy to guarantee a given bound on the error or optimize it, at the difference of [START_REF] Verriest | Robust observer-controller for delay-differential systems[END_REF][START_REF] Sename | Observer-based H ∞ control for time-delay systems: a new LMI solution[END_REF]. The closed-loop system given by the interconnection of the control law (20) and system (1) is governed by the expressions:

ẋ(t) = A cl (λ, ρ)x(t) + A h (λ, ρ)x h (t) +B(λ, ρ)K h (ρ)x d (t) + E(λ, ρ)w(t) z(t) = C cl (λ, ρ)x(t) + C h (λ, ρ)x h (t) +D(λ, ρ)K h (ρ)x d (t) + F (λ, ρ)w(t) (21)
where

x h (t) = x(t -h(t)), x d (t) = x(t -d(t)), A cl (λ, ρ) = A(λ, ρ) + B(λ, ρ)K 0 (ρ) and C cl (λ, ρ) = C(λ, ρ) + D(λ, ρ)K 0 (ρ).
It is worth noting that this system is not a classical system with two delays. Indeed, the difficulty lies in the fact that both delays satisfy an algebraic inequality which constrains the trajectories of d(t) to evolve within a ball, of radius δ, centered around the trajectory of h(t). This additional information has to be taken into account for an efficient stability and performance analysis of the system [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF]. To this aim, the following preliminary result from [START_REF] Shustin | On delay-derivative-dependent stability of systems with fastvarying delays[END_REF] is used:

4.1.
Let us define the operator ∆(•) as

∆(z 0 (t)) = 2 √ 7δ t-h(t) t-d(t) z 0 (τ )dτ with (h, d) ∈ H × D δ (22)
For any input signal ξ ∈ L 2 , the output ∆(ξ) is also in L 2 and we have ||∆(ξ)|| L2 ≤ ||ξ|| L2 .

Using this operator, we can turn system (21) into an uncertain single-delay system (i.e. with d(t) only) depending explicitly on the delay error bound δ:

ẋ(t) = A cl (λ, ρ)x(t) + A hcl (λ, ρ)x d (t) +E(λ, ρ)w(t) + √ 7 2 δA h (λ, ρ)w 0 (t) z(t) = C cl (λ, ρ)x(t) + C hcl (λ, ρ)x d (t) +F (λ, ρ)w(t) + √ 7 2 δC h (λ, ρ)x d (t) z 0 (t) = ẋ(t) w 0 (t) = ∆(z 0 (t)) (23)
where

x d (t) = x(t-d(t)), A cl (λ, ρ) = A(λ, ρ)+B(λ, ρ)K 0 (ρ), A hcl (λ, ρ) = A h (λ, ρ)+B(λ, ρ)K h (ρ), C cl (λ, ρ) = C(λ, ρ) + D(λ, ρ)K 0 (ρ) and C hcl (λ, ρ) = C h (λ, ρ) + D(λ, ρ)K h (ρ).
Finally, according to the previous discussion, the main result of the paper is given below:

Theorem 4.2. There exists a parameter dependent state-feedback control of the form (20) which robustly asymptotically stabilizes system (2) for all (h, d, ρ, λ) ∈ H × D δ × P × Λ if there exist continuously differentiable matrix functions

P i : U ρ → S n ++ , matrix functions S i : U ρ → S n ++ , constant matrices Q i , R i ∈ S n ++ , X ∈ R n×n , Y 0 , Y h : U ρ → R m×n and a scalar γ > 0 such that the parameter dependent LMIs Ω 1i Ω 2i Ω 3i ≺ 0 ( 24
)
hold for all (ρ, ν, i) ∈ U ρ × co{U ν } × {1, . . . , N } and where

Ω 1i =          Ω 11i Ω 12i Ω 13i Ω 14i E i (ρ) 0 Ω 22i R i √ 7 2 δR i 0 Ω 26i Ω 33i √ 7 2 δΩ 33i 0 Ω 36i Ω 44i 0 √ 7 2 δC hi (ρ) T -γI p F i (ρ) T -γI q          Ω 2i =         X S i (ρ) h M R i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0         Ω 3i =   -P i (ρ) -S i (ρ) -h M R i -S i (ρ) 0 -R i   Ω 11i = -X S Ω 12i = P i (ρ) + A i (ρ)X + B i (ρ)Y 0 (ρ) Ω 13i = A hi (ρ)X + B i (ρ)Y h (ρ) Ω 14i = √ 7 2 δA hi (ρ)X Ω 22i = ∂P i ∂ρ ν -P i (ρ) + Q i -R i Ω 33i = -(1 -µ)Q i -R i Ω 26i = (C i (ρ)X + D i (ρ)Y 0 (ρ)) T Ω 36i = (C hi (ρ)X + D i (ρ)Y h (ρ)) T Ω 44i = 7 4 δ 2 Ω 33i -S i (ρ)
In such a case, the controller can be computed using K 0 (ρ) = L 0 (ρ)X -1 and K h (ρ) = L h (ρ)X -1 . Moreover the closed-loop system satisfies ||z|| L2 < γ||w|| L2 for all (ρ, h, d, λ)

∈ P × H × D δ × Λ.
Proof : The proof is given in Appendix A.

Remark 4.2. Here, the derivative of the error is not restricted and is allowed to be arbitrarily large. If for some reason, the derivative is bounded from above by one, the latter results can be refined. Indeed, a sharper bound on ||∆|| L2-L2 can be shown to be 1. Moreover, the derivative bound can also be taken into account through the introduction of another operator (the delay operator) similarly as in [START_REF] Gu | Stability of Time-Delay Systems[END_REF].

It is important to note that when the delay is exactly known (i.e. δ = 0), then Theorem 4.2 reduces to Theorem 4.1. This is stated in the following proposition: Proposition 4.2. When δ = 0, then LMIs ( 24) is equivalent to LMIs [START_REF] Ivȃnescu | Dynamical compensation for timedelay systems: an LMI approach[END_REF] provided that the matrix S(λ, ρ) 0 is chosen sufficiently small (e.g. according to the 2-norm).

Proof : The proof is only sketched since it relies on simple arguments and easy calculations. First, set δ = 0 in [START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF], this creates 0 entries on the 4 th row and column of ( 24) except for the diagonal value which is -S i (ρ). Since S i (ρ) is positive definite we can remove the 4 th row and column from [START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF]. Now, we just need to analyze the impact of the remaining terms depending on S i (ρ) which are located on the 8 th row and column of [START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF]. A Schur complement on the block [START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays[END_REF][START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays[END_REF] leads to a matrix sum of the form Λ i (ρ) + Υ i (ρ) ≺ 0 where Λ i (ρ) is exactly LMI [START_REF] Ivȃnescu | Dynamical compensation for timedelay systems: an LMI approach[END_REF] and

Υ i (ρ) =          S i (ρ) 0 . . . 0 -S i (ρ) 0 0 0 . . . 0 0 0 . . . . . . . . . . . . . . . . . . 0 0 . . . 0 0 0 -S i (ρ) 0 . . . 0 S i (ρ) 0 0 0 . . . 0 0 0         
It can be shown that this matrix has n positive eigenvalues (those of S i (ρ)) and α -n zero eigenvalues where α is the dimension of Υ i (ρ). Thus, choosing S i (ρ) as small as necessary, it is possible to approximate arbitrarily well LMI [START_REF] Ivȃnescu | Dynamical compensation for timedelay systems: an LMI approach[END_REF] by LMI [START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF] with δ = 0.

