Keywords: Geometric optimal control, Minimum time problems, Singular arcs

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Tangency property and prior-saturation points in planar minimal time problems

Térence Bayen,

Tangency property and prior-saturation points in planar minimal time problems

Térence Bayen * Olivier Cots * *

INTRODUCTION

In this paper, we consider minimal time problems governed by single-input control-affine-systems in the plane ẋ(t) = f (x(t)) + u(t) g(x(t)), |u(t)| ≤ 1, where f, g : R 2 → R 2 are smooth vector fields. Syntheses for such problems have been investigated a lot in the literature (see, e.g., [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF]; [START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF]; [START_REF] Piccoli | Classification of generic singularities for the planar time-optimal synthesis[END_REF]; [START_REF] Sussmann | Regular synthesis for time-optimal control of single-input real analytic systems in the plane[END_REF]). An exhaustive description of the various encountered singularities can be found in [START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF], as well as an algorithm leading to the determination of optimal paths. It is worth mentioning that even though many techniques exist in this setting, the computation of an optimal feedback synthesis (global) remains in general difficult because of the occurence of geometric loci such as singular arcs, switching curves, cut-loci... Our aim in this work is to study properties of the minimal time synthesis when saturation occurs on a singular arc 1 . Saturation means that the singular control u s (which allows the associated trajectory to stay on the singular locus) becomes non admissible, that is, |u s | > 1. Such a situation naturally appears in several application models, see, e.g., [START_REF] Bayen | Timeoptimal control of concentration changes in the chemostat with one single species[END_REF][START_REF] Bayen | Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc[END_REF]; [START_REF] Ledzewicz | Antiangiogenic therapy in cancer treatment as an optimal control problem[END_REF]; [START_REF] Rapaport | Dynamical modeling and optimal control of landfills[END_REF]; [START_REF] Bakir | Geometric optimal control techniques to optimize the production of chemical reactors using temperature control[END_REF]. The occurence of such a phenomenon implies the following (non-intuitive) property that, if a singular arc is optimal, then it should leave the singular locus at a so-called prior-saturation point before reaching the saturation point. This property has been studied in the literature in various situations such as for control-affine systems in dimension 2 and 4 (see, e.g., [START_REF] Schättler | A synthesis of timeoptimal controls in the presence of saturated singular arcs[END_REF]; [START_REF] Schättler | Geometric optimal control[END_REF]; [START_REF] Bonnard | Toward a geometric theory in the time-minimal control of chemical batch reactors[END_REF]; [START_REF] Bayen | Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc[END_REF] and references herein).

1 Singular arcs appear when the switching function vanishes over a time interval.

Our main goal in this paper is to provide new qualitative properties on the minimum time synthesis when saturation occurs:

• We shall first give a set of conditions that guarantee occurence of a prior-saturation point showing that the system leaves the singular arc at this point (before reaching the saturation point) with the maximal value for the control, see Proposition 5. This last arc is usually called bridge following the terminology as in [START_REF] Bonnard | Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance[END_REF][START_REF] Bonnard | Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance[END_REF] (see also [START_REF] Bonnard | Time minimal synthesis for planar systems in the neighborhood of a terminal manifold of codimension one[END_REF]; [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF]). • We shall also introduce a shooting function that allows an effective computation of the prior-saturation point. This mapping is used to show our main result (Theorem 15) which can be stated as follows: when the system exhibits a switching curve emanating from the prior-saturation point, then this curve is tangent to the bridge (in the cotangent bundle) at this point.

The tangency property (in the state space) has been pointed out in several application models (see, e.g., [START_REF] Bayen | Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc[END_REF]; [START_REF] Bonnard | Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance[END_REF]). To the best of our knowledge, this fundamental property in the synthesis has not been addressed previously in this general setting in the literature.

The paper is structured as follows: in Section 2, we introduce the saturation phenomenon via Pontryagin's Principle. In Section 3, we provide a set of conditions involving the target set and the system ensuring occurence of the prior-saturation phenomenon. In Section 4, we show the tangency property between the switching curve emanating from a prior-saturation point and the bridge. Finally, we depict this geometrical property in Section 5 for a fed-batch model [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]; [START_REF] Bayen | Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc[END_REF]. Because of brevity, some proofs of the results presented hereafter can be found in [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF].

SATURATION PHENOMENON

The purpose of this section is to recall some facts about minimum time control problems in the plane that will allow us to introduce the saturation phenomenon. Throughout the paper, the standard inner product in R 2 is written a•b for a, b ∈ R 2 , and a ⊥ denotes the vector a ⊥ := (-a 2 , a 1 ) orthogonal to a.

Pontryagin's Principle

We start by applying the classical optimality conditions provided by the Pontryagin Maximum Principle (PMP), see [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]. Let f, g : R 2 → R 2 be two vector fields of class C ∞ and the system ẋ(t) = f (x(t)) + u(t) g(x(t)),

(1) with admissible controls in the set U := {u : [0, +∞) → [-1, 1] ; u meas.}. Given an initial point x 0 ∈ R 2 and a non-empty closed subset T ⊂ R 2 , we focus on the problem of driving (1) in minimal time from x 0 to the target set T :

inf u∈U T u s.t. x u (T u ) ∈ T , (2)
where x u (•) denotes the unique solution of (1) associated with the control u such that x u (0) = x 0 , and T u ∈ [0, +∞] is the first entry time of x u (•) into the target set T . We suppose hereafter that optimal trajectories exist and we apply the PMP on (2). The Hamiltonian associated with (2) is the function

H : R 2 × R 2 × R × R → R defined as H(x, p, p 0 , u) := p • f (x) + u p • g(x) + p 0 .
If u is an optimal control and x u is the associated trajectory steering x 0 to the target set T in time T u ≥ 0, the following conditions are fulfilled:

• There exist p 0 ≤ 0 and an absolutely continuous function p : [0, T u ] → R 2 satisfying the adjoint equation ṗ(t) = -∇ x H(x u (t), p(t), p 0 , u(t)) a.e. t ∈ [0, T u ].

(3) • The pair (p 0 , p(•)) is non-zero.

• The optimal control u satisfies the Hamiltonian condition almost everywhere over [0,

T u ] u(t) ∈ argmax ω∈[-1,1] H(x u (t), p(t), p 0 , ω). (4)
• At the terminal time, the transversality condition 2 p(T u ) ∈ -N T (x u (T u )) is fulfilled.

In the sequel, we only consider normal extremals that is, we take p 0 = -1, and we shall then write H(x, p, u) in place of H(x, p, p 0 , u). Since T u is free and (1) is autonomous, the Hamiltonian H is zero along any extremal: for a.e. t ∈ [0, T u ],

H = p(t) • f (x u (t)) + u(t)p(t) • g(x u (t)) + p 0 = 0. (5) The switching function φ is defined as φ(t) := p(t) • g(x u (t)), t ∈ [0, T u ],
(6) and it gives us (thanks to (4)) the following control law:

φ(t) > 0 ⇒ u(t) = +1, φ(t) < 0 ⇒ u(t) = -1. (7)
A switching time is an instant t c ∈ (0, T u ) such that the control u is discontinuous at time t c . We say that the

(t) = p(t) • [f, g](x u (t)), t ∈ [0, T u ], φ(t) = p(t) • [f, [f, g]](x u (t)) + u(t) p(t) • [g, [f, g]](x u (t))
almost everywhere over [0, T u ]. The notation [f, g](x) stands for the Lie bracket of f and g at point x. The singular locus ∆ SA (in the state space) is defined as the (possibly empty

) subset of R 2 ∆ SA := {x ∈ R 2 ; det(g(x), [f, g](x)) = 0}. (8) For future reference, we set δ SA (x) := det(g(x), [f, g](x)) for x ∈ R 2 . Note that if an extremal is singular over a time interval [t 1 , t 2 ], then one has x u (t) ∈ ∆ SA for any t ∈ [t 1 , t 2 ] because p(•) must be non-zero and orthogonal to the vector space span{g(x u (t)), [f, g](x u (t))} over [t 1 , t 2 ].
The singular control u s is then the value of the control for which the trajectory stays on the singular locus ∆ SA . Supposing then that φ = φ = 0 over [t 1 , t 2 ] gives:

u s (t) := - p(t) • [f, [f, g]](x u (t)) p(t) • [g, [f, g]](x u (t)) , t ∈ [0, T u ], (9)
provided that p(t) • [g, [f, g]](x u (t)) is non zero for t ∈ [t 1 , t 2 ]
. This expression of the singular control does not guarantee that u s is admissible, that is, u s (t) ∈ [-1, 1]: [START_REF] Bonnard | Time minimal synthesis for planar systems in the neighborhood of a terminal manifold of codimension one[END_REF]; [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF]). • When we have |u s (t)| > 1 for some instant t, we say that a saturation phenomenon occurs and that the corresponding points of the singular locus are parabolic (see [START_REF] Bonnard | Time minimal synthesis for planar systems in the neighborhood of a terminal manifold of codimension one[END_REF]; [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF]).

• When we have u s (t) ∈ [-1, 1], the point x u (t) is said hyperbolic if p(t) • [g, [f, g]](x u (t)) > 0, and elliptic if p(t) • [g, [f, g]](x u (t)) < 0 (see
Our purpose in what follows is precisely to investigate properties of the synthesis of optimal paths when saturation occurs.

Singular control and saturation phenomenon

We start by recalling classical expressions of the singular control that will allow us to properly define saturation points. The collinearity set associated with (1) is the (possibly empty) subset of R 2 defined as

∆ 0 := {x ∈ R 2 ; det(f (x), g(x)) = 0}. (10) Define two functions δ 0 , ψ : R 2 → R as δ 0 (x) := det(f (x), g(x)), x ∈ R 2 , and ψ(x) := - det(g(x), [f, [f, g]](x)) det(g(x), [g, [f, g]](x)) , x ∈ R 2 . ( 11
)
Using the PMP, we obtain the following expression of u s . Lemma 1. Suppose that ∆ SA = ∅, that the mapping

x → det(g(x), [g, [f, g]](x)
) is non-zero over ∆ SA , and consider a singular arc defined over an interval [t 1 , t 2 ].

Then, one has:

u s (t) = ψ(x(t)), t ∈ [t 1 , t 2 ], (12) where x(•) is the corresponding singular trajectory such that x(t) ∈ ∆ SA for t ∈ [t 1 , t 2 ].
Proof. The expression of u s as a feedback control is classical [START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF], [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF]), see [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF] for more details.

To introduce the notion of saturation point, it is convenient to consider a parametrization of ∆ SA as follows. When ∆ SA ∩ ∆ 0 is non-empty, ∆ SA \∆ 0 can be divided into several subsets (called components hereafter):

∆ SA \∆ 0 = k∈K γ k ,
where K is an index set. By using the implicit function Theorem, we can show that each component of ∆ SA can be parametrized by a one-to-one parametrization andψ(ζ(τ )) > 1 for any τ ∈ J such that τ > τ * . As well, we can define saturation points x such that ψ(x ) = -1.

