Abderrahman Iggidr
email: abderrahman.iggidr@inria.fr

Mohamed Oumoun
email: m.oumoun@uca.ac.ma

Max O Souza
email: maxsouza@id.uff.br

Continuous Simultaneous Stabilization of Single-Input Nonlinear Stochastic Systems *

Keywords: Nonlinear stochastic systems, continuous stochastic stabilization, simultaneous stabilization, control Lyapunov function, state feedback

published or not. The documents may come

Continuous simultaneous stabilization of single-input nonlinear stochastic systems

Introduction

Many real systems are nonlinear and are also prone to stochastic phenomena. These features naturally attracted the attention of the control literature to stabilization problems for stochastic nonlinear systems, and some interesting results have been obtained during over the past few decades-see [START_REF] Florchinger | Time-Varying Stabilizers for Stochastic Systems with no Unforced Dynamics[END_REF][START_REF] Haddad | Universal Feedback Controllers and Inverse Optimality for Nonlinear Stochastic Systems[END_REF][START_REF] Hu | Exponential Stability of Stochastic Differential Equations with Impulse Effects at Random Times[END_REF][START_REF] Shaoli Jin | Global Output Feedback Stabilization for Stochastic Nonlinear Systems with Function Control Coefficients[END_REF][START_REF] Oumoun | Continuous feedback stabilization for a class of affine stochastic systems[END_REF][START_REF] Zhang | Finite-Time Stabilization of Stochastic High-Order Nonlinear Systems[END_REF][START_REF] Zhao | Nonsmooth Stabilization of a Class of Markovian Jump Stochastic Nonlinear Systems with Parametric and Dynamic Uncertainties[END_REF] and references therein. These results were obtained using classical stochastic theory as found, for instance, in [START_REF] Khas'minskii | Stochastic Stability of Differential Equations[END_REF] and [START_REF] Mao | Stochastic Differential Equations and Their Applications[END_REF]. This theory requires the systems coefficients to be continuously differentiable, in order to guarantee the local Lipschitz condition.

However, many stochastic models arising in applications do not satisfy such a local Lipschitz condition. For example, stochastic models for a sequencing-batch reactor and for a chemostat (see [START_REF] Fontbona | Stochastic Modeling and Control of Bioreactors[END_REF]); some stochastic financial and biological models (see [START_REF] Lewis | Option Valuation Under Stochastic Volatility II[END_REF][START_REF] Mao | Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching[END_REF]). These models have coefficients involving the term x(t). In order to relax this restriction, Li & Liu [START_REF] Li | Global stability and stabilization of more general stochastic nonlinear systems[END_REF] generalized the classical theorems on global stability to cover stochastic nonlinear systems without a local Lipschitz condition. Using the results in [START_REF] Li | Global stability and stabilization of more general stochastic nonlinear systems[END_REF], several authors have obtained sufficient stabilizability conditions -e.g. [START_REF] Gao | Global state feedback stabilization of stochastic high-order nonlinear systems with high-order and low-order nonlinearities[END_REF][START_REF] Oumoun | Continuous feedback stabilization for a class of affine stochastic systems[END_REF] -allowing closed-loop systems to be merely continuous without requiring them to satisfy a local Lipschitz condition. It should be pointed out that all results mentioned above are about stability and stabilization in probability problems. Indeed, there are fewer results on the simultaneous stabilization of stochastic nonlinear systems, due to the difficulties involved in synthesis.

As it is well known, simultaneous stabilization problems arise due to several reasons: system's uncertainty, failure modes or systems with various modes of operation. The aim of this problem is to determine a single controller which simultaneously stabilizes a finite collection of systems. Current applications of simultaneous stabilization include many real systems, such as Hamiltonian systems [START_REF] Cai | Simultaneous stabilization of Port-Hamiltonian systems subject to actuation saturation and input delay[END_REF][START_REF] Wang | Simultaneous stabilization of a set of nonlinear port-controlled Hamiltonian systems[END_REF], power converters [START_REF] Chaudhuri | Simultaneous stabilization technique for damping controller design in power systems[END_REF], chemical reaction systems [START_REF] Zhu | Simultaneous H 2 /H ∞ stabilization for chemical reaction systems based on orthogonal complement space[END_REF][START_REF] Zhu | Network-based simultaneous H ∞ stabilization for chemical reaction systems with multiple packet dropouts[END_REF], and descriptor systems [START_REF] Chen | Simultaneous stabilization for uncertain descriptor systems with input saturation[END_REF] to name a few.

Therefore, the investigation of simultaneous stabilization problems is quite relevant from a practical point of view, and many important results have been obtained for linear systems: see [START_REF] Galindo | Simultaneous stabilization of full state information Linear Time-Invariant systems[END_REF][START_REF] Guan | Simultaneous stabilization of linear systems[END_REF][START_REF] Kohan-Sedgh | Conservatism reduction in simultaneous output feedback stabilisation of linear systems[END_REF][START_REF] Li | Some Open Problems on Simultaneous Stabilization of Linear Systems[END_REF][START_REF] Yu | Sufficient conditions for simultaneous stabilization of three linear systems within the framework of nest algebras[END_REF] and references therein. For nonlinear systems, the problem is more involved, and there are fewer results in the literature -e.g. [START_REF] Dastaviz | it Simultaneous stabilization for a collection of uncertain time-delay systems using sliding-mode output feedback control[END_REF][START_REF] Iggidr | Simultaneous Stabilization of Single-Input Nonlinear Systems With Bounded Controls[END_REF][START_REF] Wu | Simultaneous stabilization for a collection of single-input nonlinear systems[END_REF][START_REF] Wu | Simultaneous stabilization for a collection of nonlinear control systems in canonical form[END_REF][START_REF] Zhang | Simultaneous stabilization of a collection of singleinput nonlinear systems based on CLFs[END_REF]. All these works deal with the simultaneous stabilization of linear and nonlinear deterministic systems.

To the authors' best knowledge, there are even fewer results on simultaneous stabilization of stochastic nonlinear systems -an exception being [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF]. The goal of this paper is to contribute to this problem and to investigate the simultaneous stabilization in probability of a collection of single-input nonlinear stochastic systems, with a drift that is affine in the control. The authors showed in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF] that the simultaneous stabilization problem of a collection of single-input nonlinear stochastic systems, {S i } m i=1 , can be reduced to a simpler one, provided the following condition holds: for each system S i there exists a stochastic control Lyapunov function. In this case, the problem is then reduced to finding a single continuous feedback u satisfying that for all x ∈ IR n \ {0}, one has a i (x) + b i (x)u(x) < 0, with known functions, a i and b i , i = 1, 2, ..., m. Under the additional assumption that, for each x ∈ IR n , {b i } m i=1 is single signed, an explicit feedback is constructed in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF].

The aim of this paper is to present a new sufficient condition for simultaneous stabilizability that improves the condition derived in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF]. This new condition will allow the removal of the aditional single signed condition on {b i } m i=1 , in order to be able to design simultaneous stabilizers for a wider class of a collection of stochastic differential systems. Moreover, this condition is also necessary, if the corresponding closed-loop systems satisfy some additional regularity condition, such as a Lipschitz condition. The main tools used in this work are the generalised stochastic Lyapunov theorem proved by Li & Liu [START_REF] Li | Global stability and stabilization of more general stochastic nonlinear systems[END_REF], the technique of stochastic control Lyapunov function and the converse stability theorem of Kushner [START_REF] Kushner | Converse theorems for stochastic Lyapunov functions[END_REF].

The remainder of this paper is organized as follows: Section 2 gives some preliminary material on asymptotic stability in probability of weak solutions. Section 3 formulates the stochastic nonlinear systems and the control design problem. Section 4 presents the design scheme of the state-feedback controller, and gives the main results of the paper. Finally, in section 5, we apply the results obtained in Section 4 to two examples that cannot be stabilized using the results available in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF].