This shows that the main result embeds naturally (by construction) the case of controller with exact memory. Another important fact is when δ = h M , we get a result which is close to the memoryless case. This will be illustrated in the examples.

Remark 4.3. The LMI conditions of Theorems 3.1, 4.1 and 4.2 must be satisfied for all (ρ, ν) ∈ U ρ × co{U ν }. However, it is possible to reduce the computational complexity through the reduction of parameter set from U ρ ×co{U ν } to U ρ ×U ν . This is done using the following particular structure for the matrix P (λ, ρ) = P 0 (ρ) + N i=1 λ i P i . In this case, the resulting LMI conditions involve no product between λ and ρ and hence LMIs have to be checked only on the set U ρ × co{U ν }.

Examples

This section is devoted to examples and discussions on the provided approach. It will be illustrated that the current approach improves result of the literature in the control of LPV time-delay systems. Moreover, the connection between memoryless and exact-memory controllers 0 0.5 0.9 0.99

h M [37] 1.2 1.2 1.2 ∼ 1 h M [9]
-79.1511 -h M Theorem 4.1 929.1372 371.0928 6.9218 2.9325 Table 1: Evolution of the delay margin h M with respect to the bound on the delay derivative µ for a maximal allowable γ < 10 through δ-memory resilient controllers will be illustrated. Let us consider the following system borrowed from [35] and modified in [START_REF] Zhang | Delay-dependent stability analysis and H ∞ control for state-delayed LPV system[END_REF]:

ẋ(t) = 0 1 + φ sin(t) -2 -3 + σ sin(t) x(t) + φ sin(t) 0.1 -0.2 + σ sin(t) -0.3 x(t -h(t)) + 0.2 0.2 w(t) + φ sin(t) 0.1 + σ sin(t) u(t) z(t) = 0 10 0 0 x(t) + 0 0.1 u(t) (25)
where φ = 0.2 and σ = 0.1. Define ρ(t) := sin(t), h M = 0.5 and µ = 0.5 as in [START_REF] Zhang | Delay-dependent stability analysis and H ∞ control for state-delayed LPV system[END_REF].

Example 1: Memoryless State-Feedback (Small delay)

First, a memoryless control law is computed using Theorem 4.1. The parameter dependent decision matrices are chosen to be

P (ρ) = P 0 + P 1 ρ + P 2 ρ 2 Y 0 (ρ) = Y 0 0 + Y 1 0 ρ + Y 2 0 ρ 2
Verifying the LMI of Theorem 4.1 over a grid of N g = 41 points yields a minimal value γ * = 1.9089 which is better than all results obtained before. In [START_REF] Zhang | Delay-dependent stability analysis and H ∞ control for state-delayed LPV system[END_REF], a minimal value of γ = 3.09 is found while in [START_REF] Briat | Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach[END_REF] an optimal value of γ = 2.27 is obtained (using a nonlinear approach). The controller computed using Theorem 4.1 is given by

K 0 (ρ) = -1.0535 -2.9459ρ + 1.9889ρ 2 -1.1378 -2.6403ρ + 1.9260ρ 2 T
It is worth noting that the results are even better while the controller has smaller coefficients than in the other approaches [START_REF] Zhang | Delay-dependent stability analysis and H ∞ control for state-delayed LPV system[END_REF][START_REF] Briat | Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach[END_REF]. It is hence expected to have a smaller control input which should remain within acceptable bounds, even in presence of disturbances. The influence of µ on the delay-margin (with L 2 performance constraint) is detailed in Table 1 where, as expected, the delay margin decreases as the value of µ increases. Moreover, the results of [START_REF] Zhang | Delay-dependent stability analysis and H ∞ control for state-delayed LPV system[END_REF] are more conservative than those obtained using Theorem 4.1 since the delay-margin is always smaller than 1.4 (for γ < 10) for any value of µ.

As a final remark, the maximal value for h M obtained for µ = 0 is large and this suggests that the system might be delay-independent stabilizable in the constant delay case.

Example 2: State Feedback with Exact Memory (Large delay)

Still considering system [START_REF] Niculescu | Delay effects on stability[END_REF] but with h M = 10 and µ = 0.9, an instantaneous state-feedback controller of the form u(t) = K 0 (ρ)x(t) is sought. Theorem 4.1 yields: K 0 (ρ) = 0.5724 -6.3679ρ -1.4898ρ This controller ensures a closed-loop L 2 performance level of 4.1641. The gain of performance resulting from the use of a controller involving a delayed term is evident. However, this controller is non implementable due to the practical impossibility of knowing the exact delay value at any time. This motivates the synthesis of a memory-resilient controller.

Example 3: δ-Memory Resilient Controller Synthesis (Large delay)

Finally, a δ-memory resilient state-feedback controller stabilizing system (25) is sought. Using Theorem 4.2, the achieved minimal closed-loop L 2 performance with respect to δ is plotted in Figure 1. As expected the minimal L 2 performance level grows as the delay uncertainty radius increases. This illustrates that the achieved closed-loop performance is deteriorated when the delay is badly known.

Moreover, there are two remarkable values for the worst-case L 2 gain, respectively obtained for δ = 0 and δ = h M . For these particular values we have:

γ| δ=0 = 4.1658 K 0 (ρ)| δ=0 = 1.0542 -2.8895ρ -0.4827ρ 2 -0.7714 -1.8912ρ + 0.1216ρ 2 T K h (ρ)| δ=0 = -0.6885 + 0.5849ρ + 0.1116ρ 2 -0.0817 + 0.3148ρ + 0.0667ρ 2 T ( 27
)
γ| δ=10 = 13.0604

K 0 (ρ)| δ=10 = 0.4422 -6.3469ρ -1.3619ρ 2 -0.9475 -4.1219ρ -0.6140ρ 2 T K h (ρ)| δ=10 = -0.0163 -0.0005ρ + 0.0127ρ 2 -0.0007 -0.0006ρ + 0.0011ρ 2 T ( 28
)
When the delay is exactly known (i.e. δ = 0), the L 2 performance index and the controller are very close (quite identical) to the ones obtained using Theorem 4.1 which considers exact-memory controllers. This illustrates well Remark 4.2.

On the other hand, when δ = h M , it could be considered that the delay is actually unknown since for any value for h(t), the implemented value d(t) may take any value into [0, h M ]. Hence, this means that the results with such a value for δ should be close to the results obtained using a memoryless control law. Comparing [START_REF] Sename | Observer-based H ∞ control for time-delay systems: a new LMI solution[END_REF] and [START_REF] Papachristodoulou | Stability analysis of linear systems with time-varying delays: Delay uncertainty and quenching[END_REF] it is possible to remark that the obtained closedloop performance level is very near to the one obtained with a memoryless control law. Moreover, the matrix gain K h (ρ) in ( 28) has a small norm making it almost identical to a memoryless controller. The delayed action is highly penalized due to a too large uncertainty on the delay knowledge. The results are summarized in Table 2.

