Weighted Homogeneity for Time-Delay Systems: Finite-Time and Independent of Delay Stability

Global delay independent stability is analyzed for nonlinear time-delay systems by applying homogeneity theory. It is shown that finite-time stability can be encountered in this class of systems under uniformity of the convergence time with respect to delay. Some additional tools for stability analysis of time-delay systems using homogeneity are also presented: in particular, it is shown that if a time-delay system is homogeneous with nonzero degree and it is globally asymptotically stable for some delay, then this property is preserved for any delay value, which is known as the independent of delay (IOD) stability.

I. INTRODUCTION

The theory of homogeneous dynamical systems has been well developed for continuous time-invariant differential equations [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Kawski | Progress in systems and control theory: New trends in systems theory[END_REF], [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF]. The main feature of a homogeneous nonlinear system is that its local behavior is the same as the global one (local attractiveness implies global asymptotic stability, for example). In addition, the homogeneous stable/unstable systems admit homogeneous Lyapunov functions [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF]. Since the subclass of nonlinear systems with a global symmetry is rather narrow, the concept of local homogeneity has been introduced [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], [START_REF] Efimov | Oscillations conditions in homogenous systems[END_REF]. Then, a local homogeneous approximation of nonlinear system can be computed representing locally the original system behavior. The advantage in this case is that the approximating dynamics is globally homogeneous and the equation complexity is decreased.

The dynamical systems subjected by a time-delay, whose models are represented by functional differential equations, find their applications in many areas of science and technology [START_REF]Applications of Time Delay Systems[END_REF], [START_REF] Erneux | Applied Delay Differential Equations[END_REF]. Analysis of delay influence on the system stability is critical for many natural and human-developed systems [START_REF] Gu | Stability of Time-Delay Systems[END_REF], [START_REF] Hale | Theory of Functional Differential Equations[END_REF], [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]. Despite of variety of applications, most of them deal with linear time-delay models, which is originated by complexity of stability analysis for time-delay systems in general (design of a Lyapunov-Krasovskii functional or a Lyapunov-Razumikhin function is a complex problem), and that constructive and computationally tractable conditions exist for linear systems only [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], [START_REF] Fridman | Introduction to Time-Delay Systems: Analysis and Control[END_REF].

An attempt to apply the homogeneity theory for nonlinear functional differential equations in order to simplify their stability analysis has been performed in [START_REF] Efimov | Homogeneity for time-delay systems[END_REF], [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF]. In those works an extension of the homogeneity theory for time-delay systems has been proposed. Applications of the conventional homogeneity framework for analysis of time-delay systems (considering delay as a kind of perturbation, for instance) have been carried out even earlier in [START_REF] Aleksandrov | On the asymptotic stability of solutions of nonlinear systems with delay[END_REF], [START_REF] Asl | Analytical solution of a system of homogeneous delay differential equations via the Lambert function[END_REF], [START_REF] Bokharaie | D-stability and delayindependent stability of homogeneous cooperative systems[END_REF], [START_REF] Diblik | Asymptotic equilibrium for homogeneous delay linear differential equations with l-perturbation term[END_REF], [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF].

Establishing stability properties, it is also important to qualify the rate of convergence in the system: exponential, asymptotic, finite-time or fixed-time [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Dorato | An overview of finite-time stability[END_REF], [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF], [START_REF] Nersesov | Finite-time stabilization of nonlinear dynamical systems via control vector lyapunov functions[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]. Frequently, the homogeneity theory is used to establish finite-time or fixed-time stability [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF]: for example, if a system (without delay) is globally asymptotically stable and homogeneous of negative degree, then it is finite-time stable. There is a recent interest to analysis of finite-time stability behavior for time-delay systems [START_REF] Karafyllis | Finite-time global stabilization by means of time-varying distributed delay feedback[END_REF], [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF], [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF], and the objective of the present paper is to show how the homogeneity theory extension proposed in [START_REF] Efimov | Homogeneity for time-delay systems[END_REF], [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF] can be used for analysis of finite-time stability for functional differential equations with nonzero degree of homogeneity.