Stochastic stability of weak solutions

The purpose of this section is to recall some basic concepts and results concerning the stability and asymptotic stability in probability of an equilibrium solution of a stochastic differential equation that will be used in the sequel. Consider the following stochastic nonlinear system:

dx = f (x)dt + h(x)dω, x(0) = x 0 ∈ IR n , ( 2.1)
where x ∈ IR n is the system state; ω is an m-dimensional independent standard Wiener process. The function f : IR n → IR n and h : IR n → IR n×m are continuous and satisfy f (0) = 0 and h(0) = 0. Clearly, the origin is an equilibrium point of system (2.1). Usually, in order to guarantee the existence and uniqueness of strong solutions of the stochastic systems (2.1), the functions f (.) and h(.) are assumed to satisfy some regularity conditions, such as local Lipschitz condition, (see Khas'minskii [START_REF] Khas'minskii | Stochastic Stability of Differential Equations[END_REF] and Mao [START_REF] Mao | Stochastic Differential Equations and Their Applications[END_REF]). In this paper, f (.) and h(.) are assumed to be continuous, but not locally Lipschtiz. It then follows that system (2.1) may not have solutions in the classical sense, as in Khas'minskii [START_REF] Khas'minskii | Stochastic Stability of Differential Equations[END_REF] and Mao [START_REF] Mao | Stochastic Differential Equations and Their Applications[END_REF]. However, system (2.1) always has weak solutions which are essentially different from the classical (or strong) solution since the former may not be pathwise unique and may be defined on a different probability space.

The following definition of weak solution of the stochastic system (2.1) is given in [START_REF] Li | Global stability and stabilization of more general stochastic nonlinear systems[END_REF], and for more details on this subject, we refer the reader to Ikeda and Watanabe [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF], Klebaner [START_REF] Klebaner | Introduction to stochastic calculus with applications[END_REF], Ondreját and Seidler [START_REF] Ondreját | A note on weak solutions to stochastic differential equations[END_REF].

Definition 2.1. [START_REF] Li | Global stability and stabilization of more general stochastic nonlinear systems[END_REF]: If there exists a continuous adapted process x(t) on a probability space (Ω x , F x , P x ) with a filtration {F t } t≥t 0 satisfying the usual conditions, and an m-dimensional {F t }adapted Brownian motion ω x (t) with P x {ω x (t 0 ) = 0} = 1, such that the initial condition x(t 0 ) has the given distribution, and for all t ∈ [t 0 , τ x +∞ )

x(t) = x(t 0 ) +

= lim ε→+∞ inf{t ≥ t 0 , x(t) ≥ ε, ∀ε > 0}.
Given a twice-differentiable function Ψ : IR n → IR, the infinitesimal generator L associated with the stochastic differential equation (2.1) for Ψ(x), is given by

LΨ(x) = ∇Ψ(x)f (x) + 1 2 Tr h(x)h(x) ∇ 2 Ψ(x) ,
where Tr{.} represents the trace of the argument. We now recall the following version of the stochastic Lyapunov theorem stated in [START_REF] Li | Global stability and stabilization of more general stochastic nonlinear systems[END_REF], that concerns stochastic systems with coefficients that are only continuous.

Lemma 2.1. [START_REF] Li | Global stability and stabilization of more general stochastic nonlinear systems[END_REF]: For system (2.1), if there exists a C 2 function V : IR n → IR + and two K ∞ class functions α and β, such that

α( x ) ≤ V (x) ≤ β( x ), LV (x) ≤ -W (x),
where W (•) is a nonnegative continuous function. Then the zero solution of system (2.1) is globally stable in probability, and for ∀x 0 ∈ IR n , every weak solution x(t) of system (2.1) satisfies

P x { lim t→+∞ W (x(t)) = 0} = 1.
Furthermore, if W (•) is positive definite, then the zero solution of system (2.1) is globally asymptotically stable in probability.

3 Problem formulation and preliminaries

Problem formulation

Consider a collection of affine stochastic systems described by

S i : dx = f i (x)dt + ug i (x)dt + h i (x)dω, i ∈ {1, ..., m} = I, ( 3.1)
where x ∈ IR n is the state and u ∈ IR is the control and ω is a standard IR m -valued Wiener process defined on a filtered probability space (Ω, F, F t , P ) with Ω being a sample space, F a σ-algebra on Ω, F t a filtration and P a probability measure. f i , g i : IR n → IR n and h i : IR n → IR n×m are continuous functions with

f i (0) = h i (0) = 0.
We deal with the design of a single feedback law p : IR n → IR by explicit formulas in such a way that all resulting closed-loop systems

dx = f i (x)dt + p(x)g i (x)dt + h i (x)dω, i ∈ I
have the origin globally asymptotically stable in probability.

Stochastic control Lyapunov function

In the following, we present the concept of stochastic control Lyapunov function (SCLF), which plays an important role in our approach to solve the simultaneous stabilization problem. Consider a single nonlinear stochastic systems

dx = f (x)dt + ug(x)dt + h(x)dω. ( 3.2)
For a C 2 function V , the infinitesimal operator L u associated with the system (3.2) is defined for

V as L u V (x) = a(x) + b(x)u, with a(x) = ∇V (x)f (x) + 1 2 Tr h(x)h(x) ∇ 2 V (x) b(x) = ∇V (x)g(x). (3.3)
The following definition is required for stating the results of this paper.

Definition 3.1. ([6, 11]). A positive definite and proper function

V ∈ C 2 (IR n , IR + ) is said: 1)
to be a stochastic control Lyapunov function (SCLF) for the stochastic differential system (3.2), if

inf u∈IR L u V (x) = inf u∈IR a(x) + b(x)u < 0, ∀x ∈ IR n \ {0}. (3.4)
2) to satisfy the small control property with system (3.2), if for each ε > 0 there is δ > 0 such that , if x = 0 satisfies x < δ, then there is some u with u < ε such that

L u V (x) = a(x) + b(x)u < 0.
Note that V is a SCLF for the system (3.2) is equivalent to

(b(x) = 0, x = 0) ⇒ a(x) < 0, ∀x ∈ IR n \ {0}. (3.5)

Main results: Design method of stabilizing controller

This section presents a constructive method of a simultaneously stabilizing feedback, based on a stochastic control Lyapunov function approach. First, we recall the continuous feedback stabilization result. Then, we present a design method.

Continuous feedback stabilization

Let (as motivated by Sontag's universal formula [START_REF] Sontag | A universal construction of Artstein's theorem on nonlinear stabilization[END_REF])

k(x) =        - a(x) + a 2 (x) + b 4 (x) b(x) , if b(x) = 0 0, if b(x) = 0, (4.1)
the authors showed in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF] the following theorem.

Theorem 4.1. [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF]: If there exists a stochastic control Lyapunov function V for the system (3.2), satisfying the small control property, then the feedback u = k(x), defined in (4.1), is continuous on IR n and globally asymptotically stabilizes in probability system (3.2).

Proof. First, notice that the functions a and b are continuous, since V ∈ C 2 (IR n , IR + ) and f , g are continuous. Together with (3.5), the continuity of the function k on IR n \ {0} is easy to verify. In addition, since V satisfies the small control property, arguing as in [START_REF] Sontag | A universal construction of Artstein's theorem on nonlinear stabilization[END_REF], we can prove that k is continuous on IR n . It follows that the coefficients of the closed-loop system deduced from the stochastic system (3.2) with the control u = k(x) are continuous on IR n as well as the associated infinitesimal operator

L k V . Second, for x ∈ IR n \ {0}, if b(x) = 0 then L k V (x) = a(x) + b(x)k(x) = -a 2 (x) + b 4 (x) < 0, and if b(x) = 0 then L k V (x) = a(x) < 0 since V is a SCLF for system (3.2), that is L k V is negative definite. Now, since V is a SCLF
, that is V is proper and positive definite, and L k V is continuous and negative definite, according to Lemma 4.3 in [START_REF] Khalil | Nonlinear systems[END_REF], there exit class K ∞ functions α 1 and α 2 and class K function ξ, defined on IR + , such that

α 1 ( x ) ≤ V (x) ≤ α 2 ( x ), ∀x ∈ IR n , L k V (x) = a(x) + b(x)k(x) ≤ -ξ( x ), ∀x ∈ IR n .
The conclusion follows with the help of Lemma 2.1.

Remark 4.1. In fact, another continuous asymptotically stabilizing controller can be found for system (3.2). If the assumptions of Theorem 4.1 hold, then the functions a and b are continuous, and since V is a SCLF for system (3.2), that is b(x) = 0 x = 0 implies a(x) < 0, it is easy to verify that the feedback

u(x) =                min 0, - a(x) b(x) -b(x) , if b(x) > 0 max 0, - a(x) b(x) -b(x) , if b(x) < 0 0, if b(x) = 0,
is continuous and globally asymptotically stabilizes system (3.2).