The above discussion illustrates that δ-memory resilient controllers define the intermediary behavior of the closed-loop system between the two extremal controllers: the memoryless and the exact-memory controllers. The emphasis of this continuity between memoryless and exact memory controller through δ-memory resilient controllers constitutes one of the main contribution of the paper. Moreover, such controllers are also more realistic, from a practical point of view, than exact-memory controllers.

Conclusion

The current paper introduces a new approach to the stabilization of LPV time-delay systems using parameter dependent state-feedback controllers. First, a delay-dependent stability test with L 2 performance analysis for LPV time-delay systems with time-varying delays is provided in terms of parameter dependent LMIs. This result is obtained from the use of a parameter dependent Lyapunov-Krasovskii functional used along with the Jensen's inequality, an approach which has proven its efficiency. Since this result is not well suited for design purpose due to multiple products between decision matrices and system matrices, a relaxed version of the result is developed. This version involves an additional 'slack' (or 'lifting') variable and avoids any nonlinear terms (multiple products). This allows to find a linearizing change of variable for the stabilization problem.

A first stabilization result is then provided and characterizes both memoryless and exactmemory controllers. However, due to the difficulty of estimating delays, latter controllers are generally non-implementable in practice. This has motivated the development of another type of controllers, called 'δ-memory resilient controllers' where the delay implementation error is taken into account in the design. It turns out that this new class of controllers connect memoryless and exact-memory controllers together, through a unique formulation. Indeed, by acting the error bound δ between the delays, it is possible to recover exact-memory (δ = 0), δ-memory resilient (δ ∈ (0, h M )) and memoryless (δ ∼ h M ) control laws successively. This part constitutes the main contribution of the paper and is illustrated in the examples.

         A T cl P + P A cl + Ṗ + Q -R P A hcl + R √ 7 2 δP A h + δR P E C T cl h M A T cl R A T cl L -Q µ -R - √ 2δ(Q µ + R) 0 C T hcl h M A T hcl R A T hcl L -7 4 δ 2 (Q µ + R) -L 0 √ 7 2 δC T h √ 7 2 h M δA T h R √ 7 2 δA T h L -γI p F T h M E T R E T L -γI q 0 0 -R 0 -L           ≺ 0 (A.1)                -(X + X T ) P + X T A cl X T A hcl √ 7 2 δX T A h X T E 0 X T L h M R Ṗ -P + Q -R R √ 7 2 δR 0 C T cl 0 0 0 -Q µ -R - √ 7 2 δ(Q µ + R) 0 C T hcl 0 0 0 -7 4 δ 2 (Q µ + R) -L 0 √ 7 2 δC T h 0 0 0 -γI p F T 0 0 0 -γI q 0 0 0 -P -L -h M R -L 0 -R                ≺ 0 (A.2)
where X(t) = col(x(t), x h (t), x d (t), w(t)), Π is defined by along with the change of variables

X ← X -1 P i (ρ) ← X -T P i (ρ)X -1 Q i ← X -T Q i X -1 R ← X -T R i X -1 S i (ρ) ← X -T S i (ρ)X -1 Y 0 (ρ) ← K 0 (ρ)X -1 Y h (ρ) ← K h (ρ)X -1 (A.7)
yields LMI [START_REF] Moon | Delay-dependent robust stabilization of uncertain state-delayed systems[END_REF]. This concludes the proof.

Figure 1 :

 1 Figure 1: Evolution of the worst-case L 2 gain of the closed-loop system with respect to maximal delay uncertainty δ

12 Π 13 P 11 =

 121311 (λ, ρ)E(λ, ρ) -(1 -µ)Q(λ) -R(λ) A cl (λ, ρ) T P (λ, ρ)S + ∂P (λ, ρ) ∂ρ ρ + Q(λ) -R(λ)Π 12 = P (λ, ρ)A h (λ, ρ) + R(λ) Π 13 = P (λ, ρ)B(λ, ρ)K h (ρ) and Γ 1 (ρ) = A cl (ρ) A h (ρ) B(ρ)K h (ρ) E(ρ) . Now according to the relation w 0 = ∆( ẋ) = 2(x h (t) -x d (t)(t) T M T Ψ(λ, ρ, ρ)M Y (t) (A.5)In order to consider the uncertain norm-bounded operator ∆ and input/output L 2 performance, the following quadratic supply-rates(ζ(t)) = ζ(t) T Φ(λ, ρ)ζ(t) defined by Φ(λ, ρ) = diag(-L(λ, ρ), L(λ, ρ), -γI p , γI q ) (A.6)is added to (A.5) where ζ(t) = col(w 0 (t), z 0 (t), w(t), z(t)) and w 0 (t) = ∆[z 0 (t)]. Here, L(λ, ρ) = L(λ, ρ) T 0 is a parameter dependent D-scaling and γ > 0 characterizes the upper bound on the 2 gain of the transfer w → z. This leads to inequality (A.1) where the dependence on ρ and λ has been dropped for clarity. Then applying the same relaxation procedure as in the proof of Theorem 3.1 we get the new inequality (A.2) with A hcl (•) = A h (•) + B(•)K h (•). Finally performing a congruence transformation with respect to diag(I 4 ⊗ X -1 , I p+q , I 3 ⊗ X -1 )

Table 2 :

 2 Comparison of the results obtained using Theorem 4.2

	2

Appendix A. Proof of theorem 4.2

Let us consider the closed-loop system [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF] and the Lyapunov-Krasovskii functional V given in [START_REF] Briat | Delay-scheduled state-feedback design for time-delay systems with time-varying delays -a LPV approach[END_REF]. Computing the derivative of (10) along the trajectories solutions of system [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF] we get


M Barbier

New Set of Codes for the Maximum-Likelihood Decoding Problem

Keywords: Maximum-likelihood decoding, perfect codes, covering radius, list decoding

. In this paper, we introduce the notion of A-covered codes, that is, codes that can be decoded through a polynomial time algorithm A whose decoding bound is beyond the covering radius. For these codes, we show that the maximum-likelihood decoding problem is reachable in polynomial time in the code parameters. Focusing on binary BCH codes, we were able to find several examples of A-covered codes, including two codes for which the maximum-likelihood decoding problem can be solved in quasi-quadratic time.