The outline is as follows. The preliminary definitions and the homogeneity for time-delay systems are given in Section 2. The scaling property of solutions for some class of homogeneous time-delay systems is investigated in Section 3, where the conditions of IOD stability are established. The conditions of finite-time stability are also given in Section 3. An example is considered in Section 4.

II. PRELIMINARIES

Consider an autonomous functional differential equation of retarded type [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]:

dx(t)/dt = f (x t ), t ≥ 0, (1)
where The representation (1) includes pointwise or distributed timedelay systems. We assume that solutions of the system (1) satisfy the initial functional condition x 0 ∈ C [-τ,0] . It is known from the theory of functional differential equations [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF] that under the above assumptions the system (1) has a unique solution x(t, x 0 ) satisfying the initial condition x 0 and x t,x0 (s) = x(t + s, x 0 ) for -τ ≤ s ≤ 0, which is defined on some finite time interval [-τ, T ) (we will use the notation x(t) to reference x(t, x 0 ) if the origin of x 0 is clear from the context).

x ∈ R n and x t ∈ C [-τ,0] is the state function, x t (s) = x(t + s), -τ ≤ s ≤ 0 (
The upper right-hand Dini derivative of a locally Lipschitz continuous functional V : C [-τ,0] → R + along the system (1) solutions is defined as follows for any φ ∈ C [-τ,0] :

D + V (φ) = lim sup h→0 + 1 h [V (φ h ) -V (φ)],
where φ h ∈ C [-τ,0] for 0 < h < τ is given by

φ h = φ(θ + h), θ ∈ [-τ, -h) φ(0) + f (φ)(θ + h), θ ∈ [-h, 0].
For a locally Lipschitz continuous function V : R n → R + the upper directional Dini derivative is defined as follows:

D + V [x t (0)]f (x t ) = lim h→0 + sup V [x t (0) + hf (x t )] -V [x t (0)] h .
A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ (0) = 0; it belongs to class K ∞ if it is also radially unbounded. A continuous function β : R + × R + → R + belongs to class KL if β(•, r) ∈ K and β(r, •) is decreasing to zero for any fixed r ∈ R + . The symbol 1, m is used to denote a sequence of integers 1, ..., m.

A. Stability definitions

Let Ω be a neighborhood of the origin in C [-τ,0] . Definition 1. [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF] The system (1) is said to be (a) stable at the origin into Ω if there is σ ∈ K such that for any x 0 ∈ Ω, |x(t, x 0 )| ≤ σ(||x 0 ||) for all t ≥ 0;

(b) asymptotically stable at the origin into Ω if it is stable into Ω and lim t→+∞ |x(t, x 0 )| = 0 for any x 0 ∈ Ω;

(c) finite-time stable at the origin into Ω if it is stable into Ω and for any x 0 ∈ Ω there exists 0 ≤ T x0 < +∞ such that x(t, x 0 ) = 0 for all t ≥ T x0 . The functional T 0 (x 0 ) = inf{T x0 ≥ 0 : x(t, x 0 ) = 0 ∀t ≥ T x0 } is called the settling time of the system [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF].

If Ω = C [-τ,0] , then the corresponding properties are called global stability/asymptotic stability/finite-time stability.

B. Homogeneity

For any r i > 0, i = 1, n and λ > 0, define the dilation matrix Λ r (λ) = diag{λ ri } n i=1 and the vector of weights r = [r 1 , ..., r n ] T .