Simultaneous stabilization

As a first step towards the construction of a simultaneously stabilizing feedback law for the collection of stochastic systems (3.1), we introduce the following sets and functions: For each i ∈ I, suppose that V i is a SCLF for system S i which satisfies the small control property, and let (x is dropped for space reason)

a i = ∇V i f i + 1 2 Tr h i h ∇ 2 V i , b i = ∇V i g i , (4.2)
and

k i (x) =        - a i (x) + a 2 i (x) + b 4 i (x) b i (x) , if b i (x) = 0 0. if b i (x) = 0. As shown in Theorem 4.1, the functions k i , i = 1, 2, ..., m are continuous on IR n . For each x ∈ IR n , let I p (x) = {i ∈ I, b i (x) > 0}, I n (x) = {i ∈ I, b i (x) < 0},
and for each i ∈ I, define the functions ϕ 1i , ϕ 2i , ψ 1i and ψ 2i on IR n as follows:

ϕ 1i (x) =      - a i (x) b i (x) if b i (x) < 0, -∞ if b i (x) ≥ 0, ϕ 2i (x) =      - a i (x) b i (x) if b i (x) > 0, +∞ if b i (x) ≤ 0, ψ 1i (x) = k i (x) if b i (x) < 0 and 0 if b i (x) ≥ 0, ψ 2i (x) = k i (x) if b i (x) > 0 and 0 if b i (x) ≤ 0. Now, let ϕ 1 (x) = max i∈I ϕ 1i (x), ϕ 2 (x) = min i∈I ϕ 2i (x), ψ 1 (x) = max i∈I ψ 1i (x), ψ 2 (x) = min i∈I ψ 2i (x). Finally, define w 1 (x) = min(ψ 1 (x), ϕ 2 (x)), w 2 (x) = max(ψ 2 (x), ϕ 1 (x)). Remark 4.2. Since a i (x) + a 2 i (x) + b 4 i (x) > 0, it follows that b i (x)k i (x) < 0, if b i (x) = 0. So, ψ 1i (x) = k i (x) > 0 if b i (x) < 0, and since ψ 1i (x) = 0 if b i (x) ≥ 0, we get 0 ≤ ψ 1i (x) for all i ∈ I and for all x ∈ IR n . Thus 0 ≤ ψ 1 (x), ∀x ∈ IR n . ψ 2i (x) = k i (x) < 0 if b i (x) > 0, and since ψ 2i (x) = 0 if b i (x) ≤ 0, we get ψ 2i (x) ≤ 0 for all i ∈ I and for all x ∈ IR n . Thus ψ 2 (x) ≤ 0, ∀x ∈ IR n .
Now, we give our sufficient stabilizability condition and we state the main results of this paper.

ϕ 1 (x) < ϕ 2 (x), x ∈ IR n is equivalent to: for all x ∈ IR n s. t. I n (x) = ∅ and I p (x) = ∅ max i∈In(x) - a i (x) b i (x) < min i∈Ip(x) - a i (x) b i (x) .
Remark 4.4. Assumption 4.1 relaxes the stabilizability condition stated in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF], where all the term b i (x), i ∈ I were supposed to have the same sign for each x ∈ IR n . However Assumption 4.1 makes the feedback control design of the collection of systems in (3.1) more complicated and cannot be directly solved by the method in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF].

Moreover, Assumption 4.1 becomes necessary under some regularity conditions. As it is well known in stochastic differential equation theory [START_REF] Khas'minskii | Stochastic Stability of Differential Equations[END_REF][START_REF] Mao | Stochastic Differential Equations and Their Applications[END_REF], in order to guarantee the existence and uniqueness of strong solutions of an unforced stochastic systems, its coefficients are assumed to satisfy some definite conditions such as Lipschitz and linear growth conditions. Now, for stochastic control systems in (3.1), assume that for each i ∈ I, f i , g i and h i satisfy the above definite conditions, we also assume that there exists a common control feedback k that satisfies the above definite conditions and globally asymptotically stabilizes the collection in (3.1), clearly, the coefficients of the resulting closed-loop system dx = f i (x)dt + k(x)g i (x)dt + h i (x)dω satisfy the Lipschitz and linear growth conditions. Then we have the following result.

Theorem 4.2. Consider the collection of systems in (3.1). If there exists a feedback k : IR n → IR such that for each i ∈ I, the closed-loop system dx = f i (x)dt + k(x)g i (x)dt + h i (x)dω, has its coefficients satisfying the above definite conditions and is globally asymptotically stable in probability, then there exists a collection of SCLFs V i for systems S i , respectively, satisfying Assumption 4.1.

Proof. Suppose that there exists a feedback k such that the closed-loop systems dx = f i (x)dt + k(x)g i (x)dt + h i (x)dω, i ∈ I, are globally asymptotically stable in probability, then by Kushner's converse Lyapunov theorem [START_REF] Kushner | Converse theorems for stochastic Lyapunov functions[END_REF], there exist C 2 positive definite and radially unbounded functions

V i , i ∈ I, such that for each i ∈ I, L u V i (x) = a i (x) + b i (x)k(x) < 0 , for all x = 0. It then follows that for each x ∈ IR n , one has: k(x) < -a i (x) b i (x) if b i (x) > 0 i ∈ I, then k(x) < ϕ 2i (x) for all i ∈ I since ϕ 2i (x) = +∞ if b i (x) ≤ 0, consequently, k(x) < ϕ 2 (x). k(x) > -a i (x) b i (x) if b i (x) < 0 i ∈ I, then k(x) > ϕ 1i (x) for all i ∈ I since ϕ 1i (x) = -∞ if b i (x) ≥ 0, consequently, k(x) > ϕ 1 (x).
Thus, ϕ 1 (x) < ϕ 2 (x), for all x ∈ IR n and this completes the proof of Theorem 4.2. Remark 4.5. Note that under the assumptions in Theorem 4.2, the existing SCLFs V i of systems in (3.1) do not necessary satisfy the stabilizability condition stated in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF]. Now, even though all the functions f i , g i and h i , i ∈ I, of the systems in (3.1) can be chosen as in Theorem 4.2, i.e satisfying the Lipschitz and linear growth conditions, and there exists a collection of SCLFs V i , i ∈ I, of system S i i ∈ I, respectively, that satisfy Assumption 4.1, it is much more difficult to construct a control feedback satisfying the Lipschitz and linear growth conditions and that globally asymptotically stabilizes in probability the collection of systems in (3.1) simultaneously. But, as shown in the following result, we are able to design a common merely continuous stabilizing feedback of systems in (3.1). (3.1). If there exists a SCLF V i , that satisfies the small control property, for every S i , i ∈ {1, ..., m}, and Assumption 4.1 holds, then the feedback

Theorem 4.3. Consider the collection of systems in

u(x) = w 1 (x) + w 2 (x) 2 (4.3)
is continuous and globally asymptotically stabilizes in probability the collection of systems in (3.1) simultaneously.

Proof. Continuity of u: We first prove the continuity of w 1 . Since k i is continuous and k i (x) = 0 whenever b i (x) = 0, it is easy to see that ψ 1i , i ∈ I, is continuous in IR n , then so is ψ 1 . Now let x 0 ∈ IR n \ {0}, we will verify that there exists η x 0 > 0, such that we either have

w 1 (x) = ψ 1 (x) for all x ∈ B(x 0 , η x 0 ) if I p (x 0 ) = ∅, or ϕ 2 is continuous on B(x 0 , η x 0 ) if I p (x 0 ) = ∅.
For each i ∈ I, from the definition of ϕ 2i and the fact that V i is a SCLF for system S i , it's not hard to verify that,

if b i (x 0 ) ≤ 0, then ∀L > 0, ∃λ i L > 0 s. t. ϕ 2i (x) > L, ∀x ∈ B(x 0 , λ i L ). Choosing λ L = min i / ∈Ip(x 0 ) λ i L , it follows for all i / ∈ I p (x 0 ) ∀L > 0, ∃λ L > 0 s. t. ϕ 2i (x) > L, ∀x ∈ B(x 0 , λ L ). (4.4) If b i (x 0 ) > 0, since the function b i is continuous, there exists δ i > 0 such that b i (x) > b i (x 0 ) 2 > 0 on B(x 0 , δ i ). It follows that ϕ 2i (x) = - a i (x) b i (x)
is continuous and bounded on B(x 0 , δ i ). By choosing

δ = min i∈Ip(x 0 )
δ i , we have, for any i ∈ I p (x 0 ), ϕ 2i is continuous on B(x 0 , δ), so for all x ∈ B(x 0 , δ)