ζ k : J → γ k , τ → ζ k (τ ) of class C 1 (where J is an interval of R), provided that the following condition is fulfilled ∀x ∈ ∆ SA , det(g(x), [g, [f, g]](x)) = 0. (13) Definition 2. Given a component parametrized by ζ, a point x * := ζ(τ * ) with τ * in the interior of J is called saturation point if ψ(x * ) = 1, ψ(ζ(τ )) ∈ (-1, 1) for any τ ∈ J such that τ < τ * ,

EXISTENCE OF A PRIOR-SATURATION POINT

In this section, we show that a prior-saturation phenomenon can occur whenever the system exhibits a saturation point. We start by introducing our main assumptions. Assumption 3. The system (1) satisfies the following hypotheses:

(i) One has ∆ 0 = ∅ and δ 0 (x) < 0 for all x ∈ R 2 . (ii) The set ∆ SA is non-empty, simply connected, and has exactly one saturation point x * with ψ(x * ) = 1. (iii) Along the singular locus, the strict (generalized)

Legendre-Clebsch optimality condition is satisfied, that is, any singular extremal

(x u (•), p(•), u(•)) defined over [t 1 , t 2 ] satisfies: ∂ ∂u d 2 dt 2 ∂H ∂u (x u (t), p(t), u(t)) > 0, ∀t ∈ [t 1 , t 2 ]. ( 14
)
(iv) If Γ -is the forward semi-orbit of (1) with u = -1 with the initial condition x * at time 0, then

T ∩ Γ -= ∅. (15) (v) The target T is reachable from every point x 0 ∈ R 2 . Remark 4. (i)
The hypothesis ∆ 0 = ∅ is not restrictive since we could restrict our analysis to a component γ of ∆ SA in place of ∆ SA . (ii) By the previous computations, we can observe that ( 14) is equivalent to det(g(x), [g, [f, g]](x)) > 0, ∀x ∈ ∆ SA , and that, under the strict Legendre-Clebsch condition, the singular arc is a turnpike, [START_REF] Bonnard | Time minimal synthesis for planar systems in the neighborhood of a terminal manifold of codimension one[END_REF].

The next proposition shows that an extremal trajectory containing a singular arc until the point x is not optimal. Proposition 5. Suppose that Assumption 3 holds true, and consider an optimal trajectory steering x 0 to the target T in time T u . Then, the corresponding extremal (x u (•), p(•), u(•)) does not contain a singular arc defined over a time interval [t 1 , t 2 ] such that x u (t 2 ) = x * .

Proof. We refer to [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF].

As an example, if x 0 := ζ(τ 0 ) belongs to the singular locus with τ 0 < τ * , and if an optimal trajectory starting from x 0 contains a singular arc, then the trajectory should leave the singular locus before reaching x * . Let us insist on the fact that this property of leaving the singular locus before reaching x * relies on the fact that the optimal trajectory should contain a singular arc. In the fed-batch model presented in Section 5, this property can be easily verified (see [START_REF] Bayen | Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc[END_REF]).

We now introduce the following definition (in line with [START_REF] Ledzewicz | Antiangiogenic therapy in cancer treatment as an optimal control problem[END_REF]; [START_REF] Schättler | A synthesis of timeoptimal controls in the presence of saturated singular arcs[END_REF]; [START_REF] Schättler | Geometric optimal control[END_REF]). Hereafter, the notation S [τ 0 ,τ0] denotes a singular arc passing through the points ζ(τ 0 ) and ζ(τ 0 ) with τ 0 ≤ τ 0 < τ * . Definition 6. Let τ 0 < τ * . A point x e := ζ(τ e ) ∈ ∆ SA with τ 0 < τ e < τ * is called a prior-saturation point if the singular arc S [τ0,τ ] ceases to be optimal for τ ≥ τ e . This definition makes sense only for initial conditions ζ(τ 0 ) with τ 0 < τ * because for τ 0 ≥ τ * , optimal controls are not singular (since the singular control is non-admissible). We highlight the dependency of x e w.r.t. initial conditions ζ(τ 0 ) ∈ ∆ SA as follows. Proposition 7. Suppose that Assumption 3 holds true and that there are τ 1 , τ 2 ∈ J with τ 1 < τ 2 < τ * such that any optimal trajectory starting from ζ(τ 0 ) with 16) Moreover, for any τ 0 ∈ [τ e , τ * ] an optimal trajectory starting at ζ(τ 0 ) leaves the singular locus at ζ(τ 0 ).

τ 0 ∈ [τ 1 , τ 2 ) contains a singular arc S [τ0,τ2] . Then, for any initial condition τ 0 ∈ [τ 1 , τ 2 ), one has x e = ζ(τ e ) with τ e := sup{τ ∈ J ; S [τ1,τ ] is optimal} ∈ [τ 2 , τ * ). (
Proof. We refer to [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF]. This property implies in particular that for every initial conditions x 0 := ζ(τ 0 ) ∈ ∆ SA such that τ 0 ∈ [τ 1 , τ 2 ], then the corresponding optimal path has a singular arc until the point x e and a switching point at this point.

TANGENCY PROPERTY AND PRIOR-SATURATION PHENOMENON

The aim of this section is to prove the tangency property as stated in Theorem 15. In the example of Section 5, this property will hold at a certain prior-saturation lift: Definition 8. Let x e be a prior-saturation point. Any point z e in the cotangent space at x e is called a prior-saturation lift of x e .

Classically, the PMP gives a set of nonlinear equations, the so-called shooting equations, that can be solved to compute the optimal extremal trajectory. We introduce some notation usually used to define this set of shooting equations, that we will be useful here to compute priorsaturation lifts; in a general frame, we can assume that the prior-saturation lifts involved in the optimal synthesis can be computed solving a set of nonlinear equations excerpted from the shooting equations.

We define the Hamiltonian lifts associated with f and g as

H f (z) := p • f (x) ; H g (z) := p • g(x),
where z := (x, p) belongs to the cotangent bundle. All the others Hamiltonian lifts in the rest of the paper are defined like this. Define also the Hamiltonians H ± := H f ±H g and H s := H f + u s H g , where u s is viewed here as a function of z:

u s (z) := - p • [f, [f, g]](x) p • [g, [f, g]](x) = - H [f,[f,g]] (z) H [g,[f,g]] (z) . ( 17
)
For any Hamiltonian H we define the Hamiltonian system #-H := (∂ p H, -∂ x H), and finally, we introduce the exponential mapping exp(tϕ)(z 0 ) as the solution at time t of the differential equation ż(s) = ϕ(z(s)) with initial condition z(0) = z 0 , where ϕ is supposed to be smooth.

Then, from a general point of view, we shall assume that prior-saturation lifts involved in the optimal synthesis will be given by solving a set of nonlinear equations of the following form:

F (t b , z b , λ) := H [f,g] (exp(-t b # - H + )(z b )) G(t b , z b , λ) , (18)
where λ ∈ R k is a vector of k ∈ N parameters, where F is a function from R 5+k to R 5+k and where G : R 5+k → R 4+k is defined by

G(t b , z b , λ) :=   H g (exp(-t b # - H + )(z b )) H + (z b ) + p 0 Ψ(z b , λ)   , (19)
with Ψ : R 4+k → R 2+k a given function and p 0 = -1 considering the normal case. We assume that all the functions F , G and Ψ are smooth. It is important to notice that the mapping Ψ does not depend on t b and that we can replace H + by H -without any loss of generality. We emphasize one more time that this form is very general.

Let (t * b , z * b , λ * ) ∈ R 5+k be a solution to the equation F = 0 and define

z e := exp(-t * b # - H + )(z * b ) (20) such that z e ∈ Σ SA := {z ∈ R 2n ; H g (z) = H [f,g] (z) = 0}.
We introduce the following assumptions at the point z e . Assumption 9. We have H [g,[f,g]] (z e ) = 0 and u s (z e ) < 1 with u s the singular control given by ( 17). Assumption 10. The matrix 4+k) . Remark 11. Assumption 9 is related to the prior-saturation phenomenon while in combination with Assumption 10, it is related to the well-posedness of the shooting system F = 0. Besides, the point z e is locally unique under these assumptions, according to the following result. Proposition 12. Suppose that Assumptions 9 and 10 hold true. Then,

∂G ∂z b (t * b , z * b , λ * ) ∂G ∂λ (t * b , z * b , λ * ) ∈ GL 4+k (R), i.e., it is invertible in R (4+k)×(
F (t * b , z * b , λ * ) ∈ GL 5+k (R).
Proof. Note that u s (z e ) < 1 plays a role in the result. We refer to [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF] for details. Lemma 13. Suppose that Assumption 10 holds true. Then, there exists ε > 0 and a C 1 -map

t b → σ(t b ) := (z b (t b ), λ(t b )) ∈ R 4+k defined over I ε := (t * b -ε, t * b + ε), that satisfies ∀t b ∈ I ε , G(t b , σ(t b )) = 0. (21) In addition, one has σ(t * b ) = (z * b , λ * ) and σ (t * b ) = 0 R 4+k .
Proof. This result is a simple application of the implicit function theorem. We refer to [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF] for details.

Let us introduce the mapping ϕ(t b ) := exp(-

t b # - H + )(z b (t b )) for t b ∈ I ε and define Σ := {ϕ(t b ) ; t b ∈ I ε }. ( 22
)
Remark 14. The curve Σ is a switching curve in the contangent bundle since one has H g (ϕ(t b )) = 0 by definition of G. However, this switching curve is not necessarily optimal, that is, the optimal synthesis, with respect to the initial condition, may not contain Σ. Let us stratify Σ according to Σ = Σ -∪ Σ 0 ∪ Σ + , with

Σ -:= {ϕ(t b ) ; t b ∈ (t * b -ε, t * b )}, Σ 0 := {ϕ(t * b )} = {z e }, Σ + := {ϕ(t b ) ; t b ∈ (t * b , t * b + ε)}.
A typical situation is when Σ -∪ Σ 0 is contained in the optimal synthesis while Σ + is not optimal for local and/or global optimality reasons. This is for instance the case in the example of Section 5.

Our first main result is given by Proposition 7 which states the existence of a prior-saturation point x e in the state space under Assumption 3. Our second main result is the following. Theorem 15. Suppose the existence of a triple (t

* b , z * b , λ * ) ∈ R 5+k such that F (t * b , z * b , λ * ) = 0, with F defined by (18) and set z e := exp(-t * b # - H + )(z * b )
. Suppose also that Assumption 10 holds true. Then, the switching curve Σ given by ( 22) is tangent at z e to the forward semi-orbit

Γ + of ż = # - H + (z) starting from z e .
Proof. This result is a consequence of Lemma 13. We refer to [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF] for details.

