Arif Bulent Ozguler
email: ozguler@ee.bilkent.edu.tr

Adamu Abdullahi

A Bülent Özgüler

Stability of Planar Piecewise Linear Systems: A Geometric Condition

Keywords: conewise linear, piecewise linear, planar, switched linear, stability, basins

published or not. The documents may come

Introduction

A thorough study of piecewise linear systems appeared to be the logical next step after the proven success of linear systems. The concern with such systems has also been application oriented because there are so many naturally hybrid, multi-modal plants all around. Examples are not limited to physical systems and sophisticated switched systems come up in the domain of social sciences, [START_REF] Özgüler | Conflictual Peacetime International Politics[END_REF][START_REF] Sezer | A dynamic allocation scheme for a multi agent Nash equilibrium[END_REF]. Since piecewise linearity is the simplest form of nonlinearity that can be imagined, such a study also promised to be very fruitful. The introduction of "conewise systems" as a worthwhile object of study has helped focus attention on this particular class of nonlinear systems, (C ¸amlibel, [START_REF] Pang | Conewise linear systems: Non-Zenoness and observability[END_REF]. One can claim that, the most complete result so far on, for instance, stability of piecewise linear systems has turned out to be on the very special case of two-state systems [START_REF] Araposthasis | Stability and controllability of planar, conewise linear systems[END_REF][START_REF] Iwatani | Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems[END_REF] (also see [START_REF] Heemels | Stability and controllability of planar bimodal linear complementarity systems[END_REF]). Although Lyapunov approach has provided many sufficient conditions for stability of more general cases [START_REF] Liberzon | Switching in Systems and Control[END_REF], the method becomes quickly stagnant by the requirement to concoct Lyapunov functions for a set of systems, [START_REF] Johansson | Piecewise Linear Control Systems: A Computational Approach[END_REF][START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]. Nevertheless, the piecewise linear interest is alive, and by now there are a number of good books and survey papers devoted to the subject [START_REF] Abdullahi | Stability of Planar Piecewise Linear Systems: A Geometric Approach[END_REF][START_REF] Johansson | Piecewise Linear Control Systems: A Computational Approach[END_REF][START_REF] Liberzon | Switching in Systems and Control[END_REF][START_REF] Lin | Stability and stabilizability of switched linear systems: A survey of recent results[END_REF][START_REF] Van Der Schaft | An Introduction to Hybrid Dynamical Systems[END_REF][START_REF] Sun | Stability of piecewise linear systems revisited[END_REF]Sun & Ge, 2011).

The result on stability of Iwatani and Hara [START_REF] Iwatani | Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems[END_REF] on planar, multi-modal systems is complete from a test of stability point of view but it is still worth a second look for many reasons. The condition offered for stability is in terms of the zeros of a subsidiary system, the relation of which to the original plant is indirect. This prevents an insightful interpretation of the condition. They devise a very useful notion of 'transitive modes' but also use a, not so natural, notion of 'weakly-transitive' modes which are also transitive so that a clear distinction between the two notions is not possible. What are the 'non-transitive modes' ? The main result of [START_REF] Araposthasis | Stability and controllability of planar, conewise linear systems[END_REF], obtained independently of [START_REF] Iwatani | Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems[END_REF], is based on a clean characterization of trajectories escaping convex cones (i.e., transitive modes) and brings the eigenvectors of each mode into the picture via the notions of "visible eigenvectors" and "stable eigenspaces". However, the result in [START_REF] Araposthasis | Stability and controllability of planar, conewise linear systems[END_REF], similar to that of [START_REF] Iwatani | Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems[END_REF], does not provide any intuition concerning "non-transitive cones." In [START_REF] Polderman | Stability and robustness of planar switching systems[END_REF], a hybrid automaton approach is followed to study the stability of planar systems and a decision algorithm that is based on "contractive cycles" that are, in essence, stable transitive trajectories are given. Also in [START_REF] Nishiyama | Optimal stable state-space partitioning for piecewise linear planar systems[END_REF] and in (Liu, Yao, Yang, Balakrishnan & Guo, 2006), integral expressions are derived to characterize the "expansion factors" when trajectories go through transitive modes.

Here, we take a different approach to the same problem and obtain a new set of necessary and sufficient conditions. Any planar piecewise linear system is shown to be globally asymptotically stable just in case each linear mode satisfies certain conditions that only depend on how its eigenvectors stand relative to the cone on which it is defined. The conditions are in terms of the eigenvalues, eigenvectors, and the cone. The improvements on both [START_REF] Iwatani | Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems[END_REF] and [START_REF] Araposthasis | Stability and controllability of planar, conewise linear systems[END_REF] are the following: i) The condition is directly in terms of the "givens" of the problem. ii) Non-transitive modes are identified. iii) Initial states and their trajectories are classified (basins of attraction and repulsion are indicated). iv) The known condition for bimodal systems is obtained as an easy corollary of the main result.

We denote the real numbers, n-dimensional real vector space, and the set of real n × m matrices by R, R n , and R n×m , respectively. The norm of a vector v ∈ R n will be denoted by |v|. The natural basis vectors in R n will be denoted by e i , i = 1, ..., n. In particular, when n = 3, we will use k := e 3 . If v, w ∈ R 3 , then v × w will denote the cross product of the vectors and v • w = v T w, their dot product, where 'T' denotes 'transpose.' If v, w ∈ R 2 , then by v × w, we mean det[v w]k, where 'det' means 'determinant,' i.e., cross product of vectors in the plane will be computed by imbedding them in the space. The set of complex n-vectors will be C n and j ∈ C will be the imaginary number. For convenience, we will use the cross product of v, w ∈ C 2 as well and define v×w := det[v w]k. By log z, z ∈ C, we denote the complex principal logarithm log z = ln |z|+j∠z with -π < ∠z ≤ π.

Planar Piecewise Linear Systems

The class of systems considered are

ẋ =          A 1 x if x ∈ S 1 , A 2 x if x ∈ S 2 , . . . . . . . . . A m x if x ∈ S m , (1)
where

A i ∈ R 2×2 and, with C i ∈ R 2×2 , S i := {x ∈ R 2 : C i x ≥ 0},
for i = 1, 2, ..., m. We assume that each C i is nonsingular and is such that det C i > 0. Note that the latter causes no loss of generality and only requires a permutation of rows of C i if necessary.

The nonsingularity assumption implies that (1) is truly multi-modal (m ≥ 2) and that the interior of each S i , int S i , is nonempty. We further assume that the interior of each pairwise intersection int S i ∩S k , i = k is empty and that S 1 ∪...∪S m = R 2 . These assumptions ensure, in the terminology of [START_REF] Iwatani | Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems[END_REF], that (1) is memoryless. Further, let

S i = s i1 s i2 := C -1 i = c T i1 c T i2 -1
so that det S i > 0. It is easy to see that, if each S i , i = 1, ..., m is strictly contained in a half-plane, then S i is a convex cone

S i = {αs i1 + βs i2 : α, β ≥ 0}
and the boundary of S i is the union two rays

B ik = {αs ik : α ≥ 0}, k = 1, 2.
Note that because det S i > 0, the cross product s i1 × s i2 points upward using the right-hand rule, i.e., positively oriented. This allows us to label B i1 and B i2 as the right and left border, respectively. If, in (1), there is a mode defined on a half-plane or a sector larger than a half-plane, then it can be split into two modes having the same dynamics (the same A-matrix) so that each is still defined on a cone. The splitting must be done with care, as we will clarify later.

Given a mode i, its eigenvalues will be denoted by λ i1 , λ i2 ∈ C and, in case of real and distinct eigenvalues, they will be indexed so that λ i1 > λ i2 .

Single Mode

We now focus on a single mode i (and temporarily discard the index i) to consider

ẋ = Ax, x ∈ S ⊂ R 2 , S = {αs 1 + βs 2 : α, β ≥ 0}, (2)
where det S > 0 for

S = [s 1 s 2 ]. Let v 1 , v 2 ∈ R 2 be such that AV = V Λ, V = v 1 v 2 ,
where Λ is equal to

λ 1 0 0 λ 2 , λ 0 1 λ , and σ -ω ω σ .
respectively, when eigenvalues are such that λ 1 > λ 2 (real and distinct), λ := λ 1 = λ 2 (real and repeated), and λ 1 = λ2 = σ+jω (non-real) with ω > 0. It follows that if the eigenvalues are distinct, then v 1 , v 2 are the eigenvectors associated with the larger and smaller eigenvalues, respectively. If they are repeated, then v 2 is an eigenvector and v 1 is a generalized eigenvector. If the eigenvalues are non-real, then v 1 + jv 2 is the eigenvector associated with σ -jω. We define

W = w T 1 w T 2 := V -1 . Note that, det V > 0 if and only if v 1 × v 2 is positively oriented.
The trajectory at t ≥ 0 of (2) starting at x(0) = b ∈ S at time 0 can be written as

x(t, b) =    e λ1t w T 1 b v 1 + e λ2t w T 2 b v 2 , e λt [w T 1 b v 1 + (t w T 1 b + w T 2 b)v 2 ], e σt {[w T 1 b cos(ωt) -w T 2 b sin(ωt)] v 1 + [w T 1 b sin(ωt) + w T 2 b cos(ωt)] v 2 }.
(3)

for the three cases. Examining the sign of the derivative of the angle of x(t, b), we can determine the direction the trajectory moves at time t.

Fact 1: Trajectory x(t, b) moves in a positive direction at time t ≥ 0 if for every real eigenvector

v k × b = 0 for k = 1, 2 and det V (v 1 × b • v 2 × b) > 0 if eigenvalues are distinct, det V > 0 otherwise. (4)
Proof. Since w T l v k = 0 for l = k, a trajectory moves radially along an eigen-direction if and only if v k × b = 0, by (3). For any other b, the angle of the vector v k × b, and hence, the direction of a trajectory is well-defined. Suppose the eigenvalues are real and distinct. Let e i denote the i-th natural vector for i = 1, 2 and let x(t, b) = ρ(t)∠ψ(t) be in polar representation. Then,

ψ(t) = (e T 2 ẋ)(e T 1 x) -(e T 1 ẋ)(e T 2 x) ρ 2 = - det V (λ 1 -λ 2 )(w T 1 b)(w T 2 b) ρ 2 e -(λ1+λ2)t so that ψ(t) > 0 if and only if det V (w T 1 b)(w T 2 b) < 0. Now, we note that v 1 × b = det V (w T 2 b)k, v 2 × b = -det V (w T 1 b)k,
where k denotes the (positively oriented) cross product of the unit vectors in x 1 and x 2 directions. It follows that ψ(t) > 0 if and only if the first condition in (4) holds. If the eigenvalues are repeated or non-real, then the expressions in (3) give

ψ(t) = det V (w T 1 b) 2 ρ 2 e -2λt , ψ(t) = det V ω[(w T 1 b) 2 + (w T 2 b) 2 ] ρ 2 e -2σt
, respectively. It follows that, in the last two cases, ψ(t) > 0 if and only if the second condition in (4) holds.