ϕ 2i (x) = - a i (x) b i (x) ≤ L 1 = max j∈Ip(x 0 ) sup y∈B(x 0 ,δ) ϕ 2j (y) . (4.5)
Now, if I p (x 0 ) = ∅, for any ε > 0, since ψ 1 is continuous and positive, take L 0 = sup y∈B(x 0 ,ε)

ψ 1 (y).
Then from (4.4), we can choose 0 < λ L 0 < ε such that

ϕ 2i (x) > L 0 , ∀x ∈ B(x 0 , λ L 0 ), ∀i ∈ I.
it then follows that

ϕ 2 (x) = min i∈I ϕ 2i (x) > L 0 ≥ ψ 1 (x), ∀x ∈ B(x 0 , λ L 0 ), which clearly implies that w 1 (x) = ψ 1 (x), for all x ∈ B(x 0 , λ L 0 ). Thus w 1 is continuous at x 0 . If I p (x 0 ) = ∅, combining (4.4
) and (4.5), and take η = min(λ L 1 , δ) yields that

ϕ 2i (x) ≤ L 1 < ϕ 2j (x), ∀x ∈ B(x 0 , η), ∀i ∈ I p (x 0 ), ∀j / ∈ I p (x 0 ),
and consequently

ϕ 2 (x) = min i∈I ϕ 2i (x) = min i∈Ip(x 0 ) ϕ 2i (x) ∀x ∈ B(x 0 , η).
From this, together with the continuity of ϕ 2i on B(x 0 , η), for all i ∈ I p (x 0 ), it follows that ϕ 2 is continuous on B(x 0 , η). Thus, w 1 = min(ψ 1 , ϕ 2 ) is continuous on B(x 0 , η) and then at x 0 , since ψ 1 is continuous on IR n . Now, we establish the continuity of w 1 at the origin in the case of small control property of V i , i ∈ I.

Far each i ∈ I, due to definition of ϕ 2i and the small control property of V i , it is not difficult to obtain that

∀ε > 0, ∃λ i > 0 such that, -ε < ϕ 2i (x), ∀x ∈ B(0, λ i ). Take λ = min i∈I λ i , it follows ∀ε > 0, ∃λ > 0 such that, -ε < ϕ 2 (x), ∀x ∈ B(0, λ). (4.6)
Continuity of ψ 1 at the origin leads to

∀ε > 0, ∃γ > 0 s. t., -ε < ψ 1 (x) < ε, ∀x ∈ B(0, γ). (4.7)
Take η = min(λ, γ), it follows from (4.6) and (4.7)that for all x ∈ B(0, η)

∀ε > 0, ∃η > 0 s. t., -ε < w 1 (x) = min(ψ 1 (x), ϕ 2 (x)) < ε.
That is, w 1 is continuous at the origin and so on IR n . The continuity of w 2 can be treated similarly and is omitted here. Thus u is continuous on IR n . Global asymptotic stability: Due to the continuity of u, we deduce that for each i ∈ I, the coefficients of the closed-loop system: dx = f i (x)dt + u(x)g i (x)dt + h i (x)dw, and the associated infinitesimal operator L u V i (x) = a i (x) + b i (x)u(x) are continuous on IR n . In view of Lemma 2.1, we only have to verify that for any i ∈ I,

L u V i (x) = a i (x) + b i (x)u(x) < 0, ∀x ∈ IR n \ {0}, (4.8)
according to the sign of b i (x), x = 0.

• If b i (x) > 0. On one hand, From the definitions of w 1 , ϕ 2 and ϕ 2i , we obtain

w 1 (x) ≤ ϕ 2 (x) ≤ ϕ 2i (x) = - a i (x) b i (x) . (4.9)
On the other hand, definitions of ψ 2 and ψ 2i give,

ψ 2 (x) ≤ ψ 2i (x) = k i (x) < -a i (x) b i (x) . From Assumption 4.1, we get ϕ 1 (x) < ϕ 2 (x) ≤ -a i (x) b i (x)
. Hence,

w 2 (x) = max(ψ 2 (x), ϕ 1 (x)) < - a i (x) b i (x) . ( 4
.10) (4.9) and (4.10) lead to

u(x) = w 1 (x) + w 2 (x) 2 < - a i (x) b i (x) . Thus, L u V i (x) = a i (x) + b i (x)u(x) < 0.
• If b i (x) < 0. Similarly to the previous case, on one hand, from definitions of w 2 , ϕ 1 and ϕ 1i , we have

w 2 (x) ≥ ϕ 1 (x) ≥ ϕ 1i (x) = - a i (x) b i (x) . ( 4

.11)

On the other hand, we have,

ψ 1 (x) ≥ ψ 1i (x) = k i (x) > -a i (x) b i (x) . By Assumption 4.1, ϕ 2 (x) > ϕ 1 (x) ≥ -a i (x)
b i (x) and then

w 1 (x) = min(ψ 1 (x), ϕ 2 (x)) > - a i (x) b i (x) . (4.12) Thus, L u V i (x) = a i (x) + b i (x)u(x) < 0, since (4.11
) and (4.12) yield

u(x) = w 1 (x) + w 2 (x) 2 > - a i (x) b i (x) . • If b i (x) = 0, then L u V i (x) = a i (x) < 0, since V i is a SCLF for the system S i .
Thus, L u V i is negative definite as desired. Finally , L u V i is continuous and negative definite, and since V i is a SCLF for system S i , i ∈ I, that is, V i is proper and positive definite, according to Lemma 4.3 in [START_REF] Khalil | Nonlinear systems[END_REF], there exit class K ∞ functions α and β and class K function ξ, defined on IR + , such that

α( x ) ≤ V (x) ≤ β( x ), ∀x ∈ IR n ,
and

L u V i (x) = a i (x) + b i (x)u(x) ≤ -ξ( x ), ∀x ∈ IR n .
The conclusion follows with help of Lemma 2.1 and this completes the proof of Theorem 4.3.

Simultaneous stabilization via bounded control

This section will show that, under appropriate assumption on SCLFs, simultaneous stabilization can be guaranteed for the collection of systems in (3.1) by bounded feedback.

Here, we assume that there exists a SCLF V for system (3.2) satisfying the small control property and with controls in

B 1 = {u ∈ IR | -1 < u < 1}, that is inf u∈B 1 L u V (x) = inf u∈B 1 a(x) + b(x)u < 0, ∀x ∈ IR n \ {0}. (4.13)
Let (as defined by Lin-Sontag's universal formula [START_REF] Lin | A universal formula for stabilization with bounded controls[END_REF])

p(x) =          - a(x) + a 2 (x) + b 4 (x) b(x) 1 + 1 + b 2 (x) , if b(x) = 0, 0, if b(x) = 0, (4.14)
where the function a and b are defined in (3.3).

Theorem 4.4. For system (3.2), suppose that there exists a SCLF V with controls in B 1 , satisfying the small control property, Then, the origin solution to (3.2) is globally asymptotically stable in probability with the continuous feedback u = p(x) defined in (4.14) which takes values in B 1 .

Proof. For the continuity of the feedback p and the global asymptotic stability of the resulting closed-loop system, the proof is similar to that of Theorem 4.1.

For the boundedness of the feedback p, let us define (as it has been done in [START_REF] Lin | A universal formula for stabilization with bounded controls[END_REF]) a function φ : To derive simultaneously stabilizing controllers, for each i ∈ I, assume that V i is a SCLF of system S i , satisfying the small control property and takes values in B 1 . Define

IR × IR → IR by φ(a, b) =        - a + √ a 2 + b 4 b 1 + √ 1 + b 2 , if b = 0, 0, if b = 0, (4.15
p i (x) =          - a i (x) + a 2 i (x) + b 4 i (x) b i (x) 1 + 1 + b 2 i (x) , if b i (x) = 0, 0, if b i (x) = 0 (4.16)
where a i and b i are as defined in (4.2). Therefore, according to Theorem 4.4, the functions p i , i ∈ I, is continuous, takes values in B 1 and globally asymptotically stabilizes system S i .

In the following, we replace the function k i by the function p i in the definitions of ψ 1i and ψ 2i .