Remark 16. It is worth to mention that the tangency property is proved in the cotangent bundle, and thus it is also true in the state space at a prior saturation point (under the assumptions of Proposition 5). Remark 17. In [START_REF] Bakir | Geometric optimal control techniques to optimize the production of chemical reactors using temperature control[END_REF], the following seminormal form called bridge model

ẋ1 = u(1 -x 2 ), ẋ2 = u(1 -x 1 ) + 1 -x 2 1 x 2 ,
is introduced. The associated singular locus is the union of two (simply connected) curves and has a singularity at their intersection. The authors solve the associated minimal time control problem from the initial condition (-1, -1) to the target (0, 0), which exhibits a priorsaturation point and a bridge connecting the two singular loci. This generic model is a good illustration of the tangency property; indeed, for the time minimal synthesis the switching curve emanating from the prior-saturation point is tangent to the bridge. However, it is important to notice that the tangency property holds in a more general framework. We present in the following section an optimal synthesis which exhibits a prior-saturation point from where starts a bridge connecting the singular locus to the extended target set, and at which the tangency property holds.

ILLUSTRATION OF THE PRIOR-SATURATION PHENOMENON

A bioreactor operated in fed-batch is described by the controlled dynamics (see [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]):

       ṡ = -µ(s) M v + s in -s + Q max (1 + u) 2v (s in -s), v = Q max 2 (1 + u), (23)
where s in and s denote respectively the input substrate and substrate concentrations, and v is the volume of the reactor3 . The parameter Q max > 0 is the maximal speed of the input pump (chosen large enough),

T u s.t. (s(T u ), v(T u )) ∈ T , (24)
where T := (0, s ref ] × {v max } is the target set, s ref s in is a given threshold, and v max > 0 denotes the maximal volume of the bioreactor. For more details about this system, we refer to [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]; [START_REF] Bayen | Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc[END_REF].

It appears that Problem (24) may exhibit a saturation phenomenon. Indeed, by using the PMP, we can check that there is a singular locus that is the line segment ∆ SA := {s * } × (0, v max ], and that the singular control can be expressed as

u s [v] := µ(s * ) [M + v(s in -s * )] (s in -s * )Q max -1,
(writing ṡ = 0 along s = s * ). It follows that there is a unique saturation point

x sat := (s * , v * ) with v * := 2Qmax µ(s * ) -M sin-s * and u s [v * ] = 1 if the following condition is fulfilled 0 < v * < v max .
(25) This typically happens when v max (the volume of water to be treated) is too large, see [START_REF] Bayen | Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc[END_REF]. Next, we suppose that (25) holds true.

At this step, we wish to know if prior-saturation occurs (according to Propositions 5 and 7). Doing so, let us check Assumption 3. One gets [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF]). Now, the singular arc is of turnpike type and Legendre-Clebsch's optimality condition holds true because µ has a unique maximum for s = s * , see [START_REF] Bayen | Optimal synthesis for the minimum time control problems of fed-batch bioprocesses for growth functions with two maxima[END_REF]. In addition, observe that, in the (s, v)-plane, trajectories of ( 23) with u = -1 are horizontal, hence, every arc with u = -1 and starting at a volume value v 0 < v max never reaches the target set T . Finally, T is reachable from D taking the control u = +1 until reaching v = v max and then u = -1 until reaching s ref .

δ 0 (s, v) = -µ(s)(M/v + s in -s)Q max /2 < 0, hence ∆ 0 ∩ D = ∅ and δ 0 < 0 in D (see
Second, let us verify the hypotheses of Proposition 7. Doing so, let v → ŝ(v) be the unique solution to the Cauchy problem

ds dv = -- µ(s) Q max M v + s in -s + s in -s v , s(v max ) = s * ,
(the solution of ( 23) with u = 1 backward in time from (s * , v max )). From [START_REF] Bayen | Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc[END_REF], if there exists v * ∈ (0, v * ) such that ŝ(v * ) = s * , then optimal paths starting at a volume value sufficiently small necessarily contain a singular arc (this actually follows using the PMP). Now, by using Cauchy-Lipschitz's Theorem, the existence of v * is easy to verify when M = 0, and thus, it is also verified for small values of the parameter M (by a continuity argumentation). To pursue our analysis, we suppose next the existence of v * ∈ (0, v * ). We are then in a position to apply Propositions 5 and 7. It follows that there is a unique volume value v e ∈ (0, v * ) such that any singular arc starting at a volume value v 0 < v e will be optimal only until v e . In addition, combining this result with a study of extremals using the PMP, we obtain that

• if the initial condition is (s * , v 0 ) with v 0 < v e , then the optimal path is of the form σ s σ b + σ -(see below for the definition of σ b + ); • if the initial condition is (s * , v 0 ) with v 0 ≥ v e , then
the optimal path is of the form σ + σ -; • for any initial condition (s in , v 0 ) with v e ≤ v 0 < v max , the optimal path is of the form σ -σ + σ -where the first switching time appears on a switching curve emanating from (s * , v e ).

To determine the prior-saturation point x e := (s * , v e ) numerically, we proceed as in Section 4. For this application model, it is convenient to introduce an extended target set as T := (0, s in ]×{v max } (observe that for initial conditions on T , optimal paths are σ -arcs). In this context, a bridge is defined as an arc σ + (denoted by σ

b + ) on [0, t b ] such that φ(0) = φ(0) = φ(t b ) = 0 and v(t b ) = v max ,
where φ is the switching function defined by (6) and t b is the time to steer x e at time 0 to the extended target set T with u = +1. To compute x e , we need to compute the extremities of the bridge together with its length. Denoting by t * b the length of the bridge and by z * b its extremity in the cotangent bundle whose projection on the state space belongs to T , the point (t * b , z * b ) is then a solution of the equation F bio = 0 with

F bio (t b , z b ) :=       H [f,g] (exp(-t b # - H + )(z b )) H g (exp(-t b # - H + )(z b )) H + (z b ) + p 0 H g (z b ) v b -v max       , (26)
where (s b , v b ) is the projection of z b on the state space and vector fields f, g are given by ( 23). From Theorem 15, the bridge is then tangent to the switching curve at x e (the projection of Σ given by ( 22) onto the state space).

To conclude this part, let us comment Fig. 1 on which the optimal synthesis is plotted in a neighborhood of the prior-saturation point:

• In black, the switching curve Σ π emanates from the prior-saturation point. It is computed using the shooting functions F bio . • The synthesis is such that trajectories are horizontal (u = -1) until reaching ∆ SA or the switching curve.

For initial conditions with a substrate concentration less than s * and v 0 ≥ v e , then u = 1 is optimal until reaching T .

CONCLUSION

Even though the tangency property between the bridge and the switching curve provides useful informations on the minimum time synthesis when prior saturation occurs (typically, under assumptions of Proposition 5), it remains valid in a larger context (under the hypotheses of Theorem 15) and not only in the framework of saturation and prior-saturation of the singular control for affine-control systems in the plane. It also appears in other settings such as in Lagrange control problems, see, e.g., [START_REF] Kalboussi | Optimal control of membrane-filtration systems[END_REF]. Future works could then investigate priorsaturation phenomenon and the tangency property in other frameworks or in dimension n ≥ 3.

Fig. 1 .

 1 Fig. 1. Minimal time synthesis for (24): the target set T = (0, s ref ]×{v max } is in black (left). The switching curve Σ π (in black) is tangent to the bridge σ b + (in red) at x e . Arcs with u = +1 (resp. u = -1) are depicted in red (resp. in blue).

  := √ KK I ∈ (0, s in ) (parameters µ h , K, K I are positive). This type of growth function models inhibition by substrate.It is worth mentioning that D := (0, s in ] × R * + is invariant by (23). For waste water treatment purpose, the problem of interest is: inf

	Qmax 2 (1 + u) represents the input flow rate, u(•) being the control
	variable with values in [-1, 1], and M ∈ R is a constant.
	The kinetics µ of the reaction is of Haldane type
	µ(s) :=	µ h s K + s + s 2 K I	,
	with a unique maximum s u∈U

*

We chose to adopt the notation (s, v) in place of (x 1 , x 2 ) as in the bioprocesses literature for fed-batch operations.

ACKNOWLEDGEMENTS

We are very grateful to E. Trélat for helpful discussions about the tangency property.


Serge Blondel
email: serge.blondel@univ-angers.fr

Radu Vranceanu
email: vranceanu@essec.edu

COVID-19 mortality and health expenditures across European countries: The positive correlation puzzle

Keywords: COVID-19, health care systems, Europe, efficiency, mortality, health policy JEL CLASSIFICATION -H12, I18

The positive correlation between health share expenditures and COVID-19 cumulative case fatalities in a cross-section of 31 European countries is puzzling. The positive relationships is also detected in weighted OLS and IV models that control for many usual suspects of the COVID-19 mortality: (1) health indicators (personal risk factors, medical resources), (2) virus ease of circulation, (3) macroeconomic variables related to the economic development and social orientation of the country. COVID-19 case fatalities are lower in countries with significant resources dedicated to health care (hospital beds and medical doctors); the contribution of virus circulation factors is less significant. Policy implications follow.

INTRODUCTION

The SARS-CoV-2 virus, leading to the COVID-19 disease, was and still is a challenge for epidemiologists and medical doctors. While its structure is relatively simple, and its genetic code was quickly deciphered, the virus was surrounded by major uncertainty regarding its dynamics of contamination, potential for muting, and mechanism of destroying human health [START_REF] Beeching | Coronavirus disease 2019 (COVID-19)[END_REF][START_REF] Adam | Special Report: The simulations driving the world's response to COVID-19. How epidemiologists rushed to model the coronavirus pandemic[END_REF]. This research note adds an economic puzzle to the long list of medical and epidemic puzzles related to the COVID-19: the positive correlation between the cumulated fatalities and the share of health expenditures to GDP across 31 European countries.

We choose to focus on European countries because the epidemy hit them at the same period, all have universal health care systems that cover all their citizens for all expenses related to major illnesses and would share common civic values to guide policymakers.

The first wave of COVID-19 hit Europe in early March 2020, reached a peak in April then gradually faded away. According to the European Center of Disease Prevention and Control, if on late April the 14-day COVID-19 death notification was as high as 105, it fell to 4.1 on August 2, 2020. Therefore, late July can be considered as indicating the end of the first wave of the COVID-19 in Europe. As of July 25th, 2020, the COVID-19 crisis caused 20.19 deaths per 100,000 people in the EU.

The probability of dying from the COVID-19 compounds effects of three significant stages: (i) being contaminated, (ii) becoming symptomatically sick and (iii) falling into a serious condition. Contamination depends on the circulation of the virus, while chances of recovery depend on the patient's own risk factors (age, obesity...) and the availability of heavy health equipment and highly skilled medical staff.

Policymakers strived to contain the most dramatic consequences of the epidemic by taking measures that apply to both the contamination stage, and the diagnostic and treatment stage. These national policies, as applied to the COVID-19 epidemic, varied markedly from one country to another. With the notable exception of Sweden, governments implemented lockdown to increase social and physical distance. Some countries moved very fast to closing borders; other countries had a more gradual approach. Testing and patient care strategies also differed from one country to another. First and foremost, countries presented quite different degrees of preparedness for such a crisis in terms of resources, organization of the health care system, and leadership [START_REF] Al Saidi | Decisive leadership is a necessity in the COVID-19 response[END_REF].