Introduction

Berlekamp, McEliece and Van Tilborg showed in [START_REF] Berlekamp | On the inherent intractability of certain coding problem[END_REF] that the maximum-likelihood decoding is a NP-hard problem for general linear codes. Guruswami and Vardy later proved in [START_REF] Guruswami | Maximum-likelihood decoding of Reed-Solomon codes is NP-hard[END_REF] that this problem applied to the family of Reed-Solomon codes is also NP-hard. We briefly recall below the maximum-likelihood problem. Definition 1.1 (Maximum-likelihood decoding problem). Let C a linear code over F q and v a F q -vector in the ambient space. The maximum-likelihood decoding problem is to find the codeword w ∈ C closest to v. Most precisely, to find w ∈ C, such as

d(w, v) = d(v, C) = min c∈C {d(v, c)}.
Clearly, if for a given code there exists an algorithm able to correct a number of errors at least equal to the covering radius, then this algorithm solves the maximum-likelihood decoding problem. We recall the covering radius definition, which is the largest distance between any vector in ambient space and the code. Definition 1.2 (Covering radius). Let C a linear code over F q . Its ambient space is a F q -vector space V . Let v ∈ V , the covering radius R of C is given by

R = max v∈V {min c∈C d(v, c)}.
In light of Wu's recent algorithmic advances in list decoding [START_REF] Wu | New list decoding algorithms for Reed-Solomon and BCH codes[END_REF], we proceed in a comparaison between covering radii and now achievable decoding bounds with such algorithm. This leads us to propose the new algorithmic notion of A-covered codes for which maximum-likelihood decoding problem can be carried out in polynomial time, and provide some examples by focussing the family of binary BCH codes. We also exhibit two codes for which the maximum-likelihood decoding problem has quasi-quadratic complexity.

A-covered codes

In the rest of this paper, we follow the standard notations of [START_REF] Cohen | Covering codes[END_REF] and shall denote by R the covering radius of a code C, and by t ⌊ d-1 2 ⌋ its error correction capacity. We now recall the definition of a perfect code. Definition 2.1 (Perfect code). A code C with capacity t and covering radius R is called a perfect code if and only if R = t.

These codes are of course very interesting from a decoding point of view since each element of their ambient spaces can be decoded. Linear perfect codes are completely classified and for each of them, we know a decoding algorithm up to t = R. The maximum-likelihood problem is consequently trivial for perfect codes. This very property prompts us to propose the notion of A-covered codes in the context of list decoding. We first introduce the following definitions: Definition 2.2 (List decoding algorithm). Let C a code and v a word in its ambient space. A is a list decoding algorithm for C up to τ A if and only if it returns all codewords w ∈ C such that d(v, w) ≤ τ A .

Definition 2.3 (Polynomial time list decoding algorithm). Let C a code, n its length, v a word in its ambient space and A a list decoding algorithm up to τ

A . A is a polynomial time list decoding algorithm if it runs in O(f (n)), where f (X) ∈ R[X].
We can now present the notion of A-covered code. As seen before, the notion of A-covered code can be seen as a computational analogue to perfect codes, albeit in the list decoding context (see Figure 1).

Unique decoding List decoding

Perfect code A-covered code It is well known that 1-error-correcting BCH codes are perfect (these are Hamming codes) and 2-error-correcting codes are quasi-perfect [START_REF] Cohen | Covering codes[END_REF]. Since Wu's method is a polynomial time list decoding algorithm [START_REF] Wu | New list decoding algorithms for Reed-Solomon and BCH codes[END_REF], it is natural to compare their covering radii and the binary Johnson bound of other binary BCH codes. Unfortunately, classifying codes is a hard problem since it requires to compute the covering radii which usually is not an easy task [START_REF] Mcloughlin | The complexity of computing the covering radius of a code. Information Theory[END_REF]. Putting together and completing data from the literature [START_REF] Cohen | Covering codes[END_REF], we still manage to obtain the list in Table 1. This table includes all primitive binary BCH codes of known covering radius. The non-primitive, Wu-covered binary BCH codes of length 17 and 23 were obtained by our own calculations.

R = t R ≤ τ A
Note that the BCH codes having 3 as their minimum distance are Hamming codes. Since those are perfect codes, the maximum-likelihood decoding problem is trivial. We also found two Reed-Muller codes of first order [START_REF] Macwilliams | The Theory of Error-Correcting Codes[END_REF]. Since the dimensions of first order Reed-Muller codes are equal to the logarithm of their lengths, naively listing all closest codewords is already a polynomial time decoding algorithm. Hence, knowing that these codes are Wu-covered is of little pratical important in solving the maximum-likelihood decoding problem, since easier polynomial time methods are already available.

By contrast, the four codes [15, 7, 5], [17, 9, 5], [23,12,[START_REF] Mcloughlin | The complexity of computing the covering radius of a code. Information Theory[END_REF] and [31,11,11] given in Table 1 do not fall into the two aforementioned families and we would expect the maximum-likelihood decoding problem to be asymptotically hard. However, the fact that they are Wu-covered implies that this problem is actually solvable in polynomial time only (in the code parameters).

are Wu-covered codes, of which, four constitute a new result and two can be decoded in time quasi-quadratic in code parameters.

Definition 2 . 4

 24 (A-covered code). Let C a code with covering radius R and A a polynomial time list decoding algorithm which decodes C up to τ A . C is an A-covered code if and only ifR ≤ τ A .Remark 2.1. Since this algorithm runs in polynomial time, the returned list is also of polynomial size. Proposition 2.1. Let C an A-covered code. The maximum-likelihood decoding problem for C, (as given by Definition 1.1) is solvable in a time polynomial in the code parameters.

Figure 1 :Figure 2 :

 12 Figure 1: Perfect code vs A-covered code

Consequently, binary BCH codes having binary Johnson bound strictly greater than their covering radii, such as binary BCH [31, 6, 15] = RM(1, 5) * and BCH [23,12,[START_REF] Mcloughlin | The complexity of computing the covering radius of a code. Information Theory[END_REF], can be decoded in quasi-quadratic time up to, and including, their covering radii.

Conclusion

Working purely from an algorithmic point of view, we proposed a new set of codes, the A-covered codes, for which we showed that the maximum-likelihood decoding problem, known as NP-hard in the general case, is solvable in polynomial time.

The main difficulty in finding such codes lies in the computation of covering radii. However there may be quite a few of those codes as we exhibited nine binary BCH codes which


B Cocquelin

Gaëlle Lucas-Leclin

P Georges

I Sagnes

D Holleville

A Garnache

Single-frequency diode-pumped Single-frequency diode-pumped semiconductor laser tuned on a Cs transition semiconductor laser tuned on a Cs transition

Lasers in Lasers in Cesium atomic clocks Cesium atomic clocks

Need for high-power and narrow-linewidth sources emitting at the Cesium D 2 line (852 nm)

⇒ a single OP-VECSEL ? • High power in Optically Pumped-VECSEL 30 W @ 980 nm, M 2 = 3 (Coherent -Photonics West '04) 1.0 W in-well pump / 0.7 W @ 850 nm, M 2 = 5 (University of Strathclyde)

• No spatial hole-burning : single-frequency in simple linear cavity 500 mW @ 1003 nm (Jacquemet et al, App.Phys. B 86, 503 (2007)) 42 mW @ 870 nm, ∆ν L ≅ 3 kHz (Holm et al, IEEE PTL 11, 1551(1999))

• Linearly polarized, circular TEM 00 beam Single-frequency diode-pumped semiconductor laser at the Cs line

Design

Design optimization optimization

• Low threshold pump intensity I th for high opt-opt efficiency ⇒ N QW = 7 is optimal for ~ 2% losses ⇒ evaluation of the spectral properties + thermal management for power scaling at 852.14 nm