Remark 4.6. As in Remark 4.2, for each i ∈ I, since a i (x) + a 2 i (x) + b 4 i (x) > 0, it folows that b i (x)p i (x) < 0, i f b i (x) = 0. So, taking into account that -1 < p i (x) < 1, for all x ∈ IR n , we have 0 < p i (x), if b i (x) < 0, and then 0 ≤ ψ 1 (x) < 1, and, p i (x) < 0, if b i (x) > 0, and then -1 < ψ 2 (x) ≤ 0. Now we can state the following theorem. Theorem 4.5. Consider the collection of systems in (3.1). If there exists a SCLF V i with control in B 1 , that satisfies the small control property, for every S i , i ∈ I, and Assumption 4.1 holds, then the feedback

u(x) = w 1 (x) + w 2 (x) 2 (4.17)
is continuous, globally asymptotically stabilizes in probability the collection of systems in (3.1) simultaneously and takes values in B 1 .

Proof. First, quite similarly to the proof of Theorem 4.3, we can prove that the feedback (4.17) is continuous and globally asymptotically stabilizes systems S i , i ∈ I.

Second, we still have to verify that the feedback (4.17) takes values in B 1 . To do this, we shall verify that for all x ∈ IR n , -1 < w 1 (x) < 1 according to the set I p (x).

• If I p (x) = ∅, then w 1 (x) = ψ 1 (x). From this together with the fact that we replaced k i by p i in the definition of ψ 1i , we get 0 ≤ w 1 (x) < 1 according to Remark 4.6.

• If I p (x) = ∅, from the definition of ϕ 2 , let i 0 ∈ I p (x) such that ϕ 2 (x) = - a i 0 (x) b i 0 (x)
. Since

p i 0 (x) ∈ B 1 and b i 0 (x) > 0, we have -1 < u i 0 (x) < - a i 0 (x) b i 0 (x) = ϕ 2 (x).
This last inequality and Remark 4.6 lead to

-1 < min(ψ 1 (x), ϕ 2 (x)) = w 1 (x) ≤ ψ 1 (x) < 1.
Thus, w 1 (x) ∈ B 1 for all x ∈ IR n . The reasoning is similar to prove that w 2 takes values in B 1 , then so is for u and this completes the proof of Theorem 4.5.

Some illustrative examples

In this section, we present two examples to illustrate the effectiveness of the results obtained in this paper to design a simultaneously stabilizing feedback for some nonlinear stochastic systems.

Example 1

Consider a system with the following three possible modes

S 1 : dx 1 = -2x 2 1 x 2 2 dt + x 1 udt dx 2 = (-x 2 -2x 1 x 3 2 )dt + x 2 udt + x 2 dω S 2 : dx 1 = -x 1 dt -udt + 2x 1 x 2 dω dx 2 = 2x 1 x 3 2 dt -x 2 udt S 3 : dx 1 = -x 1 dt -udt + 2x 1 x 2 dω dx 2 = -x 2 dt + x 2 dω
We want to find a continuous function p : IR 2 → IR 2 such that the state feedback controller u(x) = p(x) globally asymptotically stabilizes in probability the three systems S 1 , S 2 and S 3 . To this end, let

V 1 (x) = x 2 1 + x 2 2 2 , V 2 (x) = x 2 1 2 + x 4 2 4 and V 3 (x) = x 2 1 + x 2 2 2 .
Denote by L u V 1 , L u V 2 and L u V 3 the infinitesimal operator associated with system S 1 , S 2 and S 3 , respectively, we have

L u V 1 (x) = -1 2 x 2 2 -2x 1 x 2 2 (x 2 1 + x 2 2 ) + (x 2 1 + x 2 2 )u, L u V 2 (x) = -x 2 1 + 2x 1 x 2 2 (x 1 + x 4 2 ) -(x 1 + x 4 2 )u, L u V 3 (x) = -x 2 1 - x 2 2 2 + 2x 2 1 x 2 2 -x 1 u, that is a 1 (x) = -1 2 x 2 2 -2x 1 x 2 2 (x 2 1 + x 2 2 ), and b 1 (x) = x 2 1 + x 2 2 , a 2 (x) = -x 2 1 + 2x 1 x 2 2 (x 1 + x 4 2 ), and b 2 (x) = -(x 1 + x 4 2 ), a 3 (x) = -x 2 1 - x 2 2 2 + 2x 2 1 x 2 2
, and b 3 (x) = -x 1 . We can verify that V 1 , V 2 and V 3 are SCLFs for system S 1 , S 2 and S 3 , respectively, satisfying the small control property. Define

A 1 = {x ∈ IR n | x 1 < -x 4 2 } A 2 = {x ∈ IR n | x 1 = -x 4 2 and x 1 = 0} A 3 = {x ∈ IR n | -x 4 2 < x 1 < 0} A 4 = {x ∈ IR n | x 1 = 0 and x 2 = 0} A 5 = {x ∈ IR n | 0 < x 1 } Clearly, IR n \ {0} = A 1 ∪ A 2 ∪ A 3 ∪ A 4 ∪ A 5 . Moreover, I p (x) = {1, 2, 3} and I n (x) = ∅ if x ∈ A 1 , I p (x) = {1, 3} and I n (x) = ∅ if x ∈ A 2 , I p (x) = {1, 3} and I n (x) = {2} if x ∈ A 3 , I p (x) = {1} and I n (x) = {2} if x ∈ A 4 , I p (x) = {1} and I n (x) = {2, 3} if x ∈ A 5 . Now, if x ∈ A 3 ∪ A 4 ∪ A 5 , then a 1 (x) b 1 (x) - a 2 (x) b 2 (x) = - x 2 2 2(x 2 1 + x 2 2 ) - x 2 1 x 1 + x 4 2 < 0. If x ∈ A 3 , then a 3 (x) b 3 (x) - a 2 (x) b 2 (x) = x 1 + x 2 2 2x 1 - x 2 1 x 1 + x 4 2 < 0. If x ∈ A 5 , then a 1 (x) b 1 (x) - a 3 (x) b 3 (x) = - x 2 2 2(x 2 1 + x 2 2 ) -x 1 - x 2 2 2x 1 < 0. That is, Assumption 4.1, i.e. ϕ 1 (x) < ϕ 2 (x) for all x ∈ IR n , holds since I n (x) = ∅ and then, ϕ 1 (x) = -∞ for all x ∈ A 1 ∪A 2 ∪{0}.
Thus, thanks to Theorem 4.3, the feedback

u(x) = w 1 (x) + w 2 (x) 2 , ( 5.1)
is continuous on IR 2 and globally asymptotically stabilizes in probability the systems S 1 , S 2 and S 3 simultaneously, with for each i ∈ {1, 2, 3}

u i (x) =        - a i (x) + a 2 i (x) + b 4 i (x) b i (x) , if b i (x) = 0, 0, if b i (x) = 0, and
w 1 (x) = min 0, min i=1,2,3 (- a i (x) b i (x) ) , w 2 (x) = min i=1,2,3 (u i (x)), if x ∈ A 1 . w 1 (x) = min 0, min i=1,3 (- a i (x) b i (x) ) , w 2 (x) = min i=1,3 (u i (x)), if x ∈ A 2 . w 1 (x) = min u 2 (x), min i=1,3 (- a i (x) b i (x) ) , w 2 (x) = max min i=1,3 (u i (x)), - a 2 (x) b 2 (x) , if x ∈ A 3 . w 1 (x) = min u 2 (x), - a 1 (x) b 1 (x) , w 2 (x) = max u 1 (x), - a 2 (x) b 2 (x) , if x ∈ A 4 . w 1 (x) = min max i=2,3 (u i (x)), - a 1 (x) b 1 (x) ) , w 2 (x) = max u 1 (x), max i=2,3 (- a i (x) b i (x) ) , if x ∈ A 5 .
Example 2

Consider a system with the following two possible modes

S 1 :          dx 1 = (x 3 2 -2 (x 2 1 + x 2 2 ) 1 + x 2 1
)dt + (-2x 1 + x 2 )udt

dx 2 = ( x 2 1 + x 2 2 1 + x 2 1 -x 3 2 )dt + (x 1 -x 2 )udt + 1 2 x 2 2 dω S 2 :            dx 1 = (-x 1 + x 2 1 + x 2 2 1 + x 2 1 )dt + x 1 udt + √ 2 x 1 1 + x 2 1 dω dx 2 = - x 2 1 x 2 1 + x 2 1 dt + x 2 udt
Taking the following functions