That is, if the eigenvalues are non-real or repeated, then the direction is independent of the initial state b and is determined by the sign of det V only. In case of real and distinct eigenvalues, how the initial state is situated with respect to the two eigenvectors also matters. For instance, if det V > 0 then the trajectory moves in negative direction if and only if b is in between v 1 and v 2 .

Let us now classify the cases of trajectories hitting the boundary, one of the two borders of S.

Fact 2: (i) There exists (a finite)

t 1 > 0 such that x(t 1 , b) intersects B 1 if and only if    det V (v 1 × b • v 2 × b) < 0 & v 1 × b • v 1 × s 1 > 0, det V < 0 & v 2 × b • v 2 × s 1 > 0, det V < 0 (5)
respectively, when eigenvalues are such that λ 1 > λ 2 (real and distinct), λ := λ 1 = λ 2 (real and repeated), and λ 1 = λ2 = σ + jω (non-real).

(ii) There exists (a finite)

t 2 > 0 such that x(t 2 , b) intersects B 2 if and only if    det V (v 1 × b • v 2 × b) > 0 & v 1 × b • v 1 × s 2 > 0, det V > 0 & v 2 × b • v 2 × s 2 > 0, det V > 0. (6)
respectively, when eigenvalues are such that λ 1 > λ 2 (real and distinct), λ := λ 1 = λ 2 (real and repeated), and λ 1 = λ2 = σ + jω (non-real).

(iii) Let µ denote λ 1 or σ and let v denote v 2 or v 1 + jv 2 in the cases of real and nonreal eigenvalues, respectively. Then, in the situations of items (i) and (ii), we have

x(t 1 , b) = s 1 (v × b) • k (v × s 1 ) • k e µt1 , x(t 2 , b) = s 2 (v × b) • k (v × s 2 ) • k e µt2 . ( 7
)
Proof. (i) Suppose the eigenvalues are real and distinct. Such a t 1 > 0 exists just in case

c T 2 x(t 1 , b) = e λ1t1 (c T 2 v 1 )(w T 1 b) + e λ2t1 (c T 2 v 2 )(w T 2 b) = 0,
which gives

e (λ1-λ2)t1 = - (c T 2 v 2 )(w T 2 b) (c T 2 v 1 )(w T 1 b) > 1, (8)
where the inequality is by

λ 1 > λ 2 . Note that c T 2 v 1 = 0 and w T 1 b = 0 since otherwise either v 1 × s 1 = 0 or v 2 × b = 0, i.e.
, either there is a sliding mode or the initial condition is along an eigenvector. Now, the condition (8) is equivalent to

1 + (c T 2 v 2 )(w T 2 b) (c T 2 v 1 )(w T 1 b) = det V det S c T 2 b v 2 × b • v 1 × s 1 < 0 (9) by the identities (c T 2 v 1 )(w T 1 b) + (c T 2 v 2 )(w T 2 b) = c T 2 b, det S (c T 2 v 1 )k = -v 1 × s 1 , det V (w T 1 b)k = -v 2 × b.
In (9), c T 2 b > 0 because, as s 1 × s 2 is positively oriented and b is in the interior of S, s 1 × b = det S (c T 2 b)k is also positively oriented. Using the condition from (4) that det V (v 1 ×b•v 2 ×b) < 0 is necessary for an intersection with B 1 , we obtain the first condition in (5). Suppose, next, that the eigenvalues are repeated. Then,

t 1 > 0 exists if and only if c T 2 x(t 1 , b) = e λt1 [(c T 2 v 1 )(w T 1 b) + (c T 2 v 2 )(w T 2 b) + t 1 (c T 2 v 2 )(w T 1 b)] = 0, which gives t 1 = - (c T 2 v 1 )(w T 1 b) + (c T 2 v 2 )(w T 2 b) (c T 2 v 2 )(w T 1 b) = - det V det S c T 2 b v 2 × b • v 2 × s 1 > 0. ( 10
)
Using the condition from (4) that det V < 0 is necessary for trajectory to intersect B 1 and the fact that c T 2 b > 0 as in the previous case, we obtain the second condition in (5). In the final case that eigenvalues are non-real, there exists such t 1 if and only if det V < 0, by (4) and by the fact that the trajectories are always foci or centers.

(ii) The proof is analogous to the proof in (i).

(iii) Let us first consider the case of non-real eigenvalues. In the situation depicted by (i), at t = t 1 > 0, we have

c T 2 x(t, b) = e σt {[w T 1 b cos(ωt) -w T 2 b sin(ωt)] c T 2 v 1 + [w T 1 b sin(ωt) + w T 2 b cos(ωt)] c T 2 v 2 } = 0, (11)
which gives

tan(ωt 1 ) = (c T 2 v 1 )(w T 1 b) + (c T 2 v 2 )(w T 2 b) (c T 2 v 1 )(w T 2 b) -(c T 2 v 2 )(w T 1 b) = c T 2 b (c T 2 v 1 )(w T 2 b) -(c T 2 v 2 )(w T 1 b) (12)
and

x(t 1 , b) = det V det S e σt1 c T 2 v 2 [w T 1 b cos(ωt 1 ) -w T 2 b sin(ωt 1 )]s 1 . (13)
In obtaining ( 12) from (3), we have used the identity

v 1 (c T 2 v 2 ) -v 2 (c T 2 v 1 ) = c T 2 v 2 (v 1 + v 2 w T 2 s 1 w T 1 s 1 ) = c T 2 v 2 w T 1 s 1 s 1 . Noting, with ∆ := [(w T 1 b) 2 + (w T 2 b) 2 ][(c T 2 v 1 ) 2 + (c T 2 v 2 ) 2 ], that cos(ωt 1 ) = (c T 2 v 1 )(w T 2 b) -(c T 2 v 2 )(w T 1 b) ∆ , sin(ωt 1 ) = c T 2 b ∆ ,
and substituting in (13), we get

x(t 1 , b) = - det V det S e σt1 (w T 1 b) 2 + (w T 2 b) 2 (c T 2 v 1 ) 2 + (c T 2 v 2 ) 2 s 1 = - det V det S | det S| | det V | e σt1 |v 2 × b| 2 + |v 1 × b| 2 |s 1 × v 1 | 2 + |s 1 × v 2 | 2 s 1 ,
which can be expressed as in (7) since det S > 0, -det V > 0. In case of real, distinct eigenvalues, substituting the expression for e λ2t1 w T 2 b v 2 obtained from (8) into (3), we have

x(t 1 , b) = e λ1t1 w T 1 b c T 2 v 2 [v 1 (c T 2 v 2 ) -v 2 (c T 2 v 1 )] = e λ1t1 w T 1 b(v 1 + v 2 w T 2 s 1 w T 1 s 1 ) = e λ1t1 (v 2 × b) • k (v 2 × s 1 ) • k s 1 ,
which again gives (7). Finally, in case of repeated eigenvalues, substituting the expression for t 1 obtained from (10) into (3), we have

x(t 1 , b) = e λt1 w T 1 b c T 2 v 2 [v 1 (c T 2 v 2 ) -v 2 (c T 2 v 1 )] = e λt1 (v 2 × b) • k (v 2 × s 1 ) • k s 1 , (14)
giving ( 7). The derivation for the expression x(t 2 , b) in ( 7) is along the same lines.

We note that ( 8), (10), and (12) provide explicit expressions for t 1 that occur in (7). These, together with the dual expressions for t 2 , will be used in obtaining our main result. The classification of initial conditions made possible by conditions (i) and (ii) of Fact 2 are given in Figures 123, where basins leading to t 1 , t 2 , or neither are indicated relative to typical positions of the eigenvectors with respect to the region S.

Definition 1: An eigenvector v is interior to S if v or -v is in int(S); it is exterior to S if neither v nor -v is in S.
Note that eigenvectors of non-real eigenvalues are exterior since they are non-real.

Definition 2: A mode like (2) is called transitive [START_REF] Iwatani | Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems[END_REF] if either the trajectory intersects B 1 (at some finite time) for all b ∈ S or it intersects B 2 for all b ∈ S. The mode will be called negative-transitive in the former, and positive-transitive in the latter case. A mode is a source if there exists n ∈ int(S) such that for all b = αn + βs 1 with α ≥ 0, β > 0, the trajectory intersects B 1 and for all b = αn + βs 2 with α ≥ 0, β > 0, the trajectory intersects B 2 .

It is clear from Figures 1-3 that if a mode is neither transitive nor a source, then it is either a sink and all trajectories starting in (or entering into) S stay in S or it is a half-sink, that is there is a sector of S that is a sink. By Fact 2 (or, by Figures 123), it is also easy to see that Fact 3: A mode (2) is transitive if and only if the eigenvector(s) are exterior. It is a source (resp., sink) if and only if there are two eigenvectors such that the one associated with the larger (resp., smaller) eigenvalue is exterior and the other interior. It is a half-sink if and only if the eigenvector(s) are interior.

Definition 3: If a mode i is transitive, then its factor of expansion is

F i := ln |v×s1| |v×s2| + µt 2 if it is positive-transitive, ln |v×s2| |v×s1| + µt 1 if it is negative-transitive,
where t 1 , t 2 , v, and µ are as in Fact 2 associated with mode (2). In view of Fact 2.iii, the factor of expansion is the natural logarithm of the gain x sk a trajectory goes through when it starts at a border B l and traverses the whole sector hitting the other border B k .

In [START_REF] Nishiyama | Optimal stable state-space partitioning for piecewise linear planar systems[END_REF]) some regions of initial conditions of Fact 2 and in both [START_REF] Nishiyama | Optimal stable state-space partitioning for piecewise linear planar systems[END_REF] and (Liu et al. , 2006), an integral expression for F i in the non-real case have also been obtained. The "visible eigenvector" of [START_REF] Araposthasis | Stability and controllability of planar, conewise linear systems[END_REF] is one that lies inside the cone and serves the same purpose as the interior eigenvector of our Definition 1. A similar expression to that of F i above also figures in the main condition of Theorem 6 of [START_REF] Araposthasis | Stability and controllability of planar, conewise linear systems[END_REF].