Specifications already adequate for optical detection in atomic clocks

  Scheme of an atomic fountain

••⇒

  Photodetected signal

Design of Design of the the semiconductor semiconductor structure structure

• λ L = 852 nm • Barriers absorption at λ P ≤ 720 nm e b ≅ 2 µm ⇒ η P = 85%

Energy Quantum wells Bragg Mirror |E| 2 Substrate GaAs 350 µm 32.5 pairs λ/4 Al 0.22 Ga 0.78 As/AlAs R ≥ 99.95% Active Layers 29λ/4 GaAs N QW = 7 ; L QW = 8 nm Barriers Al 0.22 Ga 0.78 As InGaP Caping Layer (20 nm)

		1.0
					Reflectivity
		0.8			R max ≅ 99.97%
	• AR coating (Si 3 N 4 ) at air/SC surface for :	0.6		Stop band 70 nm
	maximum pump transmission	0.4
	+ reduction of microcavity etalon effect
		0.2
	• Structure grown by MOCVD	0.0 0.6	0.7	0.8	Photoluminescence 0.9 1.0	1.1

Acknowledgements : B. Cocquelin PhD funding by CNRS/CNES

Financial support from the DGA under contract « POSEIDA » n°0534004


Adam Kamor

Francois Mauger

Cristel Chandre

Turgay Uzer

Controlling double ionization of atoms in intense bichromatic laser pulses

Keywords: numbers: 32, 80, Rm, 05, 45, Ac

come

I. INTRODUCTION

Multiple ionization of atoms takes place at intermediate and high intensities when they are subjected to laser pulses [1]. The routes by which these multiple ionizations occur can be quite complicated. Two main routes have been identified for the double ionization of two-activeelectron systems (like the helium atom subjected to linearly polarized laser pulses): a sequential process and a nonsequential one [1]. In the sequential process the field ionizes one electron after the other in an uncorrelated way. In contrast, the electron-electron correlation plays an active role in the ionization of the second electron in the nonsequential process. More specifically, the field picks up an electron, and sends this electron to the remaining ion core to dislodge the second electron, according the so-called recollision (or 3-step) scenario [2,3].

It has been observed experimentally in a broad range of intensities that non-sequential double ionization (NSDI) [also referred to as correlated double ionization (CDI) [4]] yields can be many orders of magnitude larger than in the sequential double ionization (SDI) [also referred to as uncorrelated double ionization (UDI)] counterpart. Varying the intensity gives rise to a characteristic "knee" shape for the double ionization yield as a function of the laser intensity. This strong effect due to electronelectron correlation has been observed experimentally [5][6][7][8][9][10][11][12][13] and also theoretically using quantum [1,[14][15][16][17][18][19][20], semiclassical [21][22][23] and classical simulations [4,18,21,[24][25][26][27][28][29].

In this article, we investigate the dynamics resulting from the interaction between helium atoms and a bichromatic linearly polarized laser field using a classical model. Classical means are entirely sufficient to produce the strong electron correlation needed to explain double ionization phenomena [25]. Atomic phenomena in intense bichromatic fields can be considered as an extension of the principles of coherent control into the nonlinear non-perturbative regime of laser-atom interactions [30][START_REF] Shapiro | Principles of the Quantum Control of Molecular Processes[END_REF][START_REF] Ko | [END_REF][33][34][35][36][37][38][39][40][41][42]. Although the classical mechanisms of double ionization remain qualitatively the same as for a monochromatic laser field [29], the double ionization yields show a strong sensitivity to changes in the parameters of the laser field, namely the phase lag between the two colors and their relative amplitudes. We explain this dependence using an analysis in terms of phase space structures. Our main result is that all these processes are regulated primarily by the dynamics of the inner electron which is the electron that remains bounded after the other one has been ionized. Remarkably, this electron, rather than being a spectator, is in fact the driver in the ionization process.

We consider a classical model with one spatial dimension for each electron which corresponds to the direction of polarization of the laser field. We denote the positions of the two electrons by x and y and set the origin at the nucleus. We call the canonically conjugate momenta of x and y, p x and p y respectively. We consider soft Coulomb potentials [4, 25-27, 29, 43] defined by two softening parameters a and b for the interaction between charged particles. The Hamiltonian of the system in the dipole approximation is obtained by the sum of the kinetic energy of the electrons, the soft Coulomb potentials between the two electrons and between the electrons and the nucleus, and the interaction of the electrons with the field:

H (x, y, p x , p y , t) = p 2 x 2 + p 2 y 2 + 1 (x -y) 2 + b 2 - 2 √ x 2 + a 2 - 2 y 2 + a 2 + (x + y) E (ωt) , (1)
where E (ωt) is a bichromatic laser field, defined as:

E (ωt) = E 0 f (ωt) (sin (ωt) + α sin (kωt + δ)) , (2)
where E 0 is the amplitude of the dominant mode, ω its frequency, α < 1 the ratio of the amplitudes of the two colors of the laser field, k characterizes the mode locking between the two components, and δ is the phase lag. Since the second laser is an additional energy input to the system, the intensity is scaled with α as

I 0 ∝ E 2 0 1 + α 2 .
The system has two and a half degrees of freedom, one for each electron and one half for time dependence. The envelope function f is chosen as a trapezoid with 2 laser cycle (l.c.) ramp up, 4 laser cycle plateau, 2 laser cycle ramp down, and zero elsewhere. In what follows, the wavelength is chosen to be 780 nm (corresponding to a frequency of ω = 0.0584 a.u.) and k = 3. Except when we investigate the role of α, we set α = 0.45 for numerical simulations, which corresponds to an additional intensity input of 20%. We consider the initial state to be the ground state of He, i.e. E g = -2.24 a.u. [25,44]. We choose a = 1 to prevent auto-ionization (so that the energy surface is bounded in phase space). The second softening parameter b is chosen so as to allow a significant energy exchange during a recollision. Here b = 1 [45].

For insight into the dynamics of typical trajectories and to interpret the mechanisms at play which explain the statistical analysis, we present two kinds of studies as functions of the parameters of the field, E 0 , α and δ in what follows: In Sec. II, a statistical analysis of a large ensemble of trajectories, and in Sec. III an analysis of the phase space structures.

II. STATISTICAL ANALYSIS

Starting from the classical model (1), we perform a statistical analysis of a large ensemble of trajectories (typically 150,000) initially on the ground state (with a microcanonical distribution [29]). At the end of the pulse, a trajectory can exhibit zero, one or two ionized electrons. In our calculations, an electron is considered ionized when its energy (defined as the sum of its kinetic energy and the Coulomb interaction with the nucleus) is positive, which corresponds to an electron free from the Coulomb potential well. A convenient observable is the probability of double ionization where the two electrons have a positive energy at the end of the pulse. This probability is commonly plotted as a function of the laser intensity. A signature of NSDI is a characteristic knee shape: In the intermediate intensity range, these yields are much higher (by several orders of magnitude) than expected from the uncorrelated sequential double ionization mechanism [1]. At high intensity, the yields reach saturation, corresponding to double ionization of all the electrons of the ensemble. In what follows, we investigate the double ionization yields versus the intensity when the phase lag δ and the relative amplitude α are varied.