V 1 (x) = (x 1 + x 2 ) 2 2 + x 2 2 2 and V 2 (x) = x 2 1 + x 2 2 2 . Let L u V 1 and L u V 2
the infinitesimal operator associated with system S 1 and S 2 , respectively, we have L u V 1 (x) = a 1 (x) + b 1 (x)u and L u V 2 (x) = a 2 (x) + b 2 (x)u, with a 1 (x) = -3 4 x 4 2 -

x 1 1 + x 2 1 (x 2 1 + x 2 2 ), b 1 (x) = -x 2 1 -x 2 2 , a 2 (x) = (-x 2 1 + x 1 )(x 2 1 + x 2 2 ) 1 + x 2 1 , b 2 (x) = x 2 1 + x 2 2 .
It can be seen that V i is a SCLF for system S i (i = 1, 2). It is easy to verify that both V 1 and V 2 satisfy the the small control property (SCP). On the other hand, straightforward calculations lead to L u V 1 (x) < 0, and inf

a 1 (x) b 1 (x) = 3x 4 2 4(x 2 1 + x 2 2 ) + x 1 1 + x 2
u∈B 1 L u V 2 (x) < 0, ∀x ∈ IR 2 \ {0}.
Notice that the stabilizability condition: b 1 (x) and b 2 (x) have the same sign for all x ∈ IR 2 , stated in [START_REF] Iggidr | Simultaneous stabilization for a collection of stochastic nonlinear systems based on CLFs[END_REF] is not satisfied, since b 1 (x) < 0, and b 2 (x) > 0, x = 0. However, we have a 2 (x) b 2 (x) -a 1 (x) b 1 (x) = -

x 2 1 1+x 2 1 - 3x 4 2 4(x 2 1 +x 2
2 ) < 0, ∀x ∈ IR 2 \ {0}, that is, Assumption 4.1, i.e. ϕ 1 (x) < ϕ 2 (x) for all x ∈ IR 2 , is satisfied. Thus, according to Theorem 4.5, the feedback

u(x) = w 1 (x) + w 2 (x) 2 , ( 5.2)
is continuous on IR 2 , takes values in B 1 and globally asymptotically stabilizes in probability systems S 1 and S 2 simultaneously, with, using (4.16) and notations in Theorem 4.5

w 1 (x) =          min - a 1 (x) + a 2 1 (x) + b 4 1 (x) b 1 (x) 1 + 1 + b 2 1 (x)
, -a 2 (x) b 2 (x) if x = 0, 0, if x = 0, and

w 2 (x) =          max - a 2 (x) + a 2 2 (x) + b 4 2 (x) b 2 (x) 1 + 1 + b 2 2 (x)
, -a 1 (x) b 1 (x) if x = 0, 0, if x = 0.

Assumption 4 . 1 .

 41 For every x ∈ IR n \ {0}, ϕ 1 (x) < ϕ 2 (x).

Remark 4 . 3 .

 43 Note that Assumption 4.1:

  ) and let D = {(a, b) ∈ IR 2 | a < |b|}. Now, note that property (4.13) is equivalent to a(x) < |b(x)|, for all x ∈ IR n \ {0}, that is (a(x), b(x)) ∈ D for all x ∈ IR n \ {0}. By Lemma 2.3 of [25] we have |φ(a, b)| < 1 for all (a, b) ∈ D. Hence |p(x)| = |φ(a(x), b(x))| < 1 which concludes the proof of the Theorem 4.4.

b 1

 1 (x) < 1 and -1 < -a 2 (x) b 2 (x), for all x ∈ IR 2 \ {0}, and, since b 1 (x) < 0, b 2 (x) > 0, for all x = 0, it follows inf u∈B 1

Figure 1 :

 1 Figure 1: The responses of the closed-loop systems S1, S2 and S3 with the same feedback (5.1).

Figure 2 :

 2 Figure 2: The responses of the closed-loop systems S'1 and S'2 with the same feedback (5.2).


Banan Kerdi
email: banan.kerdi@lncmi.cnrs.fr

Mathieu Pierre

Robin Cours

Bénédicte Warot-Fonrose

Michel Goiran

Walter Escoffier
email: walter.escoffier@lncmi.cnrs.fr

High magnetic field spin-valley-split Shubnikov-de Haas oscillations in a WSe 2 monolayer

We study Shubnikov-de Haas oscillations in a p-type WSe2 monolayer under very high magnetic field. The oscillation pattern is complex due to a large spin and valley splitting, in the non-fullyresolved Landau level regime. Our experimental data can be reproduced with a model in which the main parameter is the ratio between the Zeeman energy and the cyclotron energy. The model takes into account the Landau levels from both valleys with the same Gaussian broadening, which allows to predict the relative amplitude of the resistance oscillation originating from each valley. The Zeeman energy is found to be several times larger than the cyclotron energy. It translates into a large and increasing effective Landé factor as the hole density decreases, in the continuity of the values reported in the literature at lower carrier density.

I. INTRODUCTION

Single atomic layers of semiconducting transition metal dichalcogenides (TMDCs) have received much attention over the last decade due to their promising characteristics for two-dimensional-based (2D) optoelectronic devices [1][2][3][4] and their potential for the development of valleytronics physics [5]. The crystal structure of these 2D materials, composed of one transition metal atom (W, Mo) and two chalcogen atoms (S, Se) per unit cell arranged in a honeycomb lattice, leads to a direct band gap in the visible range, located at the corner (K and K points) of the first Brillouin zone [1,2,6]. The lack of inversion symmetry of the crystal structure, combined with strong spin-orbit coupling due to the heavy transition metal atoms, translate into coupled valley and spin degrees of freedom of the charge carriers, later referred to as the |K ↑ and |K ↓ states. The presence of an out-ofplane magnetic field not only turns the energy spectrum into discrete Landau levels (LL), separated by the cyclotron energy E c , but also lifts the spin/valley degeneracy of the |K ↑ and |K ↓ states through Zeeman energy E z = g * µ B B where µ B is the Bohr magneton and g * is the effective Landé factor. In TMDC monolayers, experimental studies point to a very large value of g * since the spin, orbital, and lattice Zeeman effects add up to each other [7][8][9]. The first effect refers to the interaction between the magnetic field and the electron or hole spin, the second is linked to the spin-orbit coupling, and the last alludes to the opposite Berry curvatures in the non-equivalent K and K valleys [10]. While the effective Landé factor of excitons in TMDC monolayers has been measured by several groups [11][12][13][14], it would be desirable to obtain this fundamental parameter for the conduction and valence bands independently. g * is expected to depend on the chemical composition and the number of layers of the TMDC, on the nature (electron or hole) of the charge carriers, as well as the carrier density. In this work, we focus on its determination for holes in WSe 2 monolayers, which we extend to high carrier concentration thanks to a very high magnetic field.

In general, the ratio E z /E c , from which g * is extracted, can take any value, leading to non-equidistant Landau levels and complex Shubnikov-de Haas (SdH) oscillations in a magneto-transport experiment. Figure 1 shows the Landau level spectrum as a function of magnetic field in the case of a large spin-valley splitting (Fig. 1a) and its evolution with E z /E c (Fig. 1b). When this ratio is not an integer, the non-equidistant Landau levels translate into non 1/B-periodic quantum oscillations, as illustrated in Fig. 1c. In 2D electron gases (2DEG) realized in standard semiconducting heterostructures, the effective Landé factor is experimentally addressed by tilting the magnetic field with respect to the sample's plane. Indeed, the Zeeman energy E z depends on the magnetic field intensity regardless of its orientation whereas the cyclotron energy E c is proportional to the out-of-plane projection of the magnetic field. A coincidence angle is reached when the Zeeman energy equals the cyclotron energy, and the magneto-resistance recovers the distinctive 1/B-periodicity of single-carrier Shubnikov-de Haas oscillations. In the case of TMDCs, however, this coincidence method fails because the spin of the charge carriers is locked to the sample's out-of-plane direction [9,15]. Nonetheless, and beyond the difficulty in obtaining highmobility samples with low-resistance ohmic contacts, a few studies have reported the ratio E z /E c based on the analysis of SdH oscillations. In Ref. 9, an estimation of E z /E c rounded to integer values has been extracted from sequences of SdH oscillation minima at odd or even filling factors. In Ref. 16, the identification of the transition between mixed and polarized Landau level regimes has been exploited to deduce an electron-density dependent g * factor in high-mobility MoS 2 bilayers. Otherwise, as an alternative to magneto-transport, contactless spectroscopy using a single-electron transistor has been used to sense directly the chemical potential and infer the Landau level spectrum [8].