For any r i > 0, i = 1, n and x ∈ R n the homogeneous norm can be defined as follows

|x| r = n i=1 |x i | ρ/ri 1/ρ , ρ ≥ max 1≤i≤n r i .
For all x ∈ R n , its Euclidean norm |x| is related with the homogeneous one:

σ r (|x| r ) ≤ |x| ≤ σr (|x| r )
for some σ r , σr ∈ K ∞ . The homogeneous norm has an important property that is |Λ r (λ)x| r = λ|x| r for all x ∈ R n . Define S r = {x ∈ R n : |x| r = 1}.

For any r i > 0, i = 1, n and φ ∈ C [-τ,0] the homogeneous norm can be defined as follows

||φ|| r = n i=1 ||φ i || ρ/ri 1/ρ , ρ ≥ max 1≤i≤n r i .
There exist two functions ρ r , ρr ∈ K ∞ such that for all φ ∈ C [-τ,0] [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF]:

ρ r (||φ|| r ) ≤ ||φ|| ≤ ρr (||φ|| r ). (2)
The homogeneous norm in the Banach space has the same important property that ||Λ r (λ

)φ|| r = λ||φ|| r for all φ ∈ C [-τ,0] . In C [-τ,0] the corresponding unit sphere S r = {φ ∈ C [-τ,0] : ||φ|| r = 1}. Define B τ ρ = {φ ∈ C [-τ,0] : ||φ|| r ≤ ρ} as a closed ball of radius ρ > 0 in C [-τ,0] . Definition 2. [15] The function g : C [-τ,0] → R is called r-homogeneous (r i > 0, i = 1, n), if for any φ ∈ C [-τ,0] the relation g(Λ r (λ)φ) = λ d g(φ)
holds for some d ∈ R and all λ > 0.

The function f :

C [-τ,0] → R n is called r-homogeneous (r i > 0, i = 1, n), if for any φ ∈ C [-τ,0] the relation f (Λ r (λ)φ) = λ d Λ r (λ)f (φ)
holds for some d ≥ -min 1≤i≤n r i and all λ > 0.

In both cases, the constant d is called the degree of homogeneity.

The introduced notion of weighted homogeneity in C [-τ,0] is reduced to the standard one in R n if τ → 0. An advantage of homogeneous systems described by ordinary differential equations is that any of its solutions can be obtained from another solution under the dilation rescaling and a suitable time re-parametrization. A similar property holds for some functional homogeneous systems. Proposition 1. [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF] Let x : R + → R n be a solution of the r-homogeneous system (1) with the degree d = 0 for an initial condition x 0 ∈ C [-τ,0] . For any λ > 0 define y(t) = Λ r (λ)x(t) for all t ≥ 0, then y(t) is also a solution of (1) with the initial condition y 0 = Λ r (λ)x 0 .

III. MAIN RESULTS

This section has two parts. First, an extension of Proposition 1 is presented, and some useful tools for IOD stability analysis of nonlinear time-delay systems are introduced. Second, the links with finite-time stability are investigated using the homogeneity arguments.

A. Scaling solutions of homogeneous systems with non-zero degree

Proposition 1 establishes that all solutions of homogeneous time-delay systems of degree zero are interrelated, thus the local behavior implies global one. Some more complex relations can also be established for the case when the degree is not zero. Proposition 2. Let x(t, x 0 ) be a solution of the rhomogeneous system (1) with the degree d = 0 for an initial condition x 0 ∈ C [-τ,0] , τ ∈ (0, +∞). For any λ > 0 the functional differential equation

dy(t)/dt = f (y t ), t ≥ 0 (3) with y t ∈ C [-λ -d τ,0]
, has a solution y(t, y 0 ) = Λ r (λ)x(λ d t, x 0 )1 for all t ≥ 0 with the initial condition

y 0 ∈ C [-λ -d τ,0] , y 0 (s) = Λ r (λ)x 0 (λ d s) for s ∈ [-λ -d τ, 0]. Proof. By definition x λ d t (s) = x(λ d t + s) ∈ R n and y t (s) = Λ r (λ)x λ d t (s) for any -λ -d τ ≤ s ≤ 0, then ẏ(t) = d dt Λ r (λ)x(λ d t) = λ d Λ r (λ)f (x λ d t ) = f (Λ r (λ)x λ d t ) = f (y t )
and y(t) is a solution of (3) (dilation scales the argument of

f as x(θ) → y(λ -d θ) for θ ∈ [-τ, 0]).
In [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF], using Proposition 1 it has been shown for d = 0 that local asymptotic stability implies global one (for the ordinary differential equations even more stronger conclusion can be obtained: local attractiveness implies global asymptotic stability [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF]). In the present setting that result has the following correspondence.