I (W • cm -2 ) Probability δ = 0 δ = π/2 δ = π
FIG. 1: Probability of double ionization as a function of the intensity for α = 0.45. The red crosses correspond to δ = 0, the black circles to δ = π/2, the green squares to δ = π. We also indicate the range of variation of I (c) with δ as given by Eq. ( 14) (vertical dashed-dotted lines). The vertical magenta continuous line labels the intensity chosen in Fig. 2, 7, and 8.

A. Varying the phase lag δ

In Fig. 1 we display several double ionization probability curves as a function of the laser intensity for different values of the phase lag δ (for α = 0.45).

In the intermediate range of intensities (up to saturation of double ionization), we observe a significant variability of the double ionization probabilities and consequently of the characteristic features of the knee. More precisely, in the intermediate intensity range the level of double ionization yields greatly vary with δ by a factor of 3; however, when varying δ the range of intensity where the knee is located (around 5 × 10 14 W • cm -2 , vertical dashed-dotted lines in Fig. 1) does not change significantly. In the low intensity regime i.e., for intensities lower than 8 × 10 13 W • cm -2 there is a very weak dependence of δ in these yields. The same conclusions hold for other values of α. We note a significant impact of the phase lag in the resulting double ionization probability, meaning that adjusting this parameter can act as a control knob for both sequential and non-sequential processes. One advantage of varying the phase lag between the two components of the laser field is that no additional input of energy is required (as opposed to varying other parameters like α) since the mean value of the electromagnetic energy does not depend on δ.

In Fig. 2 we represent the double ionization yield as a function of δ for the intensity labeled by the vertical magenta continuous line in Fig. 1. The resulting curve has a bell shape with a maximum slightly shifted before δ = π. More specifically, we observe a fast variation of the double ionization yield with changes of the phase lag around δ = 0 and a flattening around δ = π. This confirms the high sensitivity of the double ionization probability to the parameter δ.

Except for very low intensity (i.e. below 8 × 10 13 W • cm -2 ) we observe qualitatively the same impact of δ on the double ionization yield (a bell shape almost centered around δ = π). The precise value of the phase lag for which the probability is maximum depends on the laser parameters.

The value of δ for which we observe the first saturation of double ionization occurs when δ = π. This value of δ does not depend on α (and is defined as soon as α is non-zero). We define I (t) (δ) the intensity, as a function of the phase lag, for which saturation of double ionization is reached. Then, at high intensity, we note a great variability of I (t) (δ) (the differences are of the order of 10 16 W•cm -2 ) which is more than one order of magnitude larger that the range of variation of the knee.

B. Varying the relative amplitude α

After varying δ at fixed α, we keep δ fixed at π in order to investigate α's influence on the double ionization probability. We chose δ = π because it corresponds to the phase lag where the peak field amplitude is maximum (see Sec. III D). In Fig. 3 we represent the double ionization probability as a function of the intensity for different values of α. As for the variation of δ, we notice no significant dependence on the parameters at low intensities (below 8 × 10 13 W • cm -2 ). In the intermedi- ate intensity range, we notice that the probability greatly varies with α (more than a factor two) whereas the intensity at which the characteristic knee is developed does not change significantly. In the meantime, the UDI component also increases with α, and we notice a decrease of the intensity I (t) where double ionization is saturated. As a consequence we notice a flattening in the knee, the two CDI and UDI curves merging. As α is increased, the second color of the laser field becomes more important, hence δ is expected to have a more prominent role in controlling the system because the variation in probability becomes larger. This is confirmed by Fig. 4 where we have represented double ionization probabilities as a function of the intensities for α = 0.1 and α = 0.75.

In summary, the statistical analysis shows these three main trends when changing the parameters of the field:

1. High sensitivity of the actual probability of double ionization, 2. High sensitivity of the intensity of saturation of the double ionization yields, 3. Low sensitivity of the intensity of knee regime (i.e. the CDI maximum).

In the next section, we analyze the dynamics of Hamiltonian (1) in terms of phase space structures and we find results that support and explain these trends.

III. NONLINEAR DYNAMICAL ANALYSIS

For the dynamical analysis of Hamiltonian (1) we follow the one performed in Ref. [29] for the monochromatic laser pulse. The analysis proceeds in the following way: First, without laser field (E 0 = 0), an outer electron and an inner one were identified from a periodic orbit analysis of the chaotic motion of typical trajectories. When the field is turned on, if the inner electron remains bounded, the atom may undergo a recollision which follows from the 3-step picture [2,3] (the field picks up the outer electron, and upon reversal of the field, brings it back to the core and the two electrons share the collision energy).

A. Effective models

The dynamics of both electrons are accurately reproduced by reduced models except during recollisions [29]. These models allow one to derive the main properties of the double ionization curves versus intensity. We begin by investigating the behavior of typical trajectories. In Fig. 5, we represent the position of each electron (upper panel) and their energy (kinetic energy plus soft Coulomb potential, lower panel) as functions of time for a typical trajectory. We observe that one electron is first quickly ionized (the "outer" electron) while the other remains close to the core (the "inner" electron). Once ionized, the outer electron is driven mainly by the laser field and experiences large excursions from the core. In contrast, the inner electron experiences a competition between Coulomb interaction with the nucleus and the laser excitation. Without loss of generality, we assign x to the outer electron and y to the inner one. In the sample trajectory of Fig. 5 the inner electron (red curves) does not ionize until it collides with the outer electron (blue curves). Double ionization occurs at approximately 2.5 laser cycles, which can be confirmed by examining the energies of both electrons in the lower panel. We see that both electrons have positive energy after this time (and thus are ionized). From this qualitative analysis, two reduced effective Hamiltonians are identified, H out for the outer and H in for the inner one:

H out (x, p x , t) = p 2 x 2 + xE (ωt) , (3)
H in (y, p y , t) = p 2 y 2 - 2
y 2 + 1 + yE (ωt) , (4)
where we have neglected the interaction with other charges for the outer electron and have allowed the inner electrons to feel both the laser field and the attraction from the nucleus. For both electrons, we ignore the electron-electron interaction term since they are assumed to be far away from each other between two recollisions. The main advantage of the above effective models is that their dimensions are lower than the one of Hamiltonian (1) (they both have one and a half degrees of freedom). Hamiltonian H out is even integrable in the sense that one can explicitly give the expression of the trajectories as functions of the initial conditions. Hamiltonian H in is not integrable but its dynamics can be analyzed qualitatively.

In what follows, we analyze these two models as the parameters of the field are varied and correlate their dynamics with the double ionization curves plotted in Sec. II in order to support and explain the observed trends.