Hence, the COVID-19 death toll varies greatly from one country to another, from zero in Cyprus to 85 per 100,000 people in Belgium (Figure 1). FIGURE 1 COVID-19 cumulated case fatalities per 100,000 people as of July 25, 2020 Significant variation is also recorded across European countries with respect to the share of health expenditures to GDP, from less than 6% in Romania, to 11.3% in France and Germany, with a maximum of 12.3% in Switzerland (2018, Eurostat) (Figure 2). What relationship should one expect between the health share to GDP and the cumulated COVID-19 deaths?

Intuitive reasoning might suggest a negative link: the more resources a country dedicates to health care, the more lives should be saved during a sanitary crisis, both at stage one, virus circulation, and stage two, treatment. For instance, the faster a country can generalize testing and isolate contaminated persons, the lower will be the fatalities.

On the other hand, a positive correlation can be driven by the relationship between another variable and each of the COVID-19 case fatalities and health expenditures. For instance, a higher proportion of old age persons is associated with both a higher share of health expenditures and a higher COVID-19 mortality. If the correlation between the health share and the death rate is still present when controlling for all other observed variables that jointly impact the two variables (age, temperatures, hospital beds, obesity, etc.), then the correlation might reveal differences in the efficiency of the health care systems. All other things being equal, the higher the efficiency of the health care system, the lower should be the health care expenses, and the lower should be the COVID19 death rate (see Online Appendix 1 for the formal explanation).1

Figure 3 presents on the horizontal axis the share of health expenditures to GDP (in 2018) and the cumulated COVID-19 deaths on the vertical axis. The distribution of the plots suggests a positive association between the two variables (correlation coefficient 0.51). A similar positive correlation, albeit nonsignificant, was detected by [START_REF] Khan | Healthcare capacity, health expenditure, and civil cociety as predictors of COVID-19 case fatalities: A global analysis[END_REF] in a sample of 86 developed and developing countries (with very different health care systems). This research notes studies whether the positive correlation between the COVID-19 death rates in Europe and the share of health care expenditures holds when controlling for the usual suspects of the COVID-19 mortality and provide a positive answer. We verify that the puzzling correlation resists to IV regressions that would correct for omitted variables and endogeneity. A lower health share coefficient in IV compared to OLS regressions suggests that the omitted variable (efficiency) hypothesis cannot be ruled out. Our analysis also reveals the factors that contributed the most to the COVID-19 first-wave mortality by country. It emphasizes the essential role of medical resources in the fight of the epidemic.

THE DATA

Our sample comprises 31 countries: the 27 EU countries, the UK and the EFTA large countries (Iceland, Norway, Switzerland). With a population of 529.06 million people in 2018, they represent a large proportion of all European countries.

Our dependent variable is the COVID-19 total deaths per 100,000 people (or death rate) from the beginning of the epidemic until July 25, 2020. This is a noisy measure, since not all deaths were correctly imputed to the COVID-19 due to test shortages and false negative tests, and some deaths determined by other causes might have been wrongly attributed to COVID-19. When studying the efficiency of the health care systems, it might be meaningful to consider the ratio of fatalities to the number of contaminated people; unfortunately, this indicator can hardly serve for international comparisons, given the initial shortage of tests in many countries, and the substantial differences in testing strategies.

The analysis considers several explanatory variables, grouped in three categories.

I. Health indicators.

A key variable is the share of health expenditures to GDP (in percentage points) (Figure 2). The Eurostat data are available for 2018 (thus the COVID-19 extra spending is not included).

Within the health care system, human resources (medical doctors and nurses) and hospital beds (ICU mainly) were key resources in fighting the COVID-19. Both human and material resources were stretched at the peak of the crisis, with large differences from one country to another. Old age and overweight were pointed out as major factors of risk (inter alia, Al Saidi et al., 2020;[START_REF] Beeching | Coronavirus disease 2019 (COVID-19)[END_REF][START_REF] Ecdpc | Coronavirus disease 2019 (COVID-19) in the EU/EEA and the UKeleventh update: resurgence of cases[END_REF]; they also contribute to higher health expenses as they are associated to many critical diseases, such as cancer or heart fragility.

II. Virus circulation.

Temperature might influence the spread of COVID-19. Population density could also have played a role. The less a country allows for travel and exchange with the rest of the world in the first stages of the epidemic, the lower the speed of contamination. As a proxy for openness, we use the share of exports in GDP.

III. Macroeconomic variables.

Public debt to GDP is a proxy for the strain imposed on governments in taking decisive action against the COVID-19. The Gini inequality index is also a proxy for the social orientation of the institutions. 2As auxiliary variables, that are useful for the regression analysis, we present the country population and its real GDP per capita or Affluence.

THE ECONOMETRIC ANALYSIS

The COVID-19 case fatality rate can be interpreted as the probability of death, from the beginning of the pandemic to the 25 th of July, for an individual living in a given country.

As other death probability models, a model based on variables measuring country averages builds on an underlying individual death probability model. It is therefore meaningful to implement weighted regressions, using the population share as a weight for each country [START_REF] Wooldridge | Introductory Econometrics: A Modern Approach[END_REF]. 4One element of interest is the relationship between the health share expenditures and the COVID-19 fatalities. As we argued before, the coefficients in OLS regressions can be biased, due to endogeneity problems (for instance, old age is determining both health expenses and is a COVID-19 risk factor), and omitted variable (the unobserved efficiency of the health system).

To correct for these potential problems, we also implement a set of IV regressions. Real GDP/capita (or Affluence) is a natural candidate to be the instrument, since it is correlated with the health share but not with the COVID-19 death rate. Why, in our sample, affluence would not be correlated with the death rate? Because all of the 31 countries run universal health care systems that cover all expenses related to the major illnesses. The treatment of COVID-19 did not require expensive drugs, only a dedicated medical staff and ICU hospital beds for the most difficult cases. 5 On the other hand, the affluence should be positively correlated with the share of health care expenditures at country level, for many reasons. In a Belassa-Samuelson framework, richer countries have higher wages, and a higher productivity in the production of tradable goods. Because health care is a non-tradable good and countries should feature a similar productivity, richer countries should have higher health shares. Other explanations were provided to this documented correlation [START_REF] Hall | The value of life and the rise in health spending[END_REF]. [START_REF] Getzen | Health care is an individual necessity and a national luxury: applying multilevel decision models to the analysis of health care expenditures[END_REF] suggests that health care services are a necessity good at the individual level (income elasticity lower than 1) and a luxury good at the country level. The correlation between the health share and the COVID-19 death rate holds even if controlling for all these "usual suspects". A one percentage point increase in the share of health expenditures is associated to approximately 4 more cumulated death per 100000 people. The coefficient of the health share expenditure in the IV regressions is lower compared to the OLS regressions, suggesting that the effect of a unobserved variable, such as the efficiency of the health system, could be an explanation. However, the coefficient is still positive and statistically significant, revealing a persistent puzzle. It is possible that our instrument is not powerful enough, albeit the Shea's partial R2 (higher than 0.60) and the F-stat>10 suggest that it is (see last lines of Table 3).

The results also show that medical resources appear to be powerful determinants of the success in the fight against the epidemic, a result that has also been emphasized by [START_REF] Khan | Healthcare capacity, health expenditure, and civil cociety as predictors of COVID-19 case fatalities: A global analysis[END_REF]. Old age also appears to be an important factor, which is in line with the standard facts known about the COVID-19 mortality. On the contrary, macroeconomic variables and variables related to the circulation of the virus are not significant. In line with the above-mentioned papers and theory, affluence is strongly correlated with the health share (which justifies its use as an instrument). The Gini index is negatively associated (more egalitarian countries spending relatively more on health care).

As expected, old age (a pull factor) and hospital beds/1,000 people (a push factor) are both associated with higher health shares in a given country. However, the number of doctors per 10,000 population is not associated to higher share of health expenditures.

DISCUSSION

We document a positive correlation between the share of health expenditures in the GDP prior to the epidemic and the death rates during the first wave, as it appears in a crosssection of 31 European countries.

We show that the correlation is robust to including the usual suspects in the case fatality regression model. The comparison of the health share coefficient between OLS and IV models suggest that unobserved variables, such as the efficiency of the health care system, can be at work. However, the "puzzle" is still present in IV models that strive to control for such a bias. This might be related to the quality of our instrumental variable, although theory and empirical tests suggest it is a satisfactory choice. Another explanation, difficult to verify, would involve possible heterogeneity in the reporting of the COVID case fatalities.

The COVID-19 death rate equation points out to the essential role played by the medical resources in taming COVID mortality, compared to personal risk and virus circulation factors. In the light of this result, to prepare for a would-be second wave, resource constrained governments might aim to invest in emergency measures to strengthen their hospitals, rather than to impose new lockdowns, which are less efficient and much more harmful for the economy.

In the long run, countries should aim to improve the efficiency of the health care systems, by allowing for additional flexibility following for instance the example of Germany. One should not neglect the lasting economic consequences of the lack of medical preparedness. 6The EU could help, by developing a common strategy, following the example of the countries which have had the best results in managing both the epidemic and the economic crisis.

FIGURE 2

 2 FIGURE 2Health expenditures/GDP (%) in 2018 or latest year available (Eurostat)

FIGURE 3

 3 FIGURE 3Health expenditures/GDP (and cumulated COVID-19 deaths per 100,000 people.

TABLE 1

 1 Summary statistics of the data over 31 countries 3

	Mean	SD	Min	Max

Source: Eurostat, except rate of obesity

(WHO, 2016)

, the medical doctors per 10,000 people (WHO, 2016, except for Bulgaria and Malta, 2015) the number of COVID deaths (Worldometers), the mean temperature (Climate Research Unit, University of East Anglia), and the Gini coefficient (World Bank).

TABLE 2

 2 COVID-19 fatality rate: OLS and IV regressions, weighted by the population

			Weighted OLS			Weighted IV
		Model 1	Model 2	Model 3	Model 1	Model 2	Model 3
	Health Expend./GDP	4.267***	5.406***	6.368***	3.905**	4.288***	4.705**
		(1.14)	(1.24)	(1.40)	(1.54)	(1.53)	(1.83)
	Obesity rate	1.436*	1.630*	1.217	1.433**	1.609**	1.369
		(0.77)	(0.89)	(1.08)	(0.68)	(0.73)	(0.86)
	Age (% above 80)	13.540***	16.418**	12.727	13.801***	17.222***	15.101*
		(2.88)	(6.53)	(9.81)	(2.71)	(5.82)	(8.54)
	Hospital beds/10,000	-0.079***	-0.081***	-0.077***	-0.079***	-0.080***	-0.077***
		(0.01)	(0.01)	(0.02)	(0.01)	(0.01)	(0.01)
	Medical doctors /10,000	-1.623***	-2.027***	-2.018***	-1.612***	-1.951***	-1.934***
		(0.42)	(0.65)	(0.71)	(0.39)	(0.54)	(0.57)
	Mean temperature		1.044	1.105		0.738	0.723
			(0.72)	(1.08)		(0.72)	(0.95)
	Density		0.004	-0.000		0.011	0.010
			(0.02)	(0.02)		(0.02)	(0.02)
	Exports / GDP		0.289	0.330		0.266	0.286
			(0.25)	(0.25)		(0.20)	(0.19)
	Public debt / GDP			0.050			0.033
				(0.17)			(0.14)
	Gini index			1.005			0.576
				(1.07)			(0.97)
	Constant	-14.634	-54.521	-74.720*	-13.167	-48.378	-59.025*
		(30.10)	(46.44)	(42.07)	(26.37)	(37.49)	(35.59)
	R²	0.804	0.840	0.846	0.803	0.837	0.841
	N	31	31	31	31	31	31
	Legend: * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Standard errors between brackets.