Condition for Stability

The planar system (1) is well-posed in the sense of Carathéodory if there exist a unique solution of the form without any sliding mode, where f (x(τ )) is the discontinuous vector field given by the right hand side of (1) and x(t 0 ) = b [START_REF] Imura | Characterization of well-posedness of piecewise linear systems[END_REF]. Fact 1 implies a geometric condition for these. The condition (ii) below says, in effect, that trajectories of every pair of adjacent modes have the same direction on their common border. ii) for every pair of adjacent modes (i, k) with common border B il it holds that

x(t, b) = b + t t0 f (x(τ ))dτ, ( 15
   det V i (v i1 × s il • v i2 × s il ) det V k (v k1 × s il • v k2 × s il ) > 0 det V i (v i1 × s il • v i2 × s il ) det V k > 0 det V i det V k > 0. ( 16
)
respectively, if eigenvalues of both modes are distinct, if eigenvalues of mode i are distinct and of k are repeated or non-real, and otherwise.

We also remark here that in splitting a mode of dynamics A into two modes (to satisfy the assumption that all modes are defined on cones), one should take care to choose the common border not to coincide with any eigenvectors of A, since otherwise there will be a sliding mode by Fact 4. The assumption of well-posedness actually puts some serious constraint on the set of systems considered since any trajectory would have to evolve in one direction only. A thorough study of systems, like (Utkin, Guldner, & Shi, 1999), in which sliding modes and chattering is allowed is thus highly desirable, as pointed out in [START_REF] Imura | Characterization of well-posedness of piecewise linear systems[END_REF]. Fact 5: Let log denote the complex principal logarithm. Define, for i = 1, ..., m,

E i := λ i1 λ i1 -λ i2 log (v i × s i1 ) • k (v i × s i2 ) • k - λ i2 λ i1 -λ i2 log (v i × s i1 ) • k (v i × s i2 ) • k ,
where v i is v i2 or v + jv i2 and vi is v i1 or v i1 -jv i2 in case of real or non-real eigenvalues, respectively, and the right hand side is computed as lim(λ i1 -λ i2 ) → 0 in case of repeated eigenvalues. Then, F i = E i when mode i is positive-transitive and F i = -E i , when negativetransitive.

Proof. We omit 'i' whenever it is clear from the context. Let us consider the negative-transitive case. Suppose the eigenvalues are real and distinct so that

F i = λ 2 λ 1 -λ 2 log (v 2 × s 1 ) • k (v 2 × s 2 ) • k - λ 1 λ 1 -λ 2 log (v 1 × s 1 ) • k (v 1 × s 2 ) • k = λ 2 λ 1 -λ 2 ln |v 2 × s 1 | |v 2 × s 2 | - λ 1 λ 1 -λ 2 ln |v 1 × s 1 )| |v 1 × s 2 | = ln |v 2 × s 2 | |v 2 × s 1 | + λ 1 λ 1 -λ 2 ln |v 1 × s 2 )||v 2 × s 1 | |v 1 × s 1 ||v 2 × s 2 | = ln |v 2 × s 2 | |v 2 × s 1 | + λ 1 t 1 , (17)
where the first equality is by

v = v 2 , v = v 1 .
The second follows by noting, for any transitive mode, that

z l := (v l × s 1 ) • k (v l × s 2 ) • k > 0 for l = 1, 2
since eigenvectors are exterior to S and that log z i = ln z i for any real, positive z i . The last equality follows by the expression for t 1 in (8

) since b = s 2 and (c T 2 v 2 )k = -v 2 × s 1 , det V (w T 1 s 2 )k = -v 2 × s 2 , det V (w T 2 s 2 )k = v 1 × s 2 .
To derive (7) as the limit of F i = -E i in the case of repeated eigenvalues, consider the third expression in ( 14) with δ := λ 1 -λ 2 . By (8),

lim δ→0 λ 1 δ ln |v 1 × s 2 ||v 2 × s 1 | |v 1 × s 1 ||v 2 s 2 | = lim δ→0 λ 1 δ ln e δt1 = λt 1 . Thus, lim δ→0 F i = ln |v 2 × s 2 | |v 2 × s 1 | + λt 1 ,
where t 1 , we interpret, is given by ( 10) with b = s 2 . Finally, suppose the eigenvalues are non-real with

λ 1 = σ + jω = λ2 . With v = v 1 + jv 2 and v = v 1 -jv 2 , let us first note that log (v × s 1 ) • k (v × s 2 ) • k = ln |v × s 1 | |v × s 2 | + jθ, log (v × s 1 ) • k (v × s 2 ) • k = ln |v × s 1 | |v × s 2 | -jθ,
where

θ := ∠ (v × s 1 ) • k (v × s 2 ) • k = ∠ (v 1 × s 1 + jv 2 × s 1 ) • k (v 1 × s 2 + jv 2 × s 2 ) • k = ∠ -c T 2 v 1 -jc T 2 v 2 c T 1 v 1 + jc T 1 v 2 = arctan{ (c T 2 v 2 )(c T 1 v 1 ) -(c T 2 v 1 )(c T 1 v 2 ) (c T 2 v 1 )(c T 1 v 1 ) + (c T 2 v 2 )(c T 1 v 2 ) } = ωt 1 . (18)
The last equality follows upon setting b = s 2 in (12) and noting that c

T 1 v 2 = -det V det S w T 1 s 2 , c T 1 v 1 = det V
det S w T 2 s 2 . Therefore,

F i = λ 2 λ 1 -λ 2 log (v × s 1 ) • k (v × s 2 ) • k - λ 1 λ 1 -λ 2 log (v × s 1 ) • k (v × s 2 ) • k = σ -jω 2jω (ln |v × s 1 | |v × s 2 | + jθ) - σ + jω 2jω (ln |v × s 1 | |v × s 2 | -jθ) = -ln |v × s 1 | |v × s 2 | + σ ω θ = ln |v × s 2 | |v × s 1 | + σt 1 .
It follows that F i is as in Definition 3 for the case of non-real eigenvalues as well. The expression for positive-transitive case is similarly derived.

Admittedly, the limit argument in case of repeated eigenvalues is heuristic and needs to be done more rigorously. Nevertheless, the mere fact that the factor of expansion in all three cases can be expressed by a single formula is very appealing.

We can now state and prove an alternative to the algebraic condition of [START_REF] Iwatani | Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems[END_REF]) for stability of (1). The condition is "geometric" since a mode being transitive, source, or (half-)sink is characterized solely in terms of the eigenvectors associated with the mode and how they stand in the phase-plane relative to the sector on which the mode is defined.

Theorem 1: A well-posed system (1) is globally asymptotically stable if and only if when all modes i = 1, ..., m are transitive ⇒ m i=1 F i < 0, when a source i exists ⇒ λ i2 < 0, when a sink or half-sink i exists ⇒ λ i1 < 0.

Proof. Suppose all modes are transitive. Then, by well-posedness, all are positive or all are negative transitive, since otherwise there will be chattering. In either case, for any b ∈ R 2 , we must have x(t(b), b) = γ(b)b for some t(b) > 0 and γ(b) > 0, i.e., the trajectory comes back to the ray passing through b after going through an expansion or contraction of size γ(b). We might as well consider the case b = s 12 , which is taking the initial state to be on the left-hand border of the first mode, without loss of generality. If the modes are negative transitive, then by (7) of Fact 2, si2) ,

γ(s 12 ) = m i=1 |v i × s i2 | |v i × s i1 | e µit(
where µ i denotes λ i1 or σ i and v i denotes v i2 or v i1 + jv i2 in the cases of mode i having real or non-real eigenvalues. The system is globally asymptotically stable if and only if γ(s 12 ) < 1, which is equivalent to F 1 + ... + F m < 0 in view of ln[γ(s 12 )] < 0 and the expressions for t(s i2 ).

Note that if all modes are positive transitive, then starting the trajectory at b = s 11 , we have γ(s 11 ) = 1/γ(s 12 ), and the same condition is again obtained. Suppose now that not all modes are transitive and a mode i with n = v i2 is a source. In this case there must be a mode k that is a sink or half-sink since otherwise a sliding mode or chattering would exist. If the system is stable, then λ i2 < 0 must clearly hold for trajectories starting along that eigenvector to converge. Conversely, λ i2 < 0 implies that trajectories starting in mode i converge to the origin if they start along v i2 or they go outside S i and enter mode k. For such trajectories to converge to the origin, it is necessary that λ k1 , λ k2 < 0. The necessity of λ k1 , λ k2 < 0 for any sink or half-sink mode k is also clear by considering trajectories that start inside the sink-sector of k. Conversely, the sufficiency of the condition follows by the fact that any trajectory starting in a transitive mode or a non-sink sector of a half-sink must end up in either a sink or in the sink sector of a half-sink.

Corollary 1: Let B 1 , B 2 ∈ R 2×2 and c ∈ R 2 be given. A well-posed bimodal system ẋ = B 1 x if c T x ≥ 0, B 2 x if c T x ≤ 0, (19)
is globally asymptotically stable if and only if when both modes have non-real eigenvalues ⇒ σ1 ω1 + σ2 ω2 < 0, when a mode, say i, has real eigenvalues λ i1 ≥ λ i2 ⇒ λ i1 < 0.

Proof. In order to be able to apply Theorem 1, we let c 0 be any vector that is not perpendicular to any of the real eigenvectors of B 1 and B 2 , if any, and such that d

:= det[c c 0 ] is positive. Let C T 1 := [c c 0 ]. Note that S 1 := C -1
1 satisfies det S 1 > 0 and neither columns are in the direction of the real eigenvectors of B 1 or B 2 , if any. The four modal system

A 1 = A 2 = B 1 , A 3 = A 4 = B 2 , C T 2 := [-c 0 c], C 3 = -C 1 , C 4 = -C 2 is well-posed,
is in the framework of (1), and is equivalent to (19). Suppose, first that, say, B 1 has real eigenvalues so that modes 1 and 2 both have those eigenvalues with the same corresponding eigenvector(s). If, say, mode 1 is a source, then eigenvalues must be distinct and v 1 of mode 1 must be exterior. This implies, since mode 2 complements 1 in a half plane, that v 1 is interior to mode 2 and v 2 is exterior, that is mode 2 is a sink. By Theorem 1, the system is stable if and only if both eigenvalues of B 1 are negative. If mode 1 is transitive, then both eigenvectors are exterior to mode 1. Thus, the eigenvector(s) are interior to mode 2 so that mode 2 is a half-sink, which again implies that eigenvalues being negative is necessary and sufficient for stability. The other possibilities for mode 1 clearly give the same result. Suppose, second, that both B 1 and B 2 have non-real eigenvalues. It must be that all four modes are transitive in the same direction, say negative, by A 1 = A 2 = B 1 , A 3 = A 4 = B 2 and by well-posedness of (19). It is easy to compute, by Fact 5 and by the expression in (18), that

F 1 + F 2 = σ 1 ω 1 π, F 3 + F 4 = σ 2 ω 2 π,
which implies by Theorem 1 that the four-modal system, and therefore ( 19), is stable if and only if σ1 ω1 + σ2 ω2 < 0. Example 1: Consider the bimodal system

A 1 = 6 3 -3 6 , A 2 = -1 1 -1 0 , c T = 0 1
This system is unstable by Corollary 1 since σ1 ω1 + σ2 ω2 = 6 3 + -0.5 √ 3/4 > 0. Stability can be achieved, for instance, by simply adding a half-sink mode of arbitrarily narrow sector. Let a = 0. Consider the three-modal system, A 3 = -2 1/a a -2 , with A 1 , A 2 as given above and C 1 = 1 0.5/a 0 0.5/a , C 2 = 0 -0.5/a 1 -0.5/a , C 3 = -0.5 0.25/a -0.5 -0.25/a . Then, λ 31 = -1, λ 32 = -3, v T 31 = -1 -a , and v T 32 = -1 a . By Fact 3, Mode 3 is a half-sink since the eigenvectors are both interior to S 3 . Since λ 31 < 0, it follows, by Theorem 1, that the three modal system is stable for all a > 0. Figure 4 illustrates the sectors of such a system and a trajectory for the value a = 0.05.