B. dynamics of the outer electron

The trajectory of the outer electron [as governed by Eq. ( 3)] is given by

x (t) = x 0 + p 0 t + E 0 ω 2 sin ωt + α k 2 sin (kωt + δ) , (5)
p x (t) = p 0 + E 0 ω cos ωt + α k cos (kωt + δ) , (6)
where x 0 + E 0 αk -2 ω -2 sin δ and p 0 + E 0 ω -1 1 + αk -1 cos δ are, respectively, the position and momentum of the outer electron at time t = 0. An important quantity is the maximum return kinetic energy of the outer electron: It gives the maximum amount of energy the outer electron can bring to the inner electron to trigger CDI. In addition, if the outer electron recombines, it gives the cutoff in harmonic generation for photon emission [46]. In order to compute this quantity we need to consider a shift of the field since the electron leaves the core at a certain phase φ of the field. The effective Hamiltonian is transformed into

H out = p 2 x 2 + xE(ωt + φ).
Maximizing the kinetic energy E max = p x (t r ) 2 /2 with respect to the recollision time t r and the initial phase φ leads to the familiar condition that the maximum return energy is associated with trajectories leaving the nucleus with zero momentum [46]. As a function of the parameters of the laser field, this maximum kinetic energy is equal to

E max = κ (α, δ) U p , (7)
where U p = E 2 0 /(4ω 2 ) is the ponderomotive energy and κ is only a function of α and δ. For α = 0, we have κ(0, δ) ≈ 3.17 [2]. In Fig. 6, we represent κ(α, δ) as a function of δ for various values of α. We notice that the amplitude of the variations of κ with δ increases with α. For a fixed value of α, the maximum return energy κ (in units of U p ) of the outer electron can vary significantly with δ (up to more than 2). For all values of α, the maximum of κ occurs at δ ≈ 1.04, while its minimum is obtained for δ ≈ 3.99. We notice that none of these values are related to the critical values of the phase lag for which we observe maximum of double ionization (around δ = π) and saturation of double ionization. Thus, the origin of the amount of double ionization does not solely rely on the dynamics of the outer electron. It has to be complemented by the dynamics of the inner electron.

C. Dynamics of the inner electron

After the ionization of the outer electron, the inner electron experiences the combined action of the field and of the attraction from the nucleus only. Its Hamiltonian is given by Eq. ( 4). The competition between these two interactions leads to two distinct behaviors related to two distinct regions. Figure 7 represents these two regions in phase space: More specifically, we plot the distance of the inner electron from the nucleus after 10 laser cycles in the space of initial conditions (y 0 , p y,0 ) at t = 0. Note that the color map is in logarithmic scale. The red and green curves are two different Poincaré sections of an inner electron trajectory, taken stroboscopically with the laser period. They indicate that the dark region is filled with invariant tori where electrons are bound to the nucleus (for all time). The light region corresponds to ionizing electrons (after a transient time which depends on the initial conditions, see the darker structures which spiral out). The rationale behind this dynamics is as follows: If the electron is close to the nucleus, then the Coulomb interaction dominates over the field and the electron is bound. If it is sufficiently far, the field dominates and the dynamics of the electron is unbounded, in a similar way as the outer electron. The maximum distance the inner electron can be from the nucleus while remaining bound is a function of the laser parameters.

D. Double ionization probability

Varying certain laser parameters leads to strong variations in the size of the bound region, which is pivotal in explaining the probability of double ionization. Hamiltonian (4) is not integrable. However, a close inspection of the dynamics of the inner electron in the bound region reveals that it is fairly insensitive to the field so we consider another effective Hamiltonian in the bound region, namely

H * in (y, p y ) = p 2 y 2 - 2 y 2 + 1 , (8)
which is integrable, its phase space being foliated by periodic orbits. There exists, at least locally, a canonical transformation which maps Hamiltonian (8) into action and angle variables. The action is defined by

A = 1 2π p y dy.
When the field is present, most of the outer invariant tori are broken, whereas most of the inner ones close to the nucleus persist. We denote by A m (E 0 , α, δ) the action of the last invariant rotational torus which roughly corresponds to the boundary of the inner region. Figure 8 illustrates the dependence of A m (E 0 , α, δ) with δ as α and E 0 are kept fixed for an intensity in the UDI regime (I = 4.78 × 10 15 W • cm -2 ). Numerically, we determine A m by computing the volume of phase space for which the inner electron never ionizes during 2 laser cycle ramp up and 4 laser cycle plateau (see Fig. 8). At least qualitatively, the same trend is observed in the whole UDI regime (as well as the CDI regime), namely an inverted bell shape with fast variations of A m around δ = 0 with a flattening and minimum centered around δ = π.

An approximation to the size y m of the inner region can be obtained by finding the local maximum of the effective potential (soft Coulomb potential plus laser excitation) at peak field amplitude. It is implicitly given by

M (α, δ) E 0 = 2y m (y 2 m + 1) 3/2 , (9)
where M (α, δ) is given by:

M (α, δ) = max τ ∈[0,2π] |sin τ + α sin (kτ + δ)| , (10)
and corresponds to the peak field amplitude of the laser. In Fig. 9 we display the peak field amplitude as a function of δ. In the bound region, the energy of the inner electron, as defined by Eq. ( 4), does not vary much. Then, it is possible to relate it to the action A through the formula [4] H

in (A) = -2 + √ 2 e γA -1 γ ,
where γ = -9 √ 2/16 (≈ -0.8). It can be inverted to express the action explicitly as a function of the energy The solid line corresponds to the statistical data, while the dashed dotted line corresponds to the theoretical model, as given by Eq. ( 12).

A = 1 γ ln 1 + γ √ 2 (H in + 2) . (11)
0 1 2 3 4 5 6 1.1 1.2 1.3 1.4 1.5 δ M (α, δ)
FIG. 9: M (α = 0.45, δ) as a function of δ, as given by Eq. (10).

The outermost position y m at peak field amplitude is associated with zero momentum, such that its energy H m is equal to

H m = H in (y m , 0, t m ) = - 2
y 2 m + 1 -E 0 M (α, δ) y m .
Finally, the critical action A m is obtained by plugging H m into Eq. ( 11)

A m = 1 γ ln 1 + γ √ 2 2 - 4y 2 m + 2 (y 2 m + 1) 3/2 , ( 12
)
where y m is solution of Eq. 9. We compare this prediction with numerical evaluation of the action in Fig. 8. A careful analysis of A m , as given by Eq. (12) shows that A m is symmetric and minimum around δ = π for all α. Sequential double ionization (UDI) corresponds to the situation when one electron is first ionized by the field, leaving the inner electron in the unbound region, so that the field sequentially ionizes the remaining electron. The amount of UDI is related to the volume of the bound region, therefore to A m : the smaller the inner region (and thus A m ) the more likely it is for the inner electron to be found in the unbound region as well. Nonsequential (CDI) double ionization corresponds to the situation when the inner electron is first left in the inner region, and is then propelled, through a recollision, in the unbound region under the impact of the returning outer electron. As a result, it is easier for the outer electron to ionize the inner one if the bound region is small. Either way, for both CDI and UDI, we are expecting higher double ionization probability when A m is minimum (i.e. around δ = π) and a strong sensitivity of the double ionization yield where A m varies fast (i.e. around δ = 0). It corresponds to what is observed in Fig. 2. The small discrepancy between δ = π and the actual maximum is due to nonlinear energy exchange processes during the recollision. In Fig. 6, we see that the kinetic energy of the outer electron upon return is not symmetric about δ = π. In particular we see that the maximum return energy is rather small around π in comparison with slightly lower values of δ which explains that the maximum is expected to be slightly smaller than π.

The rate at which A m changes with δ provides an explanation for the sensitivity of the probability to changes in δ which was discussed in Sec. II A. The action changes quickly in δ in the regions around δ = 0 and slowly around δ = π, which is the same trend observed for the rate of change in probability with δ from Fig. 2. This explains Observation 1 from the statistical analysis, in Sec. II, namely that the double ionization yield is highly sensitive to the choice of phase lag.