TABLE 3

 3 The health share equation: First equation in the 2SLS IV regressions (population weights)

		Model 1 IV	Model 2 IV	Model 3 IV
	Obesity rate	0.111	0.037	0.093
		(0.080)	(0.084)	(0.074)
	Age (% above 80)	0.962***	0.367	0.825**
		(0.272)	(0.304)	(0.299)
	Hospital beds/10,000	0.003***	0.003**	0.002*
		(0.0008)	(0.001)	(0.001)
	Medical doctors/10,000	-0.015	0.024	0.018
		(0.031)	(0.010)	(0.037)
	Mean temperature		-0.013	-0.012
			(0.073)	(0.070)
	Density		0.003	0.003*
			(0.002)	(0.001)
	Exports to GDP		-0.030**	-0.032***
			(0.013)	(0.009)
	Debt to GDP			-0.007
				(0.007)
	Gini index			-0.145*
				(0.007)
	Affluence	0.122***	0.113***	0.101***
		(0.018)	(0.028)	(0.025)
	Constant	-2.916	1.813	4.392
		(2.372)	(3.368)	(3.059)
	N	31	31	31
	R2	0.787	0.837	0.867
	Shea's Partial R2	0.727	0.637	0.615
	F	44.30	16.54	16.46
	Legend: * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Standard errors between brackets.

[START_REF] Newhouse | Commentary on Getzen's "Aggregation and the measurement of health care costs[END_REF] argues that the organization of the health care systems has an important role in determining health care expenditures.

[START_REF] Kennedy | Income distribution, socioeconomic status, and self-rated health: a U.S. multi-level analysis[END_REF] find that higher Gini coefficients across US states were associated with lower levels of overall health.

The correlation matrix is presented in the Online Appendix.

The results of the unweighted regressions are similar. See Online Appendix 4.

As a caveat, richer people could be in a better health[START_REF] Semyonov | Where wealth matters more for health: The wealth-health gradient in 16 countries[END_REF], thus could be more resistent to the COVID-19.

As an example, in the first two quarters of 2020, the GDP fell by 2% and 10.1% in Germany, compared to 5.9% and 13.8% in France (Eurostat, 2020).

ONLINE APPENDIX. APPENDIX 1 The efficiency channel

In a very simple way, the health expenditures H can be written as a function of variables related to the demand (pull) and supply (push) for health services, themselves related to the personal characteristics of the population, and public choices about how many resources to dedicate to the health sector:

(1) where X is the vector of relevant variables and A is the vector of coefficients. v is an unobserved variable, that captures the efficiency of the medical system. A larger v requires less health expenditures for the same service.

Then, denoting by Y the mortality factors specific to the COVID-19 other than those included in X, such as the temperature or the degree of country openness, the COVID-19 death rate D is given by :

(2) In this expression, the coefficient µ > 0 captures the effect of the general efficiency of the health care system on the efficiency of treating COVID-19 cases. Replacing v as implicitly defined by the former equation in the second, we obtain:

(3) When the efficiency factor is brought into the picture, death rates (D) and health expenditures (H) exhibit a positive correlation. The coefficients of variables in the set X are subject to offsetting forces, since both vectors B and A contain positive scalars.

APPENDIX 2. The correlation matrix


Amna Amri
email: amna.amri@uha.fr

Antoine Vaesken

Frederic Heim

Hybrid Textile for Heart Valve Prosthesis: In Vitro Assessment of Various Construction Designs

INTRODUCTION

Textile polyester could be an advantageous material to replace faulty valve leaflets under mini-invasive surgery. Moreover, combining woven with non-woven fibers in the leaflet material could provide both mechanical resistance and favorable tissue-fabric interaction.

In this study, fine non-woven PET fibers obtained with force spinning were assembled with a woven substrate to form hybrid textiles and valve prototypes. The aim was: (1) to produce non-woven fibers, (2) to perform statistical analysis of the fibers diameter, (3) to investigate different assembling modes between the woven and the non-woven layers, (4) to assess the characteristics and the performances of the obtained materials and valve prototypes.

MATERIALS AND METHODS

SEM images and statistical analysis were used to investigate the effect of the force spinning production parameters (nozzle diameter, spin speed, viscosity…) on the obtained fibers size. Various assembling methods were then tested:

• Sewing (sample 1) • Ultrasonic welding (sample 2) • Chemical adhesion (sample 3) • Thermal adhesion (sample 4)
Following characteristics of the obtained hybrid material were then assessed: roughness, bending stiffness, permeability and valve durability (fatigue tests).

RESULTS

Regarding force spinning, increased speed and reduced nozzle size lead to smaller fibers. Moreover, with reduced viscosity the jet undergoes a higher stretching and consequently finer fibers are formed (down to 3µm). After assembling, the characteristics of the hybrid leaflets are modified as presented in the table.

Permeability(ml/cm²/min) Bending stiffness (mN.cm) Roughness (µm)

Reference woven textile 5293 3.3 * 10 -5 0.17 Chemical adhesion seems to present an advantage providing the lowest bending stiffness. With respect to the fatigue tests, 10 Mio cycles have been already reached with all valve samples and no material delamination could be observed.

CONCLUSION

The force spinning process enables obtaining a non-woven mat with controlled fibers size, which can be assembled with a woven substrate in a durable way. The characteristics of the hybrid textile are slightly different from the reference woven material, but keep in the range of what is needed for the valve application.


Tendon mechanical behavior in the case of tendinitis and Low Level Laser Therapy

Keywords: LLLT, Stiffness, Tensile behavior, Fibers recruitment, Damage

INTRODUCTION

Tendinitis is a common disease of the tendon which is generally treated using nonsteroidal anti-inflammatory drugs (NSAID). An alternative to drugs is Low Level Laser Therapy (LLLT). The aim of this study is to compare the effect of LLLT and NSAID treatment on the mechanical properties in the case of acute inflammation of tendon. The relationship between the mechanical behavior of the tendon and its multiscale fibrous microstructure is discussed.

METHOD

Tendinitis was induced in a group of male Wistar rats using collagenase. One group (DIC) was treated using diclofenac sodium. Two groups (L1J and L3J) were treated by laser irradiation with doses of 1 J and 3 J respectively. One group (TEN) was not subjected to any treatment of tendinitis. A last group was used as control (CTL). Each tendon was tested with a tensile test machine using loading-unloading sequences of increasing amplitude until rupture. At each loading, three stiffnesses were measured: secant stiffness k s ; tangent stiffness k t ; unloading stiffness k u (Fig. 1a).

MAIN RESULTS AND DISCUSSION

Representative loaddisplacement curves are shown in Fig. 1b for the DIC and L1J groups as examples. Large differences of load levels were observed. The initially increasing evolution of k t is linked to the reorientation and recruitment of the fibers. It corresponds to the so called "toe" region. When the tendon is more stretched, k t suddenly decreases. This evolution can be explained by a gradual fracture of the fibers and could be described using a damage variable. The evolution of the unloading stiffness k u with displacement was different but could also be partially explained by damage.

Figure 1 :

 1 Figure 1: (a) Example of load-displacement curve obtained for a tendon and definition of the 3 different stiffnesses measured at each cycle. (b) Representative F-u curve, k s ,k t and k u evolution for DIC and L1J groups.


G K Parks\

E Lee\

N Un\

F Mozer\

M Wilber\

E Lucek

\ I Dandouras

H Reme

J B Cao

P Canu

N Cornilleau-Wehrlin

P Decreau^

M L Goldstein^

Philippe Escoubet^

Density holes in the upstream solar wind

Keywords: Density hole PACS: 95, 30, Qd, 96, 50 Tf 96, 60 Vg

Larmor size transient structures with depletions as large as 99% of ambient solar wind density levels occur commonly upstream of Earth's coUisionless bow shock. These "density holes" have a mean duration of -17.9 ± 10.4s but holes as short as 4s have been observed. The average fractional density depletion (5n/n) inside the holes is -0.68 ± 0.14. The density of the upstream edge moving in the sunward direction can be enhanced by five or more times the solar wind density. Particle distributions show the steepened edge can behave like a shock, and measured local field geometries and Mach number support this view. Similarly shaped magnetic holes accompany the density holes indicating strong coupling between fields and particles. The density holes are only observed with upstream particles, suggesting that back-streaming particles interacting with the solar wind are important.

INTRODUCTION

This article summarizes the characteristics of Larmor radius scale transient structures recently discovered in the upstream region of Earth's bow shock [1]. These structures, called density holes, are seen very frequently in the upstream region and may be important for understanding how the colhsionless bow shock forms.

Figure 1 shows typical data from the ion and magnetic field experiments on the European Cluster-1 (top) and Chinese Double Star (DS) satellites in the vicinity of the bow shock. The magnetosheath in the energy flux spectrogram is a broad energy band while the solar wind is indicated by the narrow red band, at -2 keV/q. Cluster 1 (DS) crossed the bow shock at -0829 UT (-0922 UT) but moved in and out and the shock and magnetosheath plasma was detected several times. The weaker fluxes above the few keV/q solar wind are the backstreaming particles. The solar wind speed on this day was -600 km/s in the -Vx direction with a density n -2.5 cm"^.

The sharp dips in n that go below the average solar wind density (Usw) are density holes. The holes are followed by overshoots above Usw (-0855 and 0901 UT). Examination of data from many crossings indicates that the holes appear in bunches close to the bow shock, but more in isolation further away. Density holes have been seen all the way to Cluster apogees (~19 RE). The bulk velocities accompanying these dips behave as they do near the bow shock: Vx decreases almost to zero, and Vy and Vz deviate substantially. T increases by an order of magnitude. The right panel of Fig. 2 shows spin resolution (4s) data of a density hole detected at 0855 UT on March 2, 2005 by Cluster 1 (Double Star also detected this hole; not shown). For this hole, n at both edges increased from the solar wind value ~2 to ~3.5 cm"^, while inside the hole, it decreased to -0.45 cm"\ The bulk T rose from ~3 x 10^ °K to -2x10^ °K and Vx decreased from about -600 km s"' to ~0 km/s while Vy increased from -100 km s"' to -600 km s"\ The solar wind parameters were, B = 87inkT/B^ -1.6, Te/T, -2, VA=B/(47mmO''^ -56 km s"' and Cs = (kTe/m,)'^' -90 km s"\ Ion density holes are accompanied by magnetic holes of nearly the same shape. The left panel of Figure 2 shows a Lagrangian view of this density hole. A timing analysis from the four Cluster spacecraft assuming a planar geometry yielded a normal (-0.65, 0.75, 0.065) for the downstream edge at 0855 UT and a speed of-530 km s"' along the normal. The normal of the upstream edge was (-0.80, 0.59, 0.11) with an antisunward speed -320 km s"\ indicating a significant relative motion. Detection times for the edges by Double Star downstream indicated that the speed of the edges could have changed. The resulting downstream edge acceleration was estimated to be -56 km s"^, and the sunward edge deceleration -23 km s"^. This imphes the hole was growing at an increasing rate (Timing errors are -10%. In the case of acceleration, the errors are larger).