Conclusions

We have derived a new set of necessary and sufficient conditions for the stability of a planar piecewise linear system. An obvious question is whether this helps in addressing the stability question in higher dimensions, like (1) in which A i , C i ∈ R n×n , with n ≥ 3. It is by now known that in, e.g, 3D, the condition of stability is highly "parameter dependent," which makes a progress difficult [START_REF] Eldem | A note on the stability of bimodal systems in R 3 with discontinuous vector fields[END_REF]S ¸ahan& Eldem, 2015). Nevertheless, we have been able to obtain some encouraging results using the approach presented in this paper.

Figure 1 .

 1 Figure 1. Basins for distinct eigenvalues, v 1 × v 2 is positively oriented.

Fact 4 :

 4 The system (1) is well-posed if and only if i) for every real eigenvector v il of mode i, it holds that v il × s il = 0, for l = 1, 2 and for all i = 1, ..

Figure 2 .

 2 Figure 2. Basins for distinct eigenvalues, v 1 × v 2 is negatively oriented.

Figure 3 .

 3 Figure 3. Basins for repeated eigenvalues, v 1 × v 2 is positively and negatively oriented.

Figure 4 .

 4 Figure 4. A typical trajectory of a three-modal system of Example 1

Acknowledgements

The authors greatly appreciate corrections and comments by V. Eldem, M. K. Camlibel, J.M. Schumacher, and M. E. Sezer on an early version of this paper.

S ¸ahan, G., [START_REF] Eldem | A note on the stability of bimodal systems in R 3 with discontinuous vector fields[END_REF]. Well posedness conditions for bimodal piecewise affine systems. Systems & Control Letters, 83 , 9-18. Utkin, V. I., Guldner, J. & Shi, J.(1999). Sliding Mode Control in Electromechanical Systems. London:

Taylor and Francis.


Geoffrey Gallé

Matthieu Nicoul

Theo Woike

Dominik Schaniel

Eric Freysz
email: e.freysz@cpmoh.u-bordeaux1.fr

Unraveling the mechanism of NO ligand photoisomerism by time-resolved infrared spectroscopy

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

Introduction

Light induced ligand isomerization first described by Jørgensen and Werner about a century ago [1 3] is a special case of photo chemistry. Upon optical electronic excitation a structural rear rangement of one or several ligands towards a metastable configuration is induced [4 7]. It can even lead to an exchange of atoms between neighboring ligands [8]. In order to understand such isomerization phenomena all processes following the elec tronic excitation need to be investigated which can be achieved by time resolved spectroscopy on a femtosecond time scale [START_REF] De Schryver | Femtochemistry[END_REF]. The ultrafast study of electronic, vibrational and structural changes reveals the correlation between the modified electron density and the structural response of matter. Consider an electronic transition that excites a molecule from the ground state (GS) into an anti bonding excited state (ES). Such a complex in a transition state can undergo a nuclear motion e.g., the rotation of a ligand. The ultrafast electronic excitation is followed by a slower nuclear re sponse on the fs ps time scale [START_REF] Sawyer | [END_REF]11]. According to the internal conversion the nuclear motion, e.g. rotation of the ligand, starts in a highly excited vibrational state of ES and in the absence of luminescence will cross over into highly excited vibrational rotational states of the novel geometry. The excess energy in this potential will be dissipated during the thermalization via vibra tional rotational states towards the ground state of the novel geometry [12]. If this novel geometry is metastable with sufficient lifetime, the structure can be determined by neutron or X ray diffraction [ 13 15]. Using UV Vis and infrared femtosecond spec troscopy all different steps of the photochemical reactions after the electronic excitation can be detected: the electronic relaxations are observed in the UV Vis spectral range and the nuclear motions are monitored in the infrared spectral range.

A typical example for such ultrafast photochemical processes is the reversible photoinduced linkage isomerization of the nitrosyl ligand in coordination compounds M NO [6,13] with M a transi tion metal. We study here the prototypic case of the [Fe(CN) 5 NO] 2 anion. As shown in Figure 1 the ground state (GS) is characterized by an almost linear Fe N O coordination. Irradiation with light in the blue green spectral range (e.g. k $ 400 500 nm) induces a charge transfer transition from the non bonding 3d xy orbital to the unoccupied anti bonding p ⁄( NO) orbital. Thereby, the system changes the symmetry from a 1 A 1 state to a 1 E twofold degenerate state. According to the Jahn Teller theorem the degeneracy of the 1 E state will be lifted. As a consequence the doubly degenerate deformational mode d(Fe N O) = 666 cm 1 can induce a rotation of the NO ligand [16 18]. The rotation of about 90°yields the side on configuration of Fe < N O (metastable state MS2) while a rotation of 180°results in the isonitrosyl configuration Fe O N (metastable state MS1). The structures of GS and of the metastable states MS1 and MS2 are known from X ray and neutron diffraction measurements at low temperatures in the static regime [19 21]. The lifetime of both states below 200 and 150 K, respectively, is nearly infinite due to the activation energies of 0.68 eV (MS1) and 0.43 eV (MS2) [22]. Both metastable states are lying about 1 eV above the ground state (see Figure 1) [23]. The transition from the excited state 1 E (ES) towards MS2 occurs radiationless with a time constant of 300 fs [24]. MS2 has a lifetime of about 300 ns ⇑ Corresponding author. Fax: +33 5 40 00 69 70. at room temperature [25]. According to these experimental results in the visible spectral range, a potential scheme as shown in Figure 1 with crossing at the saddle point between GS and MS2 at 1.43 eV was proposed [22]. Recently the competition between NO release and NO isomerisation in [Fe(CN) 5 NO] 2 in solution was studied using time resolved ps infrared spectroscopy [26]. In agreement with the previous study [24] it was concluded that the isomers are formed on a subpicosecond timescale. However, a detailed knowledge about the relaxation processes in ES and the transition process from ES into MS1 or MS2 is still lacking, which is connected with the question where and when the N O rotation starts. In order to shine more light on this relaxation from the excited 1 E state into the metastable states MS1 and MS2, we studied the rotation of the NO in Na 2 [Fe(CN) 5 NO] 2H 2 O (SNP, space group Pnnm) single crystals using polarized femtosecond infrared spectroscopy. In this manner the rotation of the NO ligand and the vibrational relaxation processes starting from the electroni cally excited state ES and ending in the metastable minimum of MS1 or MS2 can be observed. Infrared spectroscopy is especially suited for this purpose as GS, MS1, and MS2 can be clearly distin guished by the different m(NO) stretching vibration frequencies at 1948, 1835 and 1664 cm 1 , respectively. It yields an unambiguous identification of the corresponding relaxation pathways [26]. The use of single crystalline material has the advantage that we do not have to care about a possible solvent response which might mask the actual phenomena searched for [26]. Moreover in single crystals of SNP competing photochemical reactions like NO release are marginal when exciting in the visible spectral range.

Experimental set-up

The experimental set up is based on a 1 kHz Ti:Sapphire regenerative amplifier system producing 50 fs, 4.5 mJ laser pulses centered at 800 nm (Coherent Legend USP). The output of the laser system was split in two 1 mJ laser pulses which pumped two Optical Parametric Amplifiers (OPA, Model TOPAS from Light Conversion). One of these OPA was equipped with a Degenerate Frequency Generation (DFG) module for the generation of light in the spectral range of 2000 10.000 nm. The maximum output en ergy is about 5 lJ in the 4000 6000 nm spectral range. The probe pulse energy used in our experiment is at least ten times lower. The pump beam generated by the second OPA centred at k $ 500 nm and of energy of about 9 lJ was focused in the sample by a 10 cm ZnSe lens. The non collinear probe beam is focused within the sample using the same lens. The pump and probe beams overlap in the sample. We adjusted the lens so that within the sample the probe beam is more tightly focused than the pump beam. The sample used is a 270 lm thick single crystal (b cut) of Na 2 [Fe(CN) 5 NO] 2H 2 O. SNP crystallizes in the orthorhombic space group Pnnm. The polarization of the pump beam is parallel to the c axis while the probe polarization is parallel to the a axis. All measurements were performed at room temperature. Prior to the experiments, we measured the pump and probe pulse widths using an autocorrelation and a cross correlation (mixing the probe beam with the 50 fs pulses centred at 800 nm in a nonlinear LiNbO 3 crys tal) technique, yielding 120 ± 10 fs and 220 ± 50 fs for the pump and probe pulses, respectively. To measure the transient absorp tion signal at a given time delay s between the pump and the probe, we used two HgCdTe detectors, placed before and after the sample, that measure the initial (I r ) and the transmitted (I t (s)) intensity of the probe beam respectively. The detectors were cooled at liquid nitrogen temperature. The time delay s between the probe and pump pulses was controlled using a motorized translation stage that has a spatial resolution of 1.5 lm. Hence 3300 700 nm with a resolution of 6 cm 1 . The polarization was chosen along all crystallographic axes to find clearly the total sym metric NO stretching modes. They are forbidden for light parallel to the c axis. MS1 and MS2 are detected with a cryostat at T = 80 K.