E. Saturation of the double ionization probability

As explained before, CDI corresponds to the case where both electrons are left in the unbound region. An estimate of the size of the bound region is implicitly obtained through Eq. ( 9). An approximation of the amplitude of the field at which there is no inner region leads to an estimate of the intensity at which double ionization is saturated, meaning that all the trajectories double ionize. Equation ( 9) has a real solution if E 0 M is smaller than 4/(3 √ 3) which leads to

E (t) 0 (α, δ) = 4 3 √ 3M (α, δ) .
The field is maximum for δ = π and its value is M (α, π) = 1 + α. For this value of δ the critical value of the intensity at which all the trajectories double ionize is minimum. This is what is observed in Fig. 1 and4.

A numerical estimate of the range of variation of this critical intensity as a function of δ, corresponding to the amplitude of variation of intensity between E (t) 0 (α, 0) and E (t) 0 (α, π) gives 1.1 × 10 16 W • cm -2 at α = 0.45 which is what is observed in Fig. 1. A more precise approximation of the value of the intensity (as a function of the laser parameters) at which the bound region disappears is obtained by following the elliptic periodic orbit (with the period of the laser field) at the center of the elliptic island [29]. It explains Observation 2 from the statistical analysis, in Sec. II, namely that the intensity of saturated double ionization greatly varies with the choice of the laser parameters.

F. Maximum of CDI

When one electron is bound to the nucleus after the other one has ionized, the only way through which it leaves the nucleus is by an exchange of energy with the outer electron when this one comes back to the core upon sign reversal of the laser field. Through this recollision the inner electron jumps from one invariant torus to another one (depending on the amount of energy exchanged), and it can even jump from one invariant torus to the unbound region. After it enters into the unbound region, it ionizes (solely due to the action of the field). A nonsequential double ionization corresponds to the situation when the exchange of energy allows the inner electron to jump from the bound region to the unbound one, while the outer electron keeps enough energy to remain ionized. The amount of energy exchanged as well as the size of the bound region are the two main reasons for the sensitivity of the nonsequential double ionization to changes in the parameters α and δ.

Using an equal sharing of the energy of the inner and outer electron [29], we derive a prediction for the intensity when the CDI probability is maximum. At (y, p y ) = (0, 0), when the inner electron is bounded, its energy is U in = -2, while at (y, p y ) = (y m , 0), when the inner electron is on the boundary (on the invariant torus with action A m ), its energy is U in = -2/ y 2 m + 1. This leads to a difference in energy of ∆E y = 2 -2/ y 2 m + 1 which is the minimum energy required to ionize the inner electron in one recollision. This energy comes from the outer electron, and when the outer electron gives enough energy to ionize the inner electron (in one recollision) while keeping itself ionized as well.

∆E y = E max 2 , (13)
The equal sharing relation can be solved in a manner very similar to the case of the one laser system [29] by making the assumption that double ionization occurs at peak field amplitude, as given by Eq. ( 9). Combining Eq. ( 13) with Eq. ( 9) yields the following approximation for E (c) 0

E (c) 0 = 4ω √ κ - 2ω √ κ 3 2 √ M + O 2ω √ κ 3 . (14)
For numerically computed κ and M we can see from Fig. 10 that I (c) does not vary significantly with δ. Over the entire range of δ, the critical intensity I (c) varies by approximately 2×10 14 W•cm -2 which is compatible with the variations of I (c) as observed in Fig. 1. This variation is small compared to the variations of the laser intensity from 10 13 to 10 17 W • cm -2 over the entire CDI and UDI regimes. It is also small compared to the variations of I (t) which are of the order of 10 16 W • cm -2 . It confirms Observation 3 from the statistical analysis, in Sec. II, namely that the knee regime (and particularly the intensity for which CDI is maximum) is fairly insensitive to the laser parameters.

IV. CONCLUSION

Using the statistical and nonlinear dynamical analysis we showed that the laser parameters α and δ can be used to control both the probabilities of double ionization and the rates of change of these probabilities both in the CDI and UDI regimes. This control acts on the outer electron by regulating the amount of energy it can bring to the core during a recollision. For the inner electron, the control depends on its stability, by changing the size of the bound area. Those two modulations act together to determine the double ionization probability, but it is the inner electron which has the strongest impact on its variations. In a nutshell, using the inner and outer electrons reduced models, we were able to explain the three observations obtained with the statistical analysis of the system: Namely the high sensitivity of the double ionization yield and saturation of double ionization in the choice of the laser parameters and the low sensitivity of the knee regime in the same parameters. This control helps in regulating the proportion of sequential versus nonsequential processes which occur in this atomic system.

FIG. 2 :

 2 FIG. 2: Double ionization probability as a function of δ for α = 0.45 and the intensity labeled by the vertical magenta continuous line in Fig. 1 (I = 4.78 × 10 14 W • cm -2 ).

FIG. 3 :

 3 FIG.3: Double ionization probability as a function of intensity for different values of α. Blue circles correspond to α = 0 (single laser), Red crosses to α = 0.10, green squares to α = 0.45, black plus signs to α = 0.75 and magenta triangles correspond to α = 1.00. Here δ = π for all curves.

I

  FIG. 4: Double ionization probability as a function of the intensity for different values of δ. We keep α constant and equal to α = 0.1 and α = 0.75, respectively.

FIG. 5 :

 5 FIG.5: Typical nonsequential double ionization. We display the position (upper panel) and energy (as defined in the text, lower panel) of each electron as functions of time. The parameters are I = 4.78 × 10 14 W • cm -2 , α = 0.45 and δ = 0.

FIG. 6 :

 6 FIG.6: Maximum return energy of the outer electron as a function of δ in units of Up. Red squares correspond to α = 0.10, blue triangles to α = 0.45, and green circles to α = 1.00, while the dashed black line corresponds to the one laser system (α = 0).

3 FIG. 7 :

 37 FIG.7: Laminar plots of inner electron dynamics as given by Hamiltonian (4) as a function of the initial conditions (y0, py,0, t = 0). For each initial condition, we represent the distance of the electron from the nucleus after 10 laser cycles. The distance is plotted in a logarithmic scale. In the left panel δ = 0 and in the right panel δ = π. The dashed curves are Poincaré sections of two typical trajectories in the bound region taken stroboscopically with the laser period. The parameters are α = 0.45 and I = 4.78 × 10 14 W • cm -2 in both plots.

FIG. 8 :

 8 FIG.8: Action Am of the outermost invariant torus as a function of δ for α = 0.45 and I = 4.78 × 10 14 W • cm -2 . The solid line corresponds to the statistical data, while the dashed dotted line corresponds to the theoretical model, as given by Eq. (12).

• cm - 2 )FIG. 10 :

 210 FIG.10: Expansion of I (c) , where κ and M have been solved for numerically. For this plot α = 0.45, k = 3, and ω = 0.0584.

V. ACKNOWLEDGEMENTS C.C. and F.M. acknowledge financial support from the PICS program of the CNRS. This work is partially funded by NSF.