The size of the downstream edge at Cluster was deduced to be -1300 km, the upstream edge -2300 km, and the hole -3700 km (Including acceleration would make them larger). The gyroradius of 4 keV proton in B=4 nT is -1100 km. The growth time of the hole, obtained by dividing the width of the hole by the relative velocity of the two edges yields -10s (including acceleration will reduce this value). The |B| variations were in phase with the density but the B intensity at the downstream edge was -20% larger than the exterior field (-4.5 to -5.5 nT), while at the upstream edge it nearly doubled, -4 nT to -7 nT. Inside the hole, B decreased to -0.5 nT. Here Et = Eti + Et2. Measurements of B inside the hole indicated presence of small and fluctuating field (Small fluctuations are also seen in E but data have been smoothed). This density hole is a published SLAMS event and the behavior of E and B has been studied [2][3][4]. SLAMS have been studied in detail using magnetic field data but not the ions associated with them.

Power spectra obtained from two wave experiments [5,6] for the density hole discussed above are complex and not all understood (Figure 3). Wave power for 5E and 5B was considerably enhanced inside and at the edges of the density hole. For 5B, power inside the hole was enhanced from a few Hz to close to the ion cyclotron frequency (-0.01 Hz -0.1 Hz) while enhancement for 5E was observed at a few kHz. These waves are in the same frequency range as the Doppler shifted ion acoustic waves frequently observed in the solar wind. We also see enhanced power for 5E above the electron cyclotron frequency inside the hole. These waves are not yet identified since electrostatic waves above the electron cyclotron frequency have not been seen previously in the solar wind (to our knowledge).

Cluster SC 1 Mmm^fsm. FIGURE 3. Top three panels show power spectra of 5E from ~2 -50 kHz. (white hne is estimated plasma frequency) and sum of 5E in spin plane from 8 Hz -4 kHz (mV^/m^/Hz) and sum of 5B (nT^/Hz). White line is calculated electron cyclotron frequency. The bottom three panels show magnetic field components and magnitude and the spacecraft potential that is a proxy for electron density.

The waves enhanced at the edges for 5B and 5E cover the frequency range a few Hz to several hundred Hz (up to electron cyclotron frequency) and the waves include whistler mode waves. The burst of broadband emissions observed up to the electron plasma frequency are probably associated with electrostatic solitary structures traveling along B.

An ion distribution associated with the density hole at -0855 UT (March 2, 2005) is shown in Figure 4 (left panel). This velocity space plot is a cut normal to B through the main solar wind beam (centered) and shows the non-gyrotropic character of the plasma (B points out of the page.) This 3-spin (12 s) integration overlaps the upstream edge of the structure and is time-ahased, but still reveals an arc of ions in the upper left quadrant that appears to be a gyro-reflected component similar to those seen at the foot of collisionless shocks. Using the upstream sunward edge normal direction and the speed from timing analysis, we obtained a normal incidence Alfven Mach number MA = 3.3. The upstream exterior B was about (-1,-1,-3) nT, which was > 80° from the normal direction, indicating the solar wind flow into this structure was supercritical with a strongly perpendicular geometry. We thus have good reason to expect that a shock-like boundary might form, and the presence of reflected ions suggests an associated cross-boundary E-field as well (Figure 2) A feature not apparent in the left panel of Figure 4 is revealed in a distribution (right panel) obtained interior to a density hole observed at -0516 UT on 3 February 2002, identified as a SLAMS (Short Large-Amplitude Magnetic Structures) event [7]. In this plot, the slice is through a plane defined by Vsw ^ B with B pointing to the right, and again the main solar wind beam is centered. Comparisons with upstream distributions obtained a few spins later show that the main solar wind beam component is reduced in phase space density by more than an order of magnitude. consistent with the large reduction in overall density n. Also present is a significant suprathermal population that appears much like an upstream intermediate distribution. The temperature in density holes rose to >10^ K, and can be accounted for by the presence of this energetic component and the reduction of the colder solar wind population. Note that while the solar wind beam is observed to slow down and change direction within and at the edges of the density hole, the measured beam velocity differs somewhat from the computed moments, indicating relative motion of the energetic component. The latter ions may originate in backstreaming particles observed exterior to the structure, retaining some of their initial field-aligned momentum.

DISCUSSION

Several characteristics of density holes are similar to those of HFAs (Hot Flow Anomalies), HDCs (Hot Diamagnetic Cavities) and SLAMS (TABLE 1). They all have significant bulk flow deflections, are filled with heated plasmas, have enhanced edge densities, and one or both edges have compressed magnetic field. Density holes durations are similar to SLAMS but much shorter than HFAs and HDCs, and correspondingly smaller dimensions. The order of magnitude density depletions have not been reported previously for HFAs. HDCs or SLAMS. Density holes are common while HFAs and HDCs occur rarely. Only eight HFA events had been reported in ISEE observations spanning more than two years, and the combined number including HDCs observed by ISEE, AMPTE-IRM and AMPTE-UKS was about 30 [8][9][10]. We have shown one example of SLAMS event that has a density hole with "heated" back-streaming plasmas associated with it. In addition, the density holes near the shock tend to have elevated edges that account for more particles than those depleted within the hole. The heating plus the net gain in particles is consistent with what is observed across the shock transition, and reflects as expected net density increases within structures moving relative to the solar wind flow.

Density holes are observed only when back-streaming energetic particles are present although the latter can exist without the former. These intervals usually have B perturbations. However, the details involved in creating the density holes are not known. Density holes have been observed for solar wind velocities <400 to >800 km s"\ n~l-IO/cc, and T~10^ -10^ °K. While B and n are correlated, deep density holes with large edge enhancements can have shallow magnetic holes and weak B overshoots, while shallow density holes with weak edge enhancements can have deep magnetic holes and large overshoots. We will continue to study the electromagnetic and electrostatic wave data and particle observations in order to estabhsh the roles of wave particle interactions. Improved statistics on upstream plasma and beam parameters (density, speed, T, Mach number, plasma (3 and shock geometry) may reveal factors controlling their origin and development.

Figure 1 :

 1 Figure 1: Cluster 1 and the Double Star observed nearly identical density hole structures, for example, at 0856 UT. Cluster 1 was outbound and Double Star inbound. The locations of the spacecraft are shown at the bottom (GSE coordinates). The panels from top to bottom show energy flux, density, velocity and temperature.

Figure 2 :

 2 Figure 2: A Langrangian view (left) of a density hole (right) observed by Cluster 1. The E-field data were averaged and high frequency variations were removed. The density holes are accompanied by nearly identically shaped magnetic holes with reduced intensity in the hole and steepened B at the edge. The scale of B is linear while for n it is logarithmic. Temperature T inside the hole is about 100 times the solar wind T. The GSE coordinates of Cluster 1 are (12.0, 3.2, 4.9) RE.

Figure 4 :

 4 Figure 4: Ion phase space distributions of density holes. Left panel: a B-perpendicular cut through the main solar wind beam in the upstream edge of the 0855 UT, 2 March 2005 event. The field points out of the page and the horizontal axis points nearly anti-sunward. Most particles have relatively modest energies, and the velocity scale includes only +/-1250 km s'' to highlight their behavior. Right panel: a Vsw ^ B slice through a distribution obtained within the interior of the 0516 UT, 3 February 2002 density hole. B points to the right (Reprinted from [1]).

Table 1 :

 1 Upstream Transient Structures

	Property	DH	HFA/HDC	SLAMS	FC
	Duration	-18 s	-few minutes	-10-20 s	> few minutes
	Scale length	Ri	~RE	~RE	>RE
	6n/n	-0.7	<0.2	No report	<0.2
	BulkV	Vx^O	Vx^-100	No report	-Vsw
	T (hole)	-10^	-10'	No report	>103
	Overshoot	Yes	Yes	Yes	Yes
	Shock-like	Yes	Yes	Yes	No
	Occurrence	Frequent	Rare	Frequent	Rare
	E-field	Yes	No report	Yes	No report
	Upstream ions	Yes	Yes	No report	No report
	Electron hole	Yes	No report	No report	No report
	Waves	Yes	No report	Yes	No report
	Currents	Yes	No report	Inferred	No report

ACKNOWLEDGEMENT

The research at UC Berkeley is performed under the auspices of a NASA Grant No. NNG04GF23G. Cluster is a joint project of ESA and NASA and Double Star a joint project of ESA and the Chinese Space Agency.


Simoen Eddy
email: eddy.simoen@imec.be

Cretu Bogdan
email: bbogdan.cretu@ensicaen.fr

Wen Fang
email: cfangwen@ime.ac.cn

Aoulaiche Marc
email: marc.aouliache@imec.be@e

Carin Regis
email: fregis.carin@unicaen.fr

Zhao Chao
email: hzhaochao@ime.ac.cn

Claeys Cor
email: icor.claeys@imec.be

Low-frequency noise spectroscopy of bulk and border traps in nanoscale devices

Keywords: Random Telegraph Signal, silicon-on-insulator, gate oxide trap, metal-oxidesemiconductor field-effect transistor, generation-recombination noise

The principles and application of Generation-Recombination (GR) noise spectroscopy will be outlined and illustrated for the case of traps in Ultra-Thin Buried Oxide Silicon-on-Insulator nMOSFETs and for vertical polycrystalline silicon nMOSFETs. It will be shown that for scaled devices the GR noise is originating from a single defect, giving rise to a so-called Random Telegraph Signal (RTS). Several methods will be described for an accurate extraction of the RTS parameters (amplitude, up and down time constant). It will be demonstrated that besides the deeplevel parameters also the position of the trap in the channel can be derived from a numerical modeling of the RTS amplitude.