The measured overtones are for GS at T = 296 K in cm 1 : 1948, 3872, 5769, 7639, 9483, 11.301. For MS2 at T = 80 K: 1664, 3296, 4890, 6454. For MS1 at T = 80 K: 1835, 3637, 5407, 7147.

Figure 2 shows the evolution of the differential transient absorption signal DOD measured at the probe wavelengths 5100 nm (1961 cm 1 ) (GS), and 6130 nm (1631 cm 1 ) (MS2) after excitation with 500 nm. At long timescales (Figure 2a) beyond 20 ps we observe constant values of DOD that remain up to 1 ns. At 6130 nm a positive differential transient absorption DOD $ 0.02 is observed, indicating the population of the MS2 state. At 5100 nm a negative DOD $ 0.015 is observed corresponding to the bleach of the GS. Thus for timescales above 20 ps we conclude that the MS2 state is populated at the expense of the GS. Consider now the temporal evolution in the range 0 10 ps (Figure 2b). In the first 340 fs the signal is dominated by the cross correlation of pump and probe pulse inside the SNP crystal, so that no conclusion about molecular processes inside this time window can be drawn. At 340 fs we observe a strong positive DOD $ 0.14 at 6130 nm, which then relaxes with two time constants of 300 fs and 9 ps before reaching the constant value at 20 ps. At 5100 nm the DOD rises after 340 fs and reaches a maximum of DOD $ 0.03 after about 1.2 ps with a time constant of 300 fs before relaxing with a time constant of 6 ps towards its constant negative value at 20 ps.

Figure 3 shows the evolution of the differential transient absorption signal DOD measured after excitation with 500 nm at the probe wavelength 5460 nm (1835 cm 1 ) corresponding to the vibration frequency of the MS1 state. Since at the long time scale (i.e. beyond 20 ps), DOD $ 0 these data clearly indicate that the MS1 state is not populated. However, one could clearly notice that upon excitation of the sample a transient absorption is recorded. This transient absorption basically follows the trend recorded for MS2 state. It increases during the pump excitation and then relaxes with long constant times of 9 ps. We noticed that we could improve the fitting of our data considering the relaxation was proceeding with two relaxations time of 6 and 9 ps. The weight of these two contributions is 10% and 90 %, respectively.

We can thus distinguish three phases in the temporal evolution:

(i) An immediate response within the experimental resolution of 340 fs for 6130 nm and 5460 nm. (ii) A fast relaxation for 6130 nm as well as a fast increase for 5100 nm of the order of 300 fs.

(iii) A long time relaxation of the order of 9 ps for the probe wavelength centered at 6130 nm, of the order of 6 ps for the probe wavelengths centered at 5100 nm and and 6 and 9 ps for the probe wavelengths centered at 5460 nm.

Discussion

These observations allow for drawing a detailed picture of the processes leading to the photoisomerization (see Figure 3). In the first step the excitation from the GS to the excited state ES occurs, initiated by the 3dxy ? p ⁄ (NO) HOMO LUMO transition with an energy distance of 19762 cm 1 (2.45 eV, 506 nm) at T = 296 K when detected parallel to the a axis [28]. An important observa tion is that immediately after the illumination (pump pulse of 120 fs duration) with 500 nm light the NO vibration can be observed as absorption at 6130 nm (1631 cm 1 ). Thus the distance between two neighboring m(NO) vibrational levels in ES are close to those in MS2, which are known to be at 1664 cm 1 from low temperature (T < 150 K) static infrared spectroscopy [27]. We can thus conclude that in this novel electronic configuration the Fe N O is kinked with respect to the linear GS configuration which is in agreement with DFT calculations for the transition from GS to MS2 [ 16 18], which propose an Fe N O angle of 30 50°in ES. The increase of the absorption at 5460 nm is also a good indication that indeed some of the molecules in the ES state have a kick angle lar ger than 45°. Indeed in such a case, one could consider that the vibrational excited state can be decomposed into the three m(NO) vibrational levels associated to the MS1, MS2 and GS states. This explains the immediate observation of a strong absorption at 6130 and 5460 nm in ES. For the other probe wavelength 5100 nm (1961 cm 1 ) an absorptive signal can be detected only after the depopulation of the ES and the subsequent occupation of GS. Therefore, the decay time (300 fs) of ES corresponds exactly to the increase (300 fs) of MS2.

The first Fe N O bending following the electronic excitation corresponds to an intermediate excited state. The molecule can now follow two different paths: either it returns back to the linear GS configuration or it continues its rotation towards the MS2 side on configuration. Let us stress that the absence of an absorption at 5460 nm on the longer time scale discards the relaxation channel towards the MS1 state. The observation of a decrease of absorption at 6130 nm as well as an increase at 5100 nm with the same time constant of 300 fs illustrates these steps in the photoisomerization. The excited intermediate state ES is depopulated towards GS and MS2 mainly by internal conversion. In Figure 4 out of ES are illustrated for ES, GS and MS2 within a potential scheme. The cut through the hyper space is chosen in such a way that at fixed bond lengths changes the reaction coordinate corre sponds to the Fe N O angle. By the excitation with 20.000 cm 1 an m(NO) stretching vibration in ES is occupied from which the measured transition (1631 cm 1 ) to higher vibrational levels occurs. In order to facilitate an internal conversion the GS should exhibit a vibrational level of m(NO) at around 20.000 cm 1 . In order to explore the positions of the m(NO) overtones in GS and MS2 we have performed polarized absorption spectroscopy on thick SNP single crystals. For GS we found m(NO) overtones up to n = 6 For MS2 the m(NO) overtones were measured at T = 80 K since MS2 decays around 150 K due to the activation energy of 0.43 eV.

Energetically the MS2 minimum is 1 eV above GS. The measurements yielded results up to the fourth overtones (see experimental details) and the extrapolation up to n = 7 yields E(7) = 11.612 cm 1 . Adding the energetic position of the MS2 minimum 8066 cm 1 (1 eV) yields 19678 cm 1 , which is very close to the excitation energy of 20000 cm 1 . The zero point energy in MS2 is E(0) = 842 cm 1 . Thus energetically an internal conversion from ES to GS and MS2 is possible. Furthermore all the states have the same multiplicity 1A and no luminescence was found during the decay of MS1 or MS2 in the spectral range 200 1200 nm up to now. Moreover MS2 is diamagnetic and thus the occupation of a triplet state in ES is highly improbable as it would require a double spin flip, one from GS to ES and one from ES to MS. The steady sequence of overtones in MS2 and MS1 up to 0.8 eV further hints to a potential crossing between GS/ MS2 and GS/MS1. For the case of an avoided crossing we would expect a discontinuity in the sequence of overtones. Due to the fact that immediately after pulsed excitation at 500 nm we observe absorption at both 5460 and 6130 nm, the geometry of ES is very likely a kinked NO configuration which is agreement with DFT calculations that propose an Fe N O angle in the range of 40 50°as proposed by DFT calculations [16 18]. The minimum of the energetic position of ES is unknown, but we know that the crossing of GS and MS2 is at 1.43 eV [22] and so 1.06 eV below the excitation with 2.49 eV. Then a relaxation from ES into GS and MS2 by occupation of highly excited NO vibrational levels that are energetically close to the NO vibrational levels of ES can occur within s = 300 fs. A precedent relaxation within ES would need more time and we therefore exclude this possibility. This fast decoupling of the ES towards MS2 or back to GS is in accordance with the results of the time resolved study in the visi ble range [24] where an occupation of MS2 within 300 fs was found. Thus the rotation of the NO ligand both towards 90°as well a back to 0°occurs with s = 300 fs. We can now specify that after this first fast relaxation the molecules are in a highly vibrationally excited state of the corresponding ground states (GS or MS2).

From this vibrationally excited states the thermalization of the molecules towards the corresponding minima in GS or MS2 occurs with time constants of s = 6 ps (GS) and s = 9 ps (MS2). From the knowledge of the extrapolated overtones of the m(NO) stretching vibrations in GS (see experimental details) we can estimate that in GS the thermalization occurs starting from 20000 cm 1 via 10 NO levels, such that for every transition about 600 fs are needed. The fact that the absorption at 5460 nm relaxes with the same con stants time as the one we recorded for the thermalization MS2 state indicates that it is mainly the release of the energy towards the molecular and lattice vibrations that accounts for the absorp tion at 5460 nm absorption. As this excess of energy is released, the molecule reaches its mimimum position within its potential well, this absorption at 5460 nm is cancelled out.

The picture we have elaborated can now be compared to ab initio calculations including excited states [16,17,29,30]. In these studies the GS potential is calculated as a function of the Fe N O angle. The energetic positions of the metastable minima and saddle points are in general calculated too high as compared to the exper iment. The doubly degenerate excited state 1E (ES) is considered Jahn Teller active and consequently upon optical excitation the symmetry of the system is spontaneously lowered resulting in two non degenerate states 1A 00 and 1A 0 . In [17] the geometrical optimization yields an eclipsed bent Fe N O geometry for the more stable 1A 00 state with a Fe N O angle of about 40°and a Jahn Teller stabilization energy of about 0.33 eV. 1A 00 is thus about 2.1 eV above the GS. Deactivation from this relaxed excited state then occurs via a non radiative process or a conical intersection. Given that we excite with 2.5 eV the system is 0.4 eV above this relaxed excited state such that the vibrational excitation of the ex cited state 1E is limited to one or two levels (0.4 eV $ 3226 cm 1 ) supposing that their distance is 1664 cm 1 . This picture agrees well with the one illustrated in Figure 4. Notably the distance be tween the m(NO) vibrations is much smaller in the 40°configura tion compared to the 1948 cm 1 of GS. Thus the decision whether GS, MS1 or MS2 is occupied is taken in ES. To this the potentials of GS, MS1 and MS2 must exhibit crossing points, so that the vibrationally excited states of GS, MS1 or MS2 can be occupied. It can be expected that this kind of photoisomerization scheme ap plies also to MS1 and to the other compounds of the ML5NO family as well as to similar photoswitchable ligands such as N2, NO2, SO2 etc., in which the multiplicity 1A is conserved in all states and thus no intersystem crossing occurs.