K ′ K E z E c E soc µ |↑ |↓ |↓ |↑ 0 2 4 6 8
E z /E c
In this work, we performed magneto-transport measurement in a WSe 2 monolayer under very high magnetic field, allowing for partial Landau level resolution in samples with reasonable mobilities fabricated without complicated contact engineering. We analyze the complex SdH oscillation pattern using a model where E z /E c is the main fitting parameter. It takes into account all the |K ↑ and |K ↓ Landau levels at once with the same Gaussian broadening. This novel approach is particularly helpful to analyze oscillations with non-fully-resolved spin and valley splitting.

II. EXPERIMENTS

A single atomic layer of WSe 2 is exfoliated from the bulk material [START_REF]The WSe2 bulk crystal has been bought from 2Dsemiconductors[END_REF] using the micro-mechanical cleavage method and transferred onto a stamp of PDMS [START_REF] Castellanos-Gomez | [END_REF]. This flake is deposited under optical microscope control onto a 20-nm-thick flake of Boron Nitride (BN) pre-deposited on a standard Si/SiO 2 substrate with d SiO2 = 280 nm. Electron beam lithography with a 495 K (100 nm)/950 K (80 nm) PMMA bilayer mask followed by platinum sputtering (10 nm) and gold thermal evaporation (50 nm), is realized to fabricate the electrodes, as sketched in Fig. 2a. The WSe 2 flake is later confirmed to be a monolayer by photoluminescence (Fig. 2c) and bright-field scanning transmission electron microscopy (Fig. 2b). In this transistor configuration, the back-gate voltage (V g ) applied between the sample and the substrate sets the hole density.

We performed transport measurements under pulsed magnetic field up to 54 T at low temperature (4.2 K) for different values of the back-gate voltage. A constant DC current of 1 µA is passed through the sample while the longitudinal voltage is measured during a pulse of magnetic field with total duration ∼ 500 ms. The magnetoresistance displays large-amplitude oscillating features on top of a smooth background, subtracted to the data to obtain the oscillatory part ∆R xx (B) of the signal only, as shown in Fig. 3. For all gate voltages, the oscillations exhibit a complex pattern, requiring an analysis beyond spin-degenerate single-band 1/B-periodic SdH oscillations. The large spin/valley splitting is responsible for this effect, as explained in the Introduction.

III. MODELING AND SIMULATIONS

To extract the ratio E z /E c , we performed numerical simulations of the oscillatory part of the resistance ∆R xx (B). We start with the LL structure of WSe 2 , adapted from the massive Dirac fermion dispersion relation [21][22][23], which provides the set of energies E N,s at a given magnetic field B.

E N,s = -N ω c -sg * µ B B, ( 1
)
where ω c = eB/m * is the cyclotron energy, g * is the effective Landé factor, N = 0, 1, 2, ... is the LL index, and s = ± 1 2 stands for the coupled spin/valley degree of freedom, respectively |K ↑ and |K ↓ . We set s = + 1 2 for N = 0 to satisfy energy minimization argument [24]. Under quantizing magnetic field and assuming a finite energy relaxation time, the density of states can be described by a sum of area-normalized Gaussian functions centered at the energy of the LLs E N,s ,

DoS(E, B) = N,s eB h × 1 √ 2πΓ(B) ×exp -(E -E N,s ) 2 2Γ 2 (B) , (2)
where Γ(B) = e m * 2B πµ h with µ h the hole mobility at zero magnetic field [25]. The pre-factor eB/h accounts for the orbital degeneracy of the LLs. We assume that the charge density n h remains constant in the device, set by the fixed back-gate voltage V g , while the chemical potential changes according to the variation of the density of states at the Fermi energy induced by the magnetic field. For each value of the magnetic field, the chemical potential µ(B) is obtained numerically by solving

n h = +∞ -∞ DoS(E, B) × f (E, µ(B), T ) × dE, (3)
i.e., when the cumulative orbital degeneracy of the occupied LLs reaches the hole carrier density. Here,

f (E, µ, T ) = 1 1+exp (E-µ) k B T is the Fermi-Dirac distribution function. The conductivity σ xx (B) is calculated from [26] σ xx (B) = e 2 h N,s +∞ -∞ N + 1 2 × exp -(E -E N,s ) 2 2Γ 2 (B) 2 × ∂f (E, µ, T ) ∂E × dE. ( 4
)
Here, the conductivity of the system is interpreted within the two-fluid model without interaction between the charge carriers belonging to different valley/spin indices. The contributions of each quantum state to the conductivity add therefore independently. We emphasize that this model is certainly oversimplified since it cannot reproduce LL anti-crossing as investigated in Refs. 16 and 27 for MoS 2 and Ref. 8 for WSe 2 . The longitudinal resistivity ρ xx is computed by inverting the conductivity tensor, where the product σ xx × ρ xy 1 can be expanded in a Taylor series when the chemical potential is located in between two successive LLs. We obtain

ρ xx = σ xx × ρ 2 xy + σ 3 xx × ρ 4 xy + . . . , (5)
where the Hall resistivity ρ xy is computed from the relation Despite some mismatch in the relative amplitude of the oscillations, probably linked to imprecise background subtraction, the overall shape of the curves is nicely reproduced without invoking any change of E z /E c with magnetic field (see Fig. 4). The best fits provide the hole density n h , the mobility µ h and the ratio E z /E c . We used the same mobility for the |K ↑ and |K ↓ states as we suppose that the main sources of scattering (remote charged impurities and punctual lattice defects) are spin/valley independent. The effective mass is set to m * = 0.45 × m e where m e is the bare electron mass (see Supplemental Material IV) and the temperature is set constant at T = 4.2 K. The numerical simulation also includes the optimization of the mobility edge, which sets the energy threshold between localized and extended states in LLs, as a magnetic fielddependent cut-off of the Gaussian functions in Eqs. ( 4) and (6). The extracted hole density with respect to V g is in good agreement with the plane capacitor model (see Supplemental Material III) where the gate capacitance per unit area,

ρ xy = h e 2   +∞ -∞ N,s dE √ 2πΓ(B) × exp - (E -E N,s ) 2 2Γ 2 (B)   -1 . ( 6
)
n h =
C g = C -1 SiO2 + C -1 BN -1 = 11.5 nF/cm 2 , is
derived from the SiO 2 thickness of 280 nm and the BN thickness of 20 nm with relative dielectric permittivity r(SiO2) = 3.9 and r(BN) = 3.8, respectively. We obtain µ h ∼ 2000 cm 2 V -1 s -1 for n h 6.5 × 10 12 cm -2 , with a progressive drop down to µ h ∼ 1000 cm 2 V -1 s -1 as the hole density decreases. For n h < 5.7 × 10 12 cm -2 , the amplitude of the oscillations is too weak with respect to the magneto-resistance background, preventing a reliable determination of E z /E c .

IV. DISCUSSION

Our simulations suggest that the ratio E z /E c is comprised between 3 and 4 for n h = 7.5 till 5 × 10 12 cm -2 , respectively (see black dots in Fig. 5a). To establish this result, we particularly focused on the specific shape of the high-field oscillations, as shown in Fig. 4. Although a shift of -2, +2, +4, ... on the ratio E z /E c gives the same apparent splitting between the |K ↑ and |K ↓ LLs (see Fig. 1b), the simulated SdH oscillation pattern turns out different at high field. Indeed, the amplitude of the conductivity depends on the LL index [25] in Eq. ( 4). Therefore, the relative amplitude of the resistance peaks corresponding to the crossings of |K ↑ and |K ↓ LLs with the chemical potential varies according to the exact value of E z /E c . Moreover, the system is expected to undergo a transition from a mixed to a polarized LL regime at a critical field depending on E z /E c . In this study, no hint of such a transition could be clearly identified. Nevertheless, it is possible to set a higher limit on E z /E c . For instance, for V g = -97.5 V (see Supplemental Material II), we observe a low-amplitude resistance peak at 49 T which we attribute to the crossing of the chemical potential with a |K ↓ LL. For this gate voltage, the best fit gave E z /E c = 3.58. If this ratio was offset by +2, an additional large amplitude peak corresponding to a |K ↑ LL should be observed instead, indicating that the transition would occur within our experimental magnetic field range. A similar conclusion can be drawn for all gate voltages ≥ -100 V, ruling out values of E z /E c above ∼ 4 in the explored range of hole densities.