Lemma 1. Let the system (1) be r-homogeneous with degree d = 0 and globally asymptotically stable at the origin for some delay 0 < τ 0 < +∞, then it is globally asymptotically stable at the origin for any delay 0 < τ < +∞ (i.e. IOD).

Proof. Due to imposed hypothesis, for the delay τ 0 > 0 for all x 0 ∈ C [-τ0,0] there is a function σ ∈ K such that |x(t, x 0 )| r ≤ σ(||x 0 || r ) for all t ≥ 0 and lim t→+∞ |x(t, x 0 )| r = 0. Take some τ ∈ (0, +∞) and select an initial condition y 0 ∈ C [-τ,0] , then for λ = τ0 τ 1/d (this λ is well defined since d = 0) there exists x 0 ∈ C [-τ0,0] such that y 0 (s) = Λ r (λ)x 0 (λ d s) for s ∈ [-τ, 0], and y(t, y 0 ) = Λ r (λ)x(λ d t, x 0 ) for all t ≥ 0 by Proposition 2. Thus lim t→+∞ |y(t, y 0 )| r = λ lim t→+∞ |x(λ d t, x 0 )| r = λ lim t→+∞ |x(t, x 0 )| r = 0 and the solution y(t, y 0 ) is converging asymptotically to the origin. In addition, |y(t,

y 0 )| r = λ|x(λ d t, x 0 )| r ≤ λσ(||x 0 || r ) = λσ(λ -1 ||y 0 || r )
for all t ≥ 0, which implies stability of the system (1) for the delay τ . The proven convergence to the origin and stability give the global asymptotic stability of the system for an arbitrary delay τ ∈ (0, +∞).

It is well known fact for linear systems (homogeneous systems of degree d = 0) that its stability for a sufficiently small delay does not imply stability for all τ ∈ (0, +∞). For nonlinear homogeneous systems with degree d = 0 this is not the case, according to the result of Lemma 1 if they are globally stable for some delay, they preserve their stability for an arbitrary delay τ ∈ (0, +∞). This is a surprising advantage of nonlinear time-delay systems.

Further let us consider several useful consequences of Proposition 2 and Lemma 1.

Corollary 1. Let the system (1) be r-homogeneous with degree d = 0 and asymptotically stable at the origin into the set Ω = B τ ρ = {φ ∈ C [-τ,0] : ||φ|| r ≤ ρ} for some 0 < ρ < +∞ for any value of delay 0 ≤ τ < +∞, then it is globally asymptotically stable at the origin IOD.

Proof. For any τ > 0 take y 0 ∈ C [-τ,0] , y 0 / ∈ B τ ρ , then according to Proposition 2 there is 0 < λ < +∞ (λ = ρ -1 ||y 0 || r ) and x 0 ∈ B λ d τ ρ such that y 0 (s) = Λ r (λ)x 0 (λ d s) for s ∈ [-τ, 0] and y(t, y 0 ) = Λ r (λ)x(λ d t, x 0 ) for all t ≥ 0. Since x(t) converges asymptotically to the origin, the same property is satisfied for y(t) and it enters the set B τ ρ in a finite time.

Corollary 2. Let the system (1) be r-homogeneous with degree d < 0 and asymptotically stable at the origin into the set B τ ρ for some 0 < ρ < +∞ for any value of delay 0 ≤ τ ≤ τ 0 with 0 < τ 0 < +∞, then it is globally asymptotically stable at the origin IOD.