Introduction

Performing defect spectroscopy in nano-scaled structures is a challenging endeavor. One technique which appears to be very suitable for this task is Generation-Recombination (GR) noise spectroscopy [START_REF] Lukyanchikova | Sources of the Lorentzian components in the low-frequency noise spectra of submicron metal-oxide-semiconductor field-effect transistors[END_REF]. Originally, the method has been developed for rather large area semiconductor devices, like, e.g., MOSFETs [START_REF] Lukyanchikova | Sources of the Lorentzian components in the low-frequency noise spectra of submicron metal-oxide-semiconductor field-effect transistors[END_REF][START_REF] Scholz | Low frequency noise and DLTS as semiconductor device characterization tools[END_REF][START_REF] Murray | Shallow defects responsible for GR noise in MOSFET's[END_REF][START_REF] Lartigau | Low temperature noise spectroscopy of 0.1 m partially depleted silicon on insulator metal-oxidesemiconductor field effect transistors[END_REF], Charge-Coupled Devices [START_REF] Hardy | Low-frequency noise in proton damaged LDD MOSFET's[END_REF] or Junction Field-Effect Transistors [START_REF] Grassi | Low frequency noise versus temperature spectroscopy of recently designed Ge JFETs[END_REF] to study deep levels in the semiconductor depletion region. It is an alternative to the popular Deep Level Transient Spectroscopy (DLTS) technique [START_REF] Scholz | Low frequency noise and DLTS as semiconductor device characterization tools[END_REF], the main difference being that the GR centers are studied in dynamic equilibrium, i.e., the noise is due to subsequent capture and emission of free carriers, while in DLTS an emission transient is monitored after trap filling. If the GR events are mainly related to either the valence (p-type) or conduction band (n-type material), the corresponding low-frequency (LF) noise spectrum is a Lorentzian as a function of the frequency and it is the result of fluctuations in the current through the device by alternating carrier capture and emission from a deep-level trap. The advantage of GR noise spectroscopy compared with DLTS is that it can be applied to any device size, since the LF noise Power Spectral Density (PSD) generally scales with 1/A [START_REF] Haartman | Low-frequency noise in advanced MOS devices[END_REF], with A the active device area, implying that the relative noise amplitude becomes higher for smaller devices. As a consequence, GR noise may become the dominant LF noise source in small-area MOSFETs and can be caused by a single defect [START_REF] Kirton | Noise in solid-state microstructures: A new perspective on individual defects, interface states and low-frequency (1/f) noise[END_REF]. At the same time, discrete switching of the device current as a function of time becomes visible and corresponds in the most simple case to the charging (carrier capture) and discharging (emission) of a single trap. Such behavior in the time domain is called a Random Telegraph Signal (RTS) and its parameters (amplitude, up and down times) can equally be used for deep-level characterization [START_REF] Kirton | Noise in solid-state microstructures: A new perspective on individual defects, interface states and low-frequency (1/f) noise[END_REF].

In this paper, an overview will be given regarding GR noise and RTS spectroscopy in MOSFETs. It will be outlined how the main defect level parameters (activation energy E T ; capture cross section for holes (electrons)  p,n and trap concentration N T ) can be derived from the Lorentzian noise spectrum as a function of temperature T. In the case of single-defect RTSs, the same information can be extracted from time domain measurements. In addition, it will be shown that studying the gate voltage (V G ) or drain voltage (V D ) dependence of the RTS amplitude yields in many cases information on the position of the trap. Criteria will be proposed to dinstinguish Lorentzian noise belonging to an ensemble of traps in the depletion region of a semiconductor material or to a single defect in the gate dielectric. The methods will be illustrated for processing-induced defects present in the silicon film of Ultra-Thin Buried Oxide (UTBOX) Silicon-on-Insulator (SOI) MOSFETs and for RTSs present in the polysilicon layer of a vertical transistor for Non-Volatile Memory (NVM) applications [START_REF] Toledano-Luque | Spectroscopic study of polysilicon traps by means of fast capacitance transients[END_REF][START_REF] De Andrade | RTN assessment of traps in polysilicon cylindrical vertical FETs for NVM application[END_REF].

Principle of GR Noise and RTS Spectroscopy

GR Noise. As schematically shown in Fig. 1, the GR noise power spectral density (PSD) (S I ) exhibits a plateau A 0 at low frequency (f) and a 1/f 2 roll-off, corresponding to [START_REF] Lukyanchikova | Sources of the Lorentzian components in the low-frequency noise spectra of submicron metal-oxide-semiconductor field-effect transistors[END_REF][START_REF] Scholz | Low frequency noise and DLTS as semiconductor device characterization tools[END_REF][START_REF] Murray | Shallow defects responsible for GR noise in MOSFET's[END_REF][START_REF] Lartigau | Low temperature noise spectroscopy of 0.1 m partially depleted silicon on insulator metal-oxidesemiconductor field effect transistors[END_REF]:

S I =A 0 /[1+(f/f 0 ) 2 ] (1)
with f 0 =1/2 0 the characteristic frequency of the GR center and  0 the characteristic time constant, given by the Shockley-Read-Hall (SRH) expression [START_REF] Lukyanchikova | Sources of the Lorentzian components in the low-frequency noise spectra of submicron metal-oxide-semiconductor field-effect transistors[END_REF][START_REF] Scholz | Low frequency noise and DLTS as semiconductor device characterization tools[END_REF][START_REF] Murray | Shallow defects responsible for GR noise in MOSFET's[END_REF][START_REF] Lartigau | Low temperature noise spectroscopy of 0.1 m partially depleted silicon on insulator metal-oxidesemiconductor field effect transistors[END_REF]:

 0 =1/(2f 0 ) = [c n (n(x,y)+n t ) + c p (p(x,y)+p t )] -1 (2)
with c n , c p the capture coefficient for electrons and holes at position (x,y) in the channel, respectively, given by the product of the corresponding capture cross section and thermal velocity, i.e., c n = n v thn . Further, we have that n(x,y) and p(x,y) are the free electron and hole densities and n t and p t is the free electon or hole density when the Fermi level E F coincides with the trap level E T . In the depletion region of a bulk or a partially depleted MOSFET, one can neglect the minority carrier terms in Eq. (2) so that the characteristic time constant can be approximated by the sum of an emission ( e ) and a capture term ( c ):

1/ 0 =1/ e +1/ c ( 3
)
The characteristic frequency can be determined from a plot of f×S I versus f, resulting in a peak with a maximum occurring at f 0 . In general, a LF noise spectrum may consist of different Lorentzians, occurring at different f 0i , a 1/f noise component K f /f and frequency-independent white noise (PSD B w ) [START_REF] Santos | Low-frequency noise assessment in advanced UTBOX SOI n-channel MOSFETs[END_REF][START_REF] Achour | In depth static and low-frequency noise characterisation of n-channel n-FinFETs on SOI substrates at cryogenic temperature[END_REF]. Fitting of the following expression:

S I •f=B w •f+K f + A 0i •f/[1+(f/f 0i ) 2 ] ( 4
)
gives accurate values for the parameters A 0i and f 0i of the i=1,j Lorentzians in the spectrum [START_REF] Santos | Low-frequency noise assessment in advanced UTBOX SOI n-channel MOSFETs[END_REF][START_REF] Achour | In depth static and low-frequency noise characterisation of n-channel n-FinFETs on SOI substrates at cryogenic temperature[END_REF]. Measuring the f 0i values at different sample temperatures enables to construct an Arrhenius law like in Fig. 2 [START_REF] Santos | Low-frequency noise assessment in advanced UTBOX SOI n-channel MOSFETs[END_REF][START_REF] Achour | In depth static and low-frequency noise characterisation of n-channel n-FinFETs on SOI substrates at cryogenic temperature[END_REF]:

(5)

450

Gettering and Defect Engineering in Semiconductor Technology XVI In Eq. ( 5) k is the Boltzmann constant, h Planck's constant, M c the equivalent minima in the conduction band, m e * and m h * the transport effective mass for electrons and holes, respectively. It is clear that from the slope of an Arrhenius plot one derives the activation energy E C -E T and from the intercept the (electron) capture cross section. An example of an Arrhenius plot obtained on an UTBOX SOI nMOSFET is given in Fig. 2, showing an electron trap with activation energy E=E C -E T =0.42 eV and 0.45 eV, respectively. with WL the effective width times the length of the transistor; C ox the oxide capacitance density of the gate, q the elementary charge and W d the width of the depletion region. In the case of a bulk or partially depleted SOI transistor, the empirical factor B is given by 1/3 [START_REF] Lukyanchikova | Sources of the Lorentzian components in the low-frequency noise spectra of submicron metal-oxide-semiconductor field-effect transistors[END_REF][START_REF] Lartigau | Low temperature noise spectroscopy of 0.1 m partially depleted silicon on insulator metal-oxidesemiconductor field effect transistors[END_REF].

RTS.

A typical time trace of a two-level RTS in a small-area MOSFET is given in Fig. 3, defining the main parameters: the current amplitude I D , the up time t up and the down time t down . From the histogram of the drain current (I D ) values, an average amplitude I D can be derived, while the time constant ratio  c / e can be found from the ratio of the area under the two peaks in the amplitude histogram of Fig.

4 [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF]. The inverse sum of the time constants can be obtained from the corner frequency of the corresponding Lorentzian spectrum [START_REF] Kirton | Noise in solid-state microstructures: A new perspective on individual defects, interface states and low-frequency (1/f) noise[END_REF][START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF] or by constructing the probability distribution of the up (t up ) and the down (t down ) time constants. They usually follow an exponential distribution, given by [START_REF] Kirton | Noise in solid-state microstructures: A new perspective on individual defects, interface states and low-frequency (1/f) noise[END_REF][START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF]:

P(t)=1/ exp(-t/) ( 7
)
whereby the average value corresponds to the capture (up time) or emission (down time) constant.

Performing time domain analysis at different temperatures enables constructing an Arrhenius diagram from which the activation energy of carrier capture and emission is obtained [START_REF] Kirton | Noise in solid-state microstructures: A new perspective on individual defects, interface states and low-frequency (1/f) noise[END_REF].

452

Gettering and Defect Engineering in Semiconductor Technology XVI

In the case of a complex multilevel RT signal, like in Fig. 5 containing a small and a large amplitude signal, the usual time series analysis method may become inadequate. In order to discriminate multiple RTSs in the same time trace, one can rely on the statistical analysis of a socalled Time Lag Plot (TLP) which consists in the representation of the current at n th sample I D (n) versus the current at n+1 th sample I D (n+1) [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF][START_REF] Realov | Analysis of random telegraph noise in 45-nm CMOS using on-chip characterization system[END_REF][START_REF] Tsuchiya | Characterization of oxide traps participating in random telegraph noise using charging history effects in nano-scaled MOSFETs[END_REF], taking into account the autocorrelation in time series data. One normally expects the TLP points to align along a diagnonal, with higher intensity clouds corresponding with each discrete drain current level present in the time series. Improved TLP can be utilized to extract the centre of the data clouds, as shown in Fig. 6, by obtaining a value related to the probality of each data point in the cloud, which means the probality of observing the n+1 th sample after the n th sample. Moreover, a combination of the TLP with a so-called Hidden Markov modeling (HMM) approach can yield accurate RTS amplitude and time constants, even for RTSs hidden in large random noise [START_REF] Li | Characterization of Random Telegraph Noise in scaled high-/metal gate MOSFETs with SiO 2 /HfO 2 gate dielectrics[END_REF]. Fig. 5. Time series containing a large and a small RTS. Fig. 6. Improved Time Lag Plot (TLP) analysis of the time series, revealing four drain current levels corresponding with two RTSs.