Conclusion

In conclusion we observed the ultrafast photoisomerization in SNP using time resolved infrared spectroscopy. A complete description of the initial process for the NO photoisomerization could be deduced on the basis of internal conversion. The optical excitation leads to an excited state from which the rapid transition into MS2 and GS occurs within 300 fs due to vibronic coupling. The rotation of the NO ligand either to 90°or back to 0°proceeds with this time constant. After these fast ($300 fs) transitions the vibrational rotational thermalization to the GS and MS2 potential minima occurs within a few picoseconds.

  our set up makes it possible to achieve a temporal resolution of ±10 fs. By computing the logarithm of I r /I t (s), we determined the absorption signal s(s) = a  d = n  r  d with a as absorption coefficient, d thickness of the sample, n number density of the occupied state and r as the cross section. The differential transient absorption signal DOD(s), at a given delay s, was then computed by subtracting the absorption signal measured with and without the pump pulse. [OD = log(I 0 /I(t)) = log[exp(ad)] = a  d  log(e) = nrd  0.4343]. For that purpose the pump beam was chopped at 40 Hz. For our measurements, the typical measurement error was better than 10 3 optical density (OD). The overtones are measured with an absorption spectrometer in the spectral range of

Figure 1 .Figure 2 .

 12 Figure 1. Potential scheme illustrating the NO linkage isomerisation process, inferred from time-resolved visible spectroscopy [14].

Figure 3 .

 3 Figure 3. Temporal evolution of the differential transient absorption DOD after excitation with 500 nm probed at the wavelengths of the NO stretching vibration in MS1 (5460 nm). The solid line is a fitting of the data.

Figure 4 .

 4 Figure 4. Two dimensional cut through the potential hyperspace illustrating the photoisomerization process via internal conversion in SNP. The v(NO) frequencies indicated for GS (1948 cm 1 at RT) and MS2 (1664 cm 1 at 80 K) are obtained from static infrared spectroscopy [18].

( 11 .

 11 301 cm 1 ) at T = 23 °C (see experimental details). From the positions we could extrapolate with a polynom of order 4 the positions of the m(NO) overtones up to n = 10 yielding E(11) = 19.274 cm 1 . The zero point energy is E(0) = 984 cm 1 .

Acknowledgements

Financial supports by the DFG (SCHA1550/1 1), National Research Agency (ANR 09 BLAN 0212) and Conseil Régional d'Aquitaine are gratefully acknowledged.


Geraldine Sarret
email: geraldine.sarret@ujf-grenoble.fr

M.-P Isaure

Matthew A Marcus

Emiko Harada

Yong-Eui Choi

Sébastien Pairis

Sirine Fakra

Alain Manceau

Géraldine Sarret

Marie-Pierre Isaure

Géraldine Sarret

Marie-Pierre Isaure

Chemical forms of calcium in Ca,Zn-and Ca,Cd-containing grains excreted by tobacco trichomes

Keywords: biomineralization, detoxification, micro-XANES, micro-XRD

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

Introduction

Recently, a novel original mechanism of Zn and Cd detoxification was described in tobacco (Nicotiana tabacum L. cv. Xanthi). Tobacco exposed to Cd excreted Cd,Ca-containing grains through leaf trichomes (1,2), and a similar excretion of Zn,Ca-containing grains was observed under Zn exposure (3). Trichomes are specialized epidermal structures. In tobacco, they are glandular and excrete various organic substances including nicotine and resins. Such detoxification process may have implication in human health since smoking is one of the principal routes of exposure to heavy metals, and also in phytoremediation as tobacco is a candidate species for phytoextraction.

The morphology, elemental composition, mineralogy as well as Zn and Cd speciation in the grains were investigated by scanning electron microscopy coupled with energy dispersive X-ray analysis (SEM-EDX), micro-X-ray fluorescence (µXRF), micro-X-ray diffraction (µXRD), Zn K-edge micro extended X-ray absorption fine structure (µEXAFS) (3) and Cd L III edge micro Xray absorption near edge structure (µXANES) spectroscopy (Isaure et al., in prep.). Calcium was always the major component. µXRD analyses revealed the presence of calcite, and less frequently vaterite and aragonite, two other CaCO 3 polymorphs. Calcium oxalate mono and dihydrate were found occasionally. Calcite and vaterite were substituted by cations, probably Zn, Cd and possibly Mn and Mg. Zn µEXAFS confirmed the occurrence of Zn-substituted calcite, and evidenced Zn associations with other phases including organic compounds, silica and phosphate (3). Cd µXANES showed that cadmium was a Ca substituent in calcite and vaterite and/or sorbed on the surface of these minerals (Isaure et al., in prep.).

The mechanism of formation of these grains remains unclear. Biogenic minerals may result from a controlled biomineralization process leading to well defined minerals and shapes. They may also be "biologically induced", which means that an organism promotes the precipitation, but does not control the crystallization process (4). In the present case, grains could result from the exudation of a liquid containing the various metals and substances, and precipitation of the various solid phases due to water evaporation. Alternatively, they might be formed intracellularly and excreted thereafter as suggested by Choi et al. [START_REF] Choi | [END_REF]. The purpose of this study is to better characterize the structure and composition of the grains. Grains excreted by tobacco under various Zn, Cd and Ca exposure conditions were investigated by SEM-EDX, µXRD and Ca K-edge µXANES spectroscopy. Results of these complementary approaches are compared, and a hypothesis for the formation of the grains is proposed in light of these data.

Experimental

Materials

Plant culture and grain collection procedure have been described previously (3). Briefly, tobacco plants were grown in hydroponic conditions and exposed for 5 weeks to Zn or Cd, with or without a supplement of Ca (Table 1). A control condition with Ca only was also tested. Then, grains were collected by washing the leaves in ultrapure water and centrifugating the suspension.

Several Ca-containing reference compounds were used for the Ca XANES data analysis.

Calcite, vaterite and Cd-containing vaterite were synthesized at room temperature according to a modified protocol of Paquette and Reeder (6,7). Briefly, an aqueous solution (500 mL) containing 10 mM CaCl 2 and 1.8 M NH 4 Cl was placed in a glass reactor, an EPPENDORF tube containing solid ammonium carbonate was allowed to float at the surface of the solution, and the reactor was closed and kept unstirred. NH 4 Cl was used as a background electrolyte to provide a high ionic strength. Initial pH was 4.9. The decomposition of ammonium carbonate produces NH 3 (g) and CO 2 (g), which dissolve into the solution, simultaneously increasing pH and alkalinity. The supersaturation of the solution induces the formation and growth of CaCO 3 crystals. Continuous sublimation of NH 3 (g) buffers the solution around pH = 7.9. After 13 days, the reactor contained rhombohedral crystals and spherical particles attached on the surface of the glass. The two types of particles were separated and characterized by XRD. Rhombohedral crystals corresponded to calcite, and spherical particles to vaterite. For the synthesis of Cdcontaining vaterite, the same procedure was used except that after 13 days, when crystal size amounted to 150-200 µm in diameter, the CaCl 2 -NH 4 Cl solution was progressively doped with 0.1 M CdCl 2 to a total concentration of 100 µM Cd in order to incorporate Cd as a Ca substituent in vaterite (6,7). The progressive addition of CdCl 2 kept the solution undersaturated with respect to otavite (CdCO 3 ). After 7 days, the particles were collected and separated. Spherical particles were identified as Cd-containing vaterite by XRD and µXRF. The Cd content was a few tens ppm based on µXRF analysis.

Aragonite was a natural specimen. The XANES spectrum for synthetic amorphous CaCO 3 was provided by Y. Politi (8). Ca oxalate monohydrate (whewellite) was purchased from Aldrich. A solution containing 0.5 M Ca(NO 3 ) 2 at pH 2.6 was used as a reference for aqueous Ca 2+ . The phase purity of all crystalline samples was checked by XRD prior to XANES analysis.

Electron microscopy

The morphology and the chemical composition of the grains were studied by SEM-EDX using a Jeol-JSM 840A equipped with a Kevex Si(Li) detector. The chamber pressure was 10 -6 -10 -5 Torr, and the accelerating voltage 20 kV. Grains were mounted on kapton tape or on carbon tape, then fixed on carbon stubs and coated with carbon. Images were taken at a magnification of 500 to 10000. EDX spectra were recorded on prominent spots of the grains to optimize the detection.

Micro-XRF, µXRD and µXANES

The µXRF, µXRD and Ca K-edge µXANES measurements were performed on beamline 10.3.2 of the Advanced Light Source (ALS, Berkeley, CA) (9). The grains were spread on kapton tape and analyzed at room temperature and pressure. First, each grain was mapped by µXRF. Then, µXANES and µXRD and data were recorded on the Zn-richest region for the Zn and Zn + Ca treatments, on the Cd-richest region for the Cd and Cd + Ca treatment, and on the Ca-richest region for the Ca treatment. µXRF data were collected at 10 keV with a beam size of 5 × 5 µm.

The same X-ray spot size was used for Ca µXANES. Fluorescence X-ray yield was measured with a 7-element Ge solid-state detector. The spectra were recorded between 3900 to 4400 eV. µXRD patterns were recorded at an incident energy of 17 keV with a beam size of 16 × 7 µm.

More experimental details are given in (3). Ca XANES reference spectra were recorded at room temperature on beamline ID21 at the European Synchrotron Radiation Facility (ESRF, Grenoble, France). Each spectrum is the average of two to three spectra, each about 15 minutes long.

XRD and XANES data treatment

The XRD data treatment was performed as described previously (3). Briefly, after calibration with alumina, two dimensional XRD patterns were integrated to one-dimensional patterns for peak assignment. For substituted calcite crystals, the unit cell parameters a and c were refined, and the stoichiometry of the substituent was estimated from these two parameters using the Vegard law [START_REF] West | Solid State Chemistry and its Applications[END_REF] with calcite, smithsonite (ZnCO 3 ) and otavite (CdCO 3 ) as end-members for the Ca-Zn and Ca-Cd solid solutions. This approach cannot be used for substituted vaterite because of the absence of a CdCO 3 structural analogue of vaterite. XANES spectra were processed using WinXAS [START_REF] Ressler | [END_REF]. All spectra were energy calibrated with respect to calcite (inflection point for this reference set to 4042.6 eV). The collected scans were averaged; background subtracted and normalized using linear (pre-edge) or cubic (postedge) polynomials. XANES spectra were then fitted by linear combinations using calcite, aragonite, vaterite, Cd-containing vaterite, amorphous CaCO 3 , aqueous Ca 2+ and Ca oxalate monohydrate reference spectra. The quality of the fits was quantified by the normalized sumsquares residuals NSS = Σ(µ experimental -µ fit ) 2 / Σ( µ experimental ) 2 x 100, in the 4000-4150 eV range, where µ is the normalized absorbance. An energy shift of ± 0.5 eV maximum and a correction of slope were allowed to account for the energy resolution of the monochromator and for possible inconsistencies in data processing. For some spectra, linear combination (LC) fitting did not provide satisfactory results, and over-absorption effects were suspected based on the comparison of these spectra with the standards. Therefore, each spectrum was fitted with and without a correction of over-absorption using a simple model [START_REF] Manceau | Applications of Synchrotron Radiation in Low-Temperature Geochemistry and Environmental Science[END_REF]. This model assumes that the smooth part of the resonant absorption is a constant fraction of the non-resonant background (which is a reasonable assumption in the XANES region) and that the sample is infinitely thick, i.e., totally absorbs the incident beam (which is probably true considering the energy range and the size and global composition of the grains). The equation used is:

y experimental = (1 + a) / (1 + a ⋅ y corrected ) (1)
where y is the normalized resonant absorbance, which equals 0 below the edge and oscillates around 1 above the edge, and a = µ resonant / (µ non resonant + µ fluorescence ) where µ resonant , µ non resonant and µ fluorescence are the three components of the absorbance µ .