Proof. Setup ρ < ρ , take

y 0 ∈ C [-τ0,0] , y 0 / ∈ B τ0 ρ and select 1 < λ = ρ -1 ||y 0 || r for some x 0 ∈ B λ d τ0 ρ , ||x 0 || r = ρ such that y 0 (s) = Λ r (λ)x 0 (λ d s) for s ∈ [-τ 0 , 0] (note that τ = λ d τ 0 ≤ τ 0 since d < 0 and λ > 1)
, then according to Proposition 2 y(t, y 0 ) = Λ r (λ)x(λ d t, x 0 ) for all t ≥ 0. Since x(t) converges asymptotically to the origin, the same property is satisfied for y(t). To prove stability note that there is a function

σ τ ∈ K such that |x(t, x 0 )| r ≤ σ τ (||x 0 || r ) for all t ≥ 0 and x 0 ∈ B τ ρ with 0 ≤ τ ≤ τ 0 , define R = sup 0≤τ ≤τ0 σ τ (ρ) < +∞ (R is finite since ρ < ρ ) then |y(t, y 0 )| r = λ|x(λ d t, x 0 )| r ≤ λR = ρ -1 R||y 0 || r
for all t ≥ 0 and y 0 ∈ C [-τ0,0] . Therefore, the system is globally asymptotically stable for the delay τ 0 . Using the result of Lemma 1 the claim about uniformity of global asymptotic stability with respect to the delay can be recovered.

Corollary 3. Let the system (1) be r-homogeneous with degree d > 0 and the set B τ ρ for some 0 < ρ < +∞ be uniformly globally asymptotically stable for any value of delay 0 ≤ τ ≤ τ 0 , 0 < τ 0 < +∞ 2 , then (1) is globally asymptotically stable at the origin IOD.

Proof. Take ρ > ρ and λ = ||y0||r ρ ≤ 1 for any

y 0 ∈ B τ0 ρ then there exists x 0 ∈ B λ d τ0 ρ , ||x 0 || r = ρ such that y 0 (s) = Λ r (λ)x 0 (λ d s) for s ∈ [-τ 0 , 0]. Note that λ d τ 0 ≤ τ 0 for λ ≤ 1 and d > 0. According to Proposition 2 y(t, y 0 ) = Λ r (λ)x(λ d t, x 0 ) for all t ≥ 0. Since |x(t, x 0 )| r ≤ σ λ d τ0 (||x 0 || r ) for all t ≥ 0, then |y(t, y 0 )| r ≤ ρ -1 ||y 0 || r σ λ d τ0 (ρ ) for all t ≥ 0 and all y 0 ∈ B τ0 ρ . Define R = sup τ ∈[0,τ0] σ τ (ρ ), then |y(t, y 0 )| r ≤ η(||y 0 || r ) = max{ρ -1 ||y 0 || r R, σ τ0 (||y 0 || r )}
for all t ≥ 0 for any y 0 ∈ C [-τ0,0] and the system (1) is globally stable for the delay τ 0 (in this case η(s) ≥ s for all s ≥ 0). Select ε > 0 (without loosing generality assume that η -1 ( ) < ρ) and take

λ = η -1 (ε) ρ+ε ≤ ε ρ+ε < 1, then for any y 0 ∈ B τ0 ρ \ B τ0 η -1 (ε)
2 In this case for any 0 ≤ τ ≤ τ 0 , any ε > 0 and κ ≥ 0 there is 0 ≤ T ε κ,τ < +∞ such that ||xt,x 0 ||r ≤ ρ + ε for all t ≥ T ε κ,τ for any x 0 ∈ B τ κ , and |x(t, x 0 )|r ≤ στ (||x 0 ||r) for all t ≥ 0 for some function στ ∈ K∞ for all x 0 / ∈ B τ ρ .