Finally, it can be shown that for an RTS due to a trap in the gate oxide, the analysis of the time constant ratio versus gate voltage [START_REF] Kirton | Noise in solid-state microstructures: A new perspective on individual defects, interface states and low-frequency (1/f) noise[END_REF][START_REF] Ralls | Discrete resistance switching in submicrometer inversion layers: Individual interface traps and low-frequency (1/f?) noise[END_REF] or drain voltage [START_REF] Amarasinghe | Characterization of oxide traps in 0.15 m 2 MOSFETs using random telegraph signals[END_REF] yields the trap position in the gate oxide with respect to the silicon/dielectric interface or along the transistor channel, respectively.

Solid State Phenomena Vol. 242

GR Noise and RTS in UTBOX SOI MOSFETs

The structure of an UTBOX SOI MOSFET is schematically represented in Fig. 7a. The study of GR noise in such devices offers some unique opportunities given the fully-depleted nature of the ~10 nm thick silicon film [START_REF] Santos | Low-frequency noise assessment in advanced UTBOX SOI n-channel MOSFETs[END_REF][START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF][START_REF] Simoen | Lessons learned from low-frequency noise studies on fully depleted UTBOX Silicon-on-Insulator nMOSFETs[END_REF]. Usually, it is assumed that Lorentzians which do not shift markedly with gate voltage in strong inversion belong to GR centers in the silicon depletion region [START_REF] Lukyanchikova | Sources of the Lorentzian components in the low-frequency noise spectra of submicron metal-oxide-semiconductor field-effect transistors[END_REF][START_REF] Scholz | Low frequency noise and DLTS as semiconductor device characterization tools[END_REF][START_REF] Murray | Shallow defects responsible for GR noise in MOSFET's[END_REF][START_REF] Lartigau | Low temperature noise spectroscopy of 0.1 m partially depleted silicon on insulator metal-oxidesemiconductor field effect transistors[END_REF]. An example is given in Fig. 7b. In that case, T-dependent GR noise spectroscopy as presented above can be applied [START_REF] Santos | Low-frequency noise assessment in advanced UTBOX SOI n-channel MOSFETs[END_REF][START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF][START_REF] Simoen | Lessons learned from low-frequency noise studies on fully depleted UTBOX Silicon-on-Insulator nMOSFETs[END_REF]. Conversely, the Lorentzians corresponding with a gate-oxide RTS exhibit a clear gate-voltage dependence, like in Fig. 7c [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF]. However, as pointed out recently [START_REF] Luque Rodríguez | Dependence of generation-recombination noise with gate bias in FD SOI MOSFETs[END_REF], GR noise due to defects in the fully depleted film of an UTBOX SOI transistor can also show a strong variation of the corner frequency f 0 with gate bias in linear operation. This is related to the fact that both the Fermi level and the free carrier concentrations undergo a strong variation with front-(V GS ) and back-gate bias (V BS ), resulting in a gate-bias dependent change in the SRH time constant of Eq. ( 2). As a consequence, the distinction between RTS and GR noise becomes more complicated in the case of FD SOI transistors and more refined criteria have to be worked out [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF][START_REF] Santos | On the variability of low frequency noise of front/back-channels in UTBOX SOI nMOSFETs[END_REF]. A first possibility is to measure the noise in the front channel with the back channel biased in accumulation and next in the back channel current with the front channel accumulated [START_REF] Santos | On the variability of low frequency noise of front/back-channels in UTBOX SOI nMOSFETs[END_REF]. In this way, the impact of the opposite interface and oxide is screened. When the Lorentzians are only present in one type of spectra, one may assume that they originate from traps in the corresponding oxide, while if the Lorentzians are present in both cases, there is a strong probability that the GR noise is due to defects in the fully depleted silicon film [START_REF] Santos | On the variability of low frequency noise of front/back-channels in UTBOX SOI nMOSFETs[END_REF]. However, for ultra-thin films (<10 nm) the strong front-back coupling prevents accumulation of either of the interfaces so that this method may break down for deeply scaled technology nodes.

A second method relies on combining noise measurements with time domain measurements. At first sight, the presence of RTS in the drain current points to single traps in one of the gate oxides. Unfortunately, this criterion is also not sufficient for typical UTBOX SOI transistor sizes (i.e., L=0.1 m; W=1 m and t film =10 nm), since the GR noise in such devices corresponds with only a few similar traps [START_REF] Simoen | Lessons learned from low-frequency noise studies on fully depleted UTBOX Silicon-on-Insulator nMOSFETs[END_REF], as has also been derived from charge retention time measurements and modeling [START_REF] Aoulaiche | Junction field effect on the retention time for one Transistor Floating Body RAM memory[END_REF]. As a result, RTS has been found which could be ascribed to the capture and emission by a single GR center in the silicon film [START_REF] Simoen | Lessons learned from low-frequency noise studies on fully depleted UTBOX Silicon-on-Insulator nMOSFETs[END_REF]. This has led to the development of a set of criteria, which are based on the RTS parameters and summarized in Table I [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF]. One of the most important guidelines is based on the gate voltage behavior of the capture and emission time constant and their ratio. At sufficiently high drain currents,  c and  e become independent on V GS and show a ratio close to 1, which is in strong contrast with the exponential dependence of  c on V GS for an oxide RTS [START_REF] Kirton | Noise in solid-state microstructures: A new perspective on individual defects, interface states and low-frequency (1/f) noise[END_REF][START_REF] Ralls | Discrete resistance switching in submicrometer inversion layers: Individual interface traps and low-frequency (1/f?) noise[END_REF]. An additional benefit of the exploitation of film-related RTSs is that one can derive the position of the trap in the silicon film when combining with 2D or 3D device simulations [START_REF] Fang | Distinction between silicon and oxide traps using single-trap spectroscopy[END_REF].

A third possibility is to analyze the gate voltage dependence of the Lorentzian parameters based on 2D numerical simulations of the FD SOI UTBOX nMOSFET characteristics including a dedicated GR noise model [START_REF] Luque Rodríguez | Dependence of generation-recombination noise with gate bias in FD SOI MOSFETs[END_REF]. This enables to perform GR noise spectroscopy at room temperature. [START_REF] Santos | Low-frequency noise assessment in advanced UTBOX SOI n-channel MOSFETs[END_REF][START_REF] Simoen | Towards single-trap spectroscopy: Generation-Recombination noise in UTBOX SOI nMOSFETs[END_REF][START_REF] Cretu | Low frequency noise spectroscopy in rotated UTBOX nMOSFETs[END_REF], which are believed to be processing-related, e.g., by ion implantation of the source/drain regions or by dry etching damage. Moreover, it has recently been shown that the empirical B factor in Eq. ( 5) is most likely smaller for FD transistors, compared with ~1/3 for PD or bulk devices [START_REF] Cretu | Low frequency noise spectroscopy in rotated UTBOX nMOSFETs[END_REF].

GR Noise and RTS in Vertical Polysilicon MOSFETs

Vertical polysilicon n-channel transistors are potentially of interest for 3D Non-Volatile Memory (NVM)) applications [START_REF] Toledano-Luque | Spectroscopic study of polysilicon traps by means of fast capacitance transients[END_REF][START_REF] De Andrade | RTN assessment of traps in polysilicon cylindrical vertical FETs for NVM application[END_REF]. The structure is schematically represented in Fig. 8a, whereby the grain boundaries are a likely source of RTS in the drain current (Fig. 8b). Detailed noise measurements have indicated that there are two types of RTS present [START_REF] De Andrade | RTN assessment of traps in polysilicon cylindrical vertical FETs for NVM application[END_REF]: slow, large-amplitude signals, which are predominantly related to traps in the gate oxide or at the GBs and smaller amplitude signals, with faster time constants, which are believed to reside in the polysilicon channel. In order to validate this assumption, 3D device simulations have been executed, using the structure of Fig. 9, whereby the GR center location was varied along the channel between source and drain (positions A to C) and at the Si/SiO 2 interface or in the center of the cylinder (primed positions A', B' and C'). The RTS amplitude I D has been calculated as the difference in the current without I D (0) or with I D (-) a negatively charged electron trap present. As can be seen from Fig. 10, the RTS amplitude is a sensitive function of the trap location, so that a comparison with actually measured data should reveal the position of the trap along the channel and in diameter [START_REF] De Andrade | RTN assessment of traps in polysilicon cylindrical vertical FETs for NVM application[END_REF]. Again, as for oxide-related RTSs the study of the trap time constants as a function of temperature should provide the activation energy and can then be compared with DLTS results obtained on similar devices [START_REF] Toledano-Luque | Spectroscopic study of polysilicon traps by means of fast capacitance transients[END_REF].

Solid State Phenomena Vol. 242

2 Fig. 1 .Fig. 2 .

 212 Fig. 1. Schematic representation of a Lorentzian noise spectrum and definition of its main parameters.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Time series of the drain current obtained on a small-area MOSFET operated in weak inversion, showing a random two-level switching between a low and a high drain current state.

Fig. 7 .

 7 Fig. 7. (a) Schematic representation of a FD UTBOX SOI nMOSFET. (b) Lorentzian GR noise spectra for different front-gate biases in linear operation at a V DS =50 mV and back-gate bias V BS =0 V, showing little shift of the corner frequency and (c) showing a clear increase of the corner frequency with increasing front-gate voltage V GS .

Fig. 9 .

 9 Fig. 9. (a) Schematic structure of a vertical polyscilicon nMOSFET and (b) input charateristics (I D -V G ) at fixed drain voltage V D =0.5 V, showing the presence of small-amplitude RTS fluctuations.

Fig. 9 .

 9 Fig. 9. Schematic structure of a polysilicon nMOSFET with bulk trap positions.

Fig. 10 .

 10 Fig. 10. Simulated RTS amplitude as a function of gate voltage corresponding with the different trap positions between source (S) and drain (D) defined in Fig. 9.

Table 1

 1 Variation of RTS and Lorentzian parameters for a gate oxide and a film trap with front gate voltage in UTBOX SOI nMOSFETs, at constant V DS .

	Parameter	gate oxide RTN	film RTN
	 c	expon. decrease	moderate decrease
	 e	~constant	~constant
	 c / e	expon. decrease	~1
	I D /I D	certain variation + nearly constant	decrease
	S I (0)	decrease	const .+ increase
	Application of standard T-dependent GR noise spectroscopy to the UTBOX SOI nMOSFETs has
	revealed the presence of around ten different deep level centers

Summary

In this paper, it has been shown that Lorentzian noise, associated with trap-assisted GR events and measured on different kinds of MOSFETs enables the study of deep levels in deeply scaled devices. Traps either in the semiconductor, at its interface(s) or in the oxide can in principle be distinguished. The advantage of employing time domain measurements is that the RTS parameters can also be exploited for deriving the trap location of the single defect responsible for it. At the same time, scaling of the device dimensions leads to small active volumes, so that RTSs in the semiconductor itself start to contribute to the noise, which opens up new perspectives for deep-level studies.