Equation ( 1) can be solved as:

y corrected = y experimental / (1 + a (1-y experimental )) (2)
The over-absorption parameter a equals 0 in the absence of over-absorption effect, and increases with this effect. Because of the sample heterogeneity, the precise composition of the sample is not known, so a is an adjusted parameter.

Results

The morphology of the grains as observed by SEM varied considerably. The grains varied from 10 µm to 150 µm in diameter. Each grain is an aggregate of different types of particles. The first type is micro faceted crystallites (Insets in Fig. 1a and1b) and the second type is globular structures of various sizes (from 2 to 50 µm, Fig. 1c, 1d and 1e). Some grains such as Zn2 (Fig. 1c) seem visually less crystalline. The elemental composition of the grains varies from one grain to another, and within the same grain. Ca was always the major component regardless of the plant metal treatment conditions. Minor elements include Mg, Si, P, Cl, K, and Zn or Cd depending on the plant treatment (Fig. 1). The high Si peaks observed in Fig. 1a to 1d are attributed to kapton tape on which the grains were mounted. Silicon is not detected in the grain presented in Fig. 1e (mounted on carbon tape). However, Si was found in small amount in some grains (not presented in this study), as well as Mn.

Our previous study using µXRD (3) showed that grain ZnCa3 contained substituted calcite and aragonite, that grain ZnCa4 contained substituted calcite, and that grain Zn2 did not contain crystalline phases. The two-dimensional µXRD patterns for grains ZnCa5 and Zn3 show that the Bragg reflections for these two grains consist of small arcs of Debye-Scherrer rings, indicating that the grains are composed of micrometer-size mosaic crystals (Fig. 2b). All reflections correspond to substituted calcite, and the major Ca substituent is Zn based on µXRF (Fig. 2a).

Note that although Ca is the major element, the Ca Kα peak has a low intensity relative to the Zn Kα peak due to air absorption and the lower fluorescence yield of Ca vs. Zn. and 1.00 Å for Ca 2+ compared to 0.74 Å for Zn 2+ ) [START_REF] Shannon | [END_REF]. For instance, the substitution rate for grain CdCa2 could be anywhere between 30 and 80%. It was not possible to calculate the stoichiometry of Cd in vaterite because of the absence of a Cd end member (see the Experimental section).

Calcium being the major element in the grains, Ca K-edge µXANES spectroscopy was then performed to get some insights on the composition of the grains, and test for the presence of amorphous Ca phases. Figure 3a shows the XANES spectra for several Ca reference compounds.

The four CaCO 3 species (calcite, vaterite, aragonite and amorphous CaCO 3 (ACC)) have clearly distinct spectra, which enables their identification (14,15). However, Ca XANES has no sensitivity to substitutional impurities as seen for instance for low Cd-containing vaterite (Cd content lower than 1%, as estimated by µXRF) and pure vaterite exhibiting nearly identical spectra. The spectra for aqueous Ca 2+ , Ca oxalate monohydrate and ACC share some similarities.

However, the lower part of the edge is shifted to lower energy for ACC spectrum, and the maximum of the white line (zero value of the first derivative) increases from 4049.5 eV (ACC) to 4049.9 eV (Ca oxalate monohydrate) and to 4050.5 eV (aqueous Ca 2+ , Fig. 3b). The aragonite spectrum is distinct from the three previous spectra in displaying a shoulder at 4046 eV (arrow in Fig. 3), and a minimum between 4060 and 4070 eV.

Figure 4 shows the Ca µXANES spectra for the grains. They were fitted by LC using the standard spectra presented above, and results are presented in Table 2. The ZnCa3 spectrum presents strong similarities with the aragonite spectrum, but markedly lower amplitude. The drop in amplitude suggests some over-absorption effect [START_REF] Manceau | Applications of Synchrotron Radiation in Low-Temperature Geochemistry and Environmental Science[END_REF] due to a high Ca content of the analyzed spot. The fit without over-absorption correction (see materials and Methods) did not provide satisfactory result (NSS = 0.065 %, with 68% vaterite + 25% aragonite). Introducing an overabsorption correction greatly improved the fit (NSS = 0.014 % with aragonite as only component, Fig. 4). The a value found (1.38) corresponds to a very strong over-absorption effect. The fit was not improved significantly by introducing a second component (NSS decreased by less than 10%). Thus, the spot analyzed by µXANES likely contains aragonite as the sole Ca species.

The grains ZnCa4 and ZnCa5 were fitted with 98% calcite. The fit was slightly improved with a correction of over-absorption (NSS = 0.015 and 0.021 %, compared to 0.020 and 0.035 %, respectively), and the proportions were unchanged. Adding a second component did not improve significantly the fits (NSS decreased by less than 10%).

For grain Zn2 and CdCa6, fits without over-absorption correction were not satisfactory (NSS = 0.048 and 0.091 %, respectively, for two-component fits). Fair one-component fits with over-absorption correction were obtained with ACC (NSS = 0.021 and 0.024 %, respectively), but the match was not optimal for the first oscillation (around 4080 eV). Ca oxalate monohydrate and aqueous Ca 2+ provided weaker fits (NSS = 0.086 and 0.053 % for Zn2, and 0.076 and 0.043 % for CdCa6, respectively). Adding a second component did improve the reproduction of the first oscillation. Several fits of equivalent qualities were obtained with ACC as major component (about 80 and 70 % of total Ca) and various species as secondary component. NSS values were more than doubled if ACC was excluded (Table 2 and Fig. 1 in Supplemental Information).

Therefore, grains Zn2 and CdCa6 likely contain ACC as major species, and an additional species whose nature remains unknown.

For the other grains (Zn3, CdCa2, CdCa3 and Cd10), over-absorption correction did not improve the fits. Calcite was the major form in grains Zn3 and CdCa2, and fits of equivalent quality were obtained with ACC, aqueous Ca 2+ and Ca oxalate monohydrate as minor species. Aqueous Ca 2+ is unlikely in solid-state material. ACC and Ca oxalate monohydrate are more likely. In the absence of reference spectra for Ca bound to organic ligands other than oxalate, this latter compound may be considered as a proxy for Ca bound to organic compounds in general.

Therefore, this pool is referred to as "organic Ca and/or ACC" in Table 3. Vaterite was found in grains CdCa3 and Cd10. For this latter grain only, no satisfactory fit was obtained with two components (NSS = 0.046 %), so a third component was introduced. Again, a contribution of organic Ca and/or ACC was found in grains CdCa3 and Cd10.

The speciation of Ca was then compared with the mineralogy and morphology of the grains (Table 3). The percentages were rounded to the nearest ten for clarity. Except for two grains, ZnCa3 and CdCa2, the same crystalline species were identified by µXRD and Ca µXANES. Znsubstituted calcite and Cd-substituted vaterite were identified by µXRD in grains ZnCa3 and CdCa2, respectively, but not by Ca µXANES spectroscopy. This difference likely results from the larger beam size and higher penetration depth of X-rays in XRD measurements relative to µXANES (16 × 7 µm vs 5 × 5 µm, and a few tens of µm at 17 keV vs. a few µm at 4 keV). Thus, XRD probes a different material volume than Ca µXANES even when conducted on the same spot.

Discussion and Conclusion

Results showed that beside crystalline calcium carbonates (calcite, vaterite and aragonite), the tobacco grains contained ACC as well. Whewellite (Ca oxalate monohydrate) did show up in some XANES fits, but its presence could not be firmly attested. Ca oxalate mono-and dihydrate have several XRD peaks in common with calcite, but peaks at distinct positions as well. These two minerals were positively identified by µXRD in other tobacco grains based on the presence of these specific peaks (3). In the present study, no such peaks were found on the XRD patterns.

However, it is still possible that coarse crystals of Ca oxalate mono or dihydrate are present but produce non-specific peaks only, or that Bragg conditions were not met for these coarse crystals.

Therefore, the presence of Ca oxalate mono-and dihydrate cannot be ruled out. Other undetermined organic ligands may also complex Ca. In addition, mixed mineral and organic Ca compounds such as organic matter-containing calcite ( 16) may occur.

The Ca µXANES analysis showed that strong over-absorption effects may take place and decrease dramatically the amplitude of the spectra. We show that this effect is far from negligible when identifying and quantifying Ca species. For instance, for grain ZnCa3, the uncorrected spectrum was fitted by 68% vaterite + 25% aragonite, and the corrected spectrum was fitted with 100% aragonite. Note that the LCF without over-absorption correction was relatively bad, which alerted us on the possibility of an over-absorption effect.

In the environment, ACC is thermodynamically unstable and rapidly transforms into vaterite and then calcite (17). However, it is found as a stable compound in plants (e.g., cystoliths) and animals (e.g., cuticle of crustaceans, spicules of ascidiae, granules in molluscs) (18). ACC is probably stabilized by proteins, magnesium and phosphorus in these organisms (18). In our case, Mg and P were found in the grains.

Biologically controlled biomineralization leads to well defined mineral species and shapes.

Here, the variety of morphologies and of CaCO 3 crystal structures, and their coexistence with amorphous and possibly organic forms, supports the hypothesis of a biologically induced (as opposed to biologically controlled) biomineralization. This study illustrates the potential of microfocused X-ray techniques to study biomineralization processes, and strategies of metals immobilization developed by plants to cope with metal toxicity.

this paper

unpubl.

-

  Isaure et al., in prep.

Figure 1 :

 1 Figure 1: SEM imaging for the grains ZnCa3 (a), ZnCa4 (b) Zn2 (c), CdCa3 (d) and grain Ca3 (e), and EDX analyses were performed on the spots marked by a black cross.

Figure 2 :

 2 Figure 2: µXRF spectrum (a) one-and two-dimensional µXRD patterns (b) for grain ZnCa5 and Zn3. All XRD peaks were attributed to substituted calcite. The positions of three peaks for pure calcite are indicated by arrows.