there exists

x 0 ∈ B λ d τ0 λ -1 ρ such that y 0 (s) = Λ r (λ)x 0 (λ d s) for s ∈ [-τ 0 , 0]. Since again y(t, y 0 ) = Λ r (λ)x(λ d t, x 0 ) for all t ≥ 0 and ||x t,x0 || r ≤ ρ + ε for all t ≥ T ε λ -1 ρ,λ d τ0 , then ||y t,y0 || r ≤ η -1 (ε) ≤ ε for all t ≥ λ -d T ε λ -1 ρ,λ d τ0 . For y 0 ∈ B τ0
η -1 (ε) we have ||y t,y0 || r ≤ ε for all t ≥ 0, that implies uniform attractivity of the origin for (1) with the delay τ 0 . From these properties the global asymptotic stability for the delay τ 0 is deduced. Finally, the claim of the corollary for any delay τ ∈ (0, +∞) follows the conclusion of Lemma 1.

Remark 1. The requirements of corollaries 2 and 3 can be relaxed asking stability of B τ0 ρ for some 0 < ρ < +∞ for a fixed 0 < τ 0 < +∞ and for τ 0 = 0, then the corresponding stability of B τ ρ for all 0 ≤ τ ≤ τ 0 can be obtained using the arguments of Lemma 1.

In addition, due to ISS property well inherited by homogeneous systems (see [START_REF] Bernuau | Verification of iss, iiss and ioss properties applying weighted homogeneity[END_REF], [START_REF] Efimov | Development of homogeneity concept for time-delay systems[END_REF] for details), under mild conditions its stability for τ = 0 implies the same property for sufficiently small delay. Similar results, connecting ISS and stability of the systems with respect to time delays, have been established in [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF], [START_REF] Teel | A note on input-to-state stability of sampled-data nonlinear systems[END_REF].

Lemma 2. Let f (x t ) = F [x(t), x(t -τ )] in (
1) and the system (1) be r-homogeneous with degree d > 0 and globally asymptotically stable at the origin for τ = 0, then for any ρ > 0 there is 0 < τ 0 < +∞ such that (1) is asymptotically stable at the origin into B τ ρ for any delay 0 ≤ τ ≤ τ 0 . Proof. For some ρ > 0 and τ > 0 let us consider x 0 ∈ B τ ρ , then

x(t, x 0 ) = x 0 (0) + ˆt 0 F [x(s, x 0 ), x(s -τ, x 0 )]ds
is the unique solution of (1) if it is defined on the interval [0, t] for t ≥ 0 [START_REF] Hale | Theory of Functional Differential Equations[END_REF], ||x 0 || ≤ ρ r (ρ) by ( 2). Since F is locally Lipschitz, then for all ||φ|| ≤ 2ρ r (ρ) there exists k > 0 such that |F [φ(0), φ(-τ )]| ≤ k||φ||.

Select τ > 0 such that kτ < 0.5, then assuming that ||x s || ≤ 2ρ r (ρ) for all s ∈ [0, t) we obtain

|x(t, x 0 )| ≤ |x 0 (0)| + ˆt 0 |F [x(s, x 0 ), x(s -τ, x 0 )]|ds ≤ |x 0 (0)| + k ˆt 0 ||x s ||ds ≤ |x 0 (0)| + kt sup 0≤s<t ||x s || ≤ (1 + 2kt)ρ r (ρ) and |x(t, x 0 )| ≤ 2ρ r (ρ) for 0 ≤ t ≤ τ , thus ||x s || ≤ 2ρ r (ρ) for s ∈ [0, τ ].
Therefore, for any ρ > 0 there exists τ > 0 such that for x 0 ∈ B τ ρ we have

x t ∈ B τ ρ for t ∈ [0, τ ] and ρ = ρ -1 r [2ρ r (ρ)] by (2). The conditions of this lemma means that F [Λ r (λ)x, Λ r (λ)z] = λ d Λ r (λ)F [x,
z] for any x, z ∈ R n and λ ∈ (0, +∞). In addition, x = 0 for the system