Figure 5a compares our results to those of Movva et al. [9] and Gustafsson et al. [8] marked by gray and blue dots, respectively. In the first study, the E z /E c ratio takes integer values only, since the authors detected the magneto-resistance minima at even(odd)-integer values of the filling factor corresponding to odd(even)-integer E z /E c ratio [28]. In the second study, however, the E z /E c ratio is determined from the chemical potential jumps between successive LLs, detected using a singleelectron transistor coupled to the WSe 2 monolayer. Interestingly, the authors assumed a LL structure based on a Schrödinger Fermion (SCHF) model. Compared to the massive Dirac Fermion (MDF) model and regarding transport properties only, the main difference lies in the double spin/valley degeneracy of the 0th Landau level for E z = 0. In systems where the Zeeman energy is very weak compared to the LL gap, this difference translates into a shift of the SdH oscillations by half a period. This effect has been particularly put forward in graphene to distinguish its peculiar electronic properties (originating from massless Dirac fermions) from those of standard semiconducting 2DEG [29,30]. However, for systems such as TMDCs where E z can be several times larger than E c , the distinction between the two models is uncertain. Indeed, our experimental data could be fitted in a similar way considering the SCHF model with the ratio E z /E c up-shifted by 1 (see Figs. 5b and5c). Although our study provides the value of E z /E c plus or minus 1 depending on the fermion model, it is noteworthy that its variation versus n h shows the same slope compared to the few data available in the literature.

V. CONCLUSION

To conclude, we studied the Zeeman-effect-dependent magneto-resistance oscillation pattern of a WSe 2 monolayer sample under strong magnetic field at low temperature. The chemical potential lies in the valence band and the explored range of hole density ranges from 7.5 till 5 × 10 12 cm -2 . The high carrier density prevents the observation of the transition from the mixed to the polarized LL regime. However, the special shape of the quantum oscillations, which deviates from perfect 1/Bperiodicity, is a clear signature of partially overlapping LLs from the |K ↑ and |K ↓ states. These features are well reproduced using the Gaussian model for conductivity, with no interaction between charge carriers of opposite spins. We extract the ratio E z /E c , which increases as the hole density decreases, thereby enlarging the carrier density range where the enhancement of g * has been reported. For systems with negligible Zeeman energy, the magneto-transport technique is sensitive to the degeneracy of the 0th LL and therefore allows a distinction between the LL structure originating from Dirac or Schrödinger fermions. Here, a large Zeeman energy prevents an unambiguous determination of E z /E c , as the fermion type in TMDC monolayers is still controversial. In this case, this ratio is inferred ±1 depending on the fermion model, but cannot be larger than ∼ 5 in the present study and within the experimentally accessible range of hole density. The ratio E z /E c is directly linked to the effective g-factor g * = 2me m * × Ez Ec , which ranges between 14.7 and 17.3 for 3.3 < E z /E c < 3.9 and from 19.1 to 21.8 for 4.3 < E z /E c < 4.9, considering a fixed effective mass (m * = 0.45 × m e ) of the holes.

ACKNOWLEDGMENTS

We acknowledge funding from ANR under project MoS 2 ValleyControl No. ANR-14-CE26-0017 and financial support through the EUR grant NanoX No. ANR-17-EURE-0009 in the framework of the "Programme des Investissements d'Avenir". The high magnetic field measurements were performed at LNCMI-Toulouse under EMFL proposals TSC01-118 and TSC07-218. We thank Emmanuel Courtade, Cédric Robert, and their colleagues at LPCNO and Pascal Puech (CEMES) for performing room-temperature photoluminescence characterization of our samples; Thomas Blon (LPCNO) for providing access to a metallic thin-film deposition machine; and our technical staff colleagues at LNCMI-Toulouse.

High magnetic field spin-valley-split Shubnikov-de Haas oscillations in a WSe 2 monolayer

Supplemental Material

Banan Kerdi , I.

Vg (V

2 FIG. 1 .

 21 FIG. 1. Energy of LLs as a function of (a) magnetic field and (b) Ez/Ec where the red and blue lines represent the |K ↑ and |K ↓ states, respectively. (c) Simulated magnetoresistance where both a large LL broadening and a non-integer ratio Ez/Ec = 3.2 have been chosen to produce an example of complex oscillations. The simulation is performed at constant chemical potential corresponding to the solid green line depicted in panel (a). A maximum of resistance is expected each time the chemical potential crosses a LL. (d) Simplified valence band of WSe2 monolayers under magnetic field. The valleys at points K and K of the first Brillouin zone are made of two spin-split parabolic bands. For a given spin sign, the energy difference between the two bands arises from the spin-orbit coupling (SOC), with Esoc = 466 meV [17, 18]. The white lines indicate the energies of the Landau levels for Ez/Ec = 3.2. They are shown only for the two upper bands, where the chemical potential (solid green line) is located in the explored range of hole densities.

FIG. 2 .

 2 FIG. 2. (a) Sketch of the BN/WSe2 heterostructure. (b) Bright-field scanning transmission electron microscopy (BF-STEM) image realized after the measurement campaign confirming a WSe2 monolayer as the conducting channel. The h-BN substrate appears as dark/bright fringes whereas the WSe2 layer consists of dark spots surrounded by brighter ones. The interatomic distances between W and Se atoms correspond to the expected ones. (c) The photoluminescence spectrum with λexc = 633 nm is characteristic of a WSe2 monolayer. (d) Current flowing in the device for V ds = 100 mV versus the back-gate voltage.

FIG. 3 .

 3 FIG. 3.High field magneto-resistance of the sample at T = 4.2 K, after background removal, for various values of the carrier density n h (given in unit of 10 12 cm -2 ) ranging from 5.70 to 7.35 × 10 12 cm -2 . The corresponding back-gate voltage Vg is given in parentheses. An offset of 1 kΩ is set between consecutive curves for clarity. The strong deviations from a 1/B-periodic oscillatory behavior are indicative of an additional Zeeman energy scale competing with the cyclotron energy. See also Supplemental Material I where the same dataset is plotted versus filling factor.

FIG. 5 .

 5 FIG. 5. (a) Extracted values of Ez/Ec as a function the hole density. The black dots are the values obtained assuming the Landau level structure of Massive Dirac Fermions (MDF). By up-shifting these values by 1, we obtain the values corresponding to the LL structure of Schrödinger Fermions (SCHF). Ez/Ec increases as the carrier density reduces, suggesting the reinforcement of the Coulomb interactions. The Ez/Ec ratio is compared to the values found in the literature (blue and gray dots), where either the MDF or SCHF model was considered (see main text). (b), (c) Comparison of LL energies in cyclotron energy unit as a function of Ez/Ec for the (b) MDF and (c) SCHF models. This figure demonstrates that their LL structure is identical under an up-shift of Ez/Ec by 1.

IFIG. 1 .FIG. 2 .

 12 FIG.1. Experimental magneto-resistance (background subtracted) at T = 4.2 K plotted versus filling factor ν = n h h eB . The curves are successively offset by 1 kΩ for clarity. The carrier density was extracted from the fitting procedure described in the main text.

FIG. 3 .FIG. 4 .

 34 FIG.3. The hole carrier density extracted from the fitting of the magneto-resistance oscillations is compared to its theoretical evolution with back-gate voltage, based on the calculation of the back-gate capacitance with the plane capacitor model, which gives Cg = 11.5 nF/cm 2 .

  1, * Mathieu Pierre , 1 Robin Cours, 2 Bénédicte Warot-Fonrose , 2 Michel Goiran, 1 and Walter Escoffier 1, †

	1 LNCMI, Université de Toulouse, CNRS, INSA, UPS, EMFL, 31400 Toulouse, France
	2 CEMES, Université de Toulouse, CNRS, 31055 Toulouse, France

TABLE I .

 I Hole density, Ez/Ec ratio, and zero-field mobility obtained from the fit of the experimental quantum oscillations shown in Fig.2.

	)	-110	-107.5	-105	-102.5	-100	-97.5	-95	-92.5	-90
	n h ×10 12 cm -2	7.35	7.26	7.01	6.76	6.70	6.47	6.18	5.82	5.70
	Ez/Ec	3.37	3.36	3.49	3.56	3.55	3.62	3.66	3.83	3.75
	µ h cm 2 V -1 s -1	2400	2400	2400	2400	2200	1600	1400	1200	1300

III.

Comparison of the hole carrier density extracted from the fit and the plane capacitor model

		8
		7
	-2
	cm
	×10 12	6
	n h
		5	Best fit values Theoretical gate capacitance model
			-110	-100 V g (V)	-90	-80