Figure 3 :

 3 Figure 3: Ca K-edge XANES spectra for the reference compounds used in the linear combination fits (a) and zoom on the main peak for aqueous Ca2+, Ca oxalate monohydrate and ACC (b).

Figure 4 :Figure 2 19Figure 3

 423 Figure 4: Ca K-edge µXANES spectra for the tobacco grains and corresponding linear combination fits. Five spectra (Zn2, CdCa6, ZnCa3, ZnCa5 and ZnCa4) were corrected for overabsorption.

  0.95 Zn 0.05 CO 3 and Ca 0.89 Zn 0.11 CO 3 , respectively. The difference in the Zn stoichiometry coefficient obtained using parameters a and c was ± 0.01. The µXRD results for the Cd-containing grains will be presented separately together with Cd L III -XANES data (Isaure

	et al., in preparation). Briefly, the grains CdCa2 and Cd10 contain micrometer crystals of
	substituted calcite and finer (nanometer sized) crystals of substituted vaterite. Grain CdCa3
	contains nano-crystals of substituted vaterite only. Finally, no crystalline phases were detected in
	grain CdCa6. For Cd-substituted calcite, the Vegard law provides an imprecise estimation of Cd
	stoichiometry because the contrast in ionic radius between Cd and Ca is small (0.95 Å for Cd 2+

The refined unit cell parameters a and c for grain ZnCa5 and Zn3 were 4.97 and 16.97 Å, and 4.95 and 16.83 Å, respectively, compared to 4.9896 and 17.0610 Å for pure calcite. The calculated formulae using the Vegard law were Ca

Table 1 .

 1 Tobacco culture conditions and name of the grains investigated

	Treatment conditions	Cation Concentrations (mM) in the Nutrient Solution	Names Grain		Techniques and References
		Ca	Zn	Cd		SEM-EDX	µXRD	Zn µEXAFS	Cd µXANES	Ca µXANES
	Zn	0.28	0.25	0	Zn2	(3) and this paper
					Zn3	-

Table 2 . Ca µXANES Results Treatme nt Grain Name a 1 Proportion of Ca species (%) Determined by Linear Combination Fitting the µXANES Spectra Calcite Vaterite Aragonite Amorphous CaCO 3

 2

	Aqueous Ca 2+	Ca oxalate 1 H 2 O	Sum	NSS (%) 2

Acknowledgements 10 We acknowledge the ALS (Berkeley, USA) and the ESRF (Grenoble, France) for the provision of beamtime. We are grateful to Jean Susini and the staff of beamline ID21 at the ESRF for their technical support during the experiment. The operations of the Advanced Light Source at Lawrence Berkeley National Laboratory are supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences Division, of the US Department of Energy under Contract No. DE-AC02-05CH11231. We acknowledge Y. Politi and S. Weiner for sharing the ACC Ca XANES spectrum, and two anonymous reviewers.

Cd+Ca

CdCa2 85 ± 5 13 ± 5 0.012 2 Residual between fit and experimental data NSS = Σ(µ experimental -µ fit ) 2 / Σ(µ experimental ) 2 x 100 in the 4000-4150 eV range, where µ is the normalized absorbance. 3 Fit shown in Fig. 4. The error bars on the percentages correspond to the variation needed to increase NSS by 20%. 1 The error bar on the percentages varies from 5 to 15% depending on the species (see Table 2)


A Iggidr

H A Ouahbi

J C Vivalda

ON THE CONTROL OF AN EXPLOITED POPULATION OF FISH *

Keywords: difference equations, Lyapunov functions, population dynamics, stabilization, Feedback

The aim of this paper is to show how some tools of control theory can be helpfull to regulate the exploitation of a population of fish.

Introduction

In this paper, we are interested in the stabilization of an exploited population of fish arround a non trivial steady state.

The dynamic of the population is supposed to be de scribed by a discrete-time system of the form

x(t + l) = F{x{t),u{t)), (1)
where x(t) is the state variable at time k = 0.1. 2.... and u(t) is the control (here it is the fishing effort).

The problem addressed here is how to compute the fish ing eflbrt (as a feedback control) u{x) in such a way. that for a given state x ^ 0 ; one has

(i) F(x°. U(X 0 )) = x° (x° is an equilibrium point).
(ii) x is a globally asymptotically stable equilibrium point for the closed-loop system

x(t + l) -F(x(t).u(x(t))
' *i(* + i) = /(E?=i &.-*.■(*)) X2(t + 1) = xi(t)exp(-Mi -qiu(t)) , Xs(t + l) = X2(t)exp(-M 2 -q2u(t)) ^) _ x"(t + l) = J!"_i(t)ej!p(-M"_i -q"-1 u(t))
Where:

♦ bi > 0 is the number of individuals produced by indi viduals of the i age class.

♦ M{ > 0 is the natural mortality of individuals of age i.

♦ q{ > 0 is the catchability of individuals of age {.

♦ u(t) is the fishing effort at time t and is regarded as an input.

♦ / is the stock-recruitment function.

Main result

For the sake of simplicity we shall give the result for n = 3 (the calculus are exactly the same for an arbitrary n but the expression of the feedback is longer). We also suppose that only individuals of age n and over are reproductive (b 1 -b 2 = 0). So that, we consider the following system

( x 1 {t+l) = f{b 3 x 3 (t)) I x 2 {t + 1) = a?i(i)ea?p(-Mi -qiu{t)) (3) { X3 (t + 1) = x 2 {t)exp(-M 2 -q 2 u{t))
For a constant fishing effort u . system (3) has a non trivial equilibrium state and So.

ou ox ou b 3 a 1 a2 -1 /3b 3 a 1 a 2 ^3*3 0 0 0 0 ----Q. x 2 -aix 1: x 3 -a 1 a 2 x 1 1 + p 632:3
Where a,-= exp(-Mi -qiU°). This steady state belongs to Q = IR, provided that

1 f>3 > a\a 2
Theorem 2.1 for any positive constant r\ < u . sys tem ( 3) is globally asymptotically stabilizable by means of the continuous feedback law

AV(x) = V(F(x ; u 0 )) -V(x) +%(F(x,u°))!jg{x,u 0 )v(x) + Io (l-t)$£(x ; u° + tv(x)) v 2 (x)dt.
(8) Now we shall construct a feedback control v(x) in order to get AV(x) < 0 for all x £ Q. To this end. we introduce some notations. Let y? : 17 x IR x ]R ->Hbe the function defined by

f 1 d 2 V ip(x.v.w)= / {l-t)v T (x)--T (x.u° +tv(xj) w 2 (x)dt. Jo ou- (9)
For a fixed number rj satisfying 0 < r\ < vP. let A'i(rc) and K 2 (x) be any nonnegative continuous real valued func tions satisfying Ki(x)

+ K 2 (x) ^ 0. Va € Q and Ki(x) > sup \ip(x.u.w)\. Vx€Q. ( 10
) |^|<^.|u|=i which satisfies u[x) = u -\-v(x)
\v{x)\\<r). Va?eQ.

Proof. Let V be the following candidate Lyapunov func tion

We construct the feedback control according to the follow ing formula :

V(x)
and define

(Xl 0\2 , ( x 2 ~ A? , ( x 3 ~ x %) + + laia 2 ) v(x) = -K(x) ( ^F{x ; u°))^(x ;U °) F(x) V:QxE^Eand V(x : u) Fix. u V{F[x.u)), which satisfies \V(X)\<TJ. yx€Q. ( 13
) (14) (6)
We evaluate the variation of V along the closed-loop sys tem (3-4):

Tacking into account (8-13-9). the variation of V along the solutions of the closed-loop system can be written :

AV(x) = V(F{x, it 0 )) -V(x) AV(x) V(F(x.u(x)))-V(x) V(x.u{x))-V{x) V(x.u° + v{x)) -V{x)
V(x. u°) -Vix) + -far. u°) v[x) ou f 1 d 2 V + / (l-<)u(i: l t( l) +tf(i:)) v 2 {x)dt.

~KTF) V2 ( X ) + i f{x:V{x).v{x)).

On the one hand, we have

(x 3 -x°3) 2
(15)

V(F(x..u°))-V(x) = (f(b 3 X3)-x u 1 )
Notice that du 2

V(x,u°) = V(F{x,u 0 )).

(7)

o.\a 2 \

  ).

	More precisely, we consider a density-dependent model
	of a population of exploited fish which is structured in n
	age classes [liu. mag. ric] :
	*travail realise avec la contribution financiere du CNCPRST
	(Maroc) et du CNRS (France) dans le cadre du PICS n°78.
	t Corresponding author

Hence

V{F{x.u°))-V{x) b 3 (x 3 -x l) ( x 3 -xl {l + /3b 3 x 3 )(l + i3b 3 xi)J

V «i«2 2 (6 3 aia 2 ) 2 -(1 + pb 3 x 3 ) 2 (l + /?&3^) 2 , ov2

We have l + j3b 3 x°3 = 1+/?& 3 i ^-1 a 1 a 2 = b 3 a ia2 . This yields.

/3fr s aia

On the other hand.i,!?(;E. v. w) being homogeneous of degree 2 with respect to w. we have for all x £ Q such that f(ar) ^ 0:

From this and (15-16). we get

And for v(x) ^ 0.

Thanks to the construction of v(x) and K(x). we have

The closed-loop system is then Lyapunov stable. On the other hand.

AV(x) = 0 <£> x 3 -x 3 and v(x) -0.

It is easy to show that the largest invariant set contained in {a? € 9.

is reduced to {x } so. by Lasalle Invariance Principle [las]. the equilibrium x is a globally asymptotically stable equi librium point for the closed-loop system.

[bev] R.J.H. Beverton and S.J. Holt. On the dynamics of exploited fish populations. Chapman fe Hall. New York. First edition 1957.

[fis] M. E. Fisher and B. S. Goh. Stability results for delayed-recuitment models in population dynamics. J. Math. Bio., 19 : 147-156. (1984).

[igg] A. Iggidr and M. Bensoubaya. New results on the stability of discrete-time systems and applications to control problems. Journal of Mathematical Analysis and Applications. 219. 392-414, (1998).

[las] J.P. LASALLE. The stability and control of discret processes. Springer-Verlag. New York. 1986.

[liu] L. Liu and J. E. Cohen. Equilibrium and local sta bility in a logistic matrix model for age-structured populations. /. Math. Bio., 25. 73-88, (1987).

[mag] P. Magal and D. Pelletier A fixed point theorem with application to a model of population dynamics. /. Difference Equ. Appl.. 3, No.l, 65-87, (1997).

[ric] W. E. Ricker. Stock and recruitment. J. Fish. Res. Board Can., 11, 559-623, (1954).