ẋ = F (x, x) (4)
is globally asymptotically stable. Note that the system ( 4) is also homogeneous of degree

d > 0 (F [Λ r (λ)x, Λ r (λ)x] = λ d Λ r (λ)F [x,
x] for any x ∈ R n and λ ∈ (0, +∞)), then according to [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] there is a differentiable and r-homogeneous

Lyapunov function V : R n → R + of degree v > 0 such that a = -sup ξ∈Sr D + V (ξ)F (ξ, ξ) > 0, 0 < b = sup |ξ|r≤1 ∂V (ξ) ∂ξ < +∞, c 1 = inf ξ∈Sr V (ξ), c 2 = sup ξ∈Sr V (ξ), c 1 |x| v r ≤ V (x) ≤ c 2 |x| v r ∀x ∈ R n .
Consider derivative of this function calculated for the system (1) for some φ ∈ C [-2τ,0] . To this end define λ 1 = |φ(0)| r and λ 2 = ||φ|| r with φ(0) = Λ r (λ 1 )ξ for some ξ ∈ S r and φ = Λ r (λ 2 )ϕ for some ϕ ∈ S r ⊂ C [-2τ,0] respectively, λ 2 ≥ λ 1 by definition of the norms and |ϕ(0)| r ≤ |ξ| r = 1 (a necessity to define two dilation transformations with λ 1 and λ 2 is originated by the fact that the state vectors of ( 1) and ( 4) belong to different spaces, R n and C [-τ,0] ):

D + V (φ(0))F [φ(0), φ(-τ )] = D + V (φ(0))F [φ(0), φ(0)] +D + V (φ(0)){F [φ(0), φ(-τ )] -F [φ(0), φ(0)]} = λ d+v 1 D + V (ξ)F [ξ, ξ] + λ d+v 2 D + V (ϕ(0)){F [ϕ(0), ϕ(-τ )] -F [ϕ(0), ϕ(0)]}.
Since d > 0 and the function F is locally Lipschitz with respect to both arguments, then for ϕ ∈ S r there exists L > 0 such that

|F [ϕ(0), ϕ(-τ )] -F [ϕ(0), ϕ(0)| ≤ L|ϕ(0) -ϕ(-τ )|.
Let us consider the solutions of (1) for t ≥ τ (we have shown above that for x 0 ∈ B τ ρ we have x t ∈ B τ ρ for t ∈ [0, τ ] for any ρ > 0, ρ = ρ -1 r [2ρ r (ρ)] and a properly selected τ < 0.5/k), then there exists M > 0 such that |ϕ(0)ϕ(-τ )| ≤ M τ , where M = sup |z|r≤ρ ,|y| R ≤ρ |F [z, y]|. In order to prove asymptotic stability, the Lyapunov-Razumikhin approach will be applied [START_REF] Gu | Stability of Time-Delay Systems[END_REF]. To this end, assume for some

γ > 1 that c 1 sup θ∈[-2τ,0] |φ(θ)| v r ≤ sup θ∈[-2τ,0] V [φ(θ)] < γV [φ(0)] ≤ γc 2 |φ(0)| v r , therefore sup θ∈[-2τ,0] |φ(θ)| r ≤ (γc -1 1 c 2 ) 1/v |φ(0)| r = (γc -1 1 c 2 ) 1/v λ 1 , then λ 2 = ||φ|| r = n i=1 sup θ∈[-2τ,0] |φ i (θ)| ρ/ri 1/ρ ≤ n i=1 sup θ∈[-2τ,0] |φ(θ)| ρ r 1/ρ = n 1/ρ sup θ∈[-2τ,0] |φ(θ)| r ≤ Rλ 1 , where R = n 1/ρ (γc -1 1 c 2 ) 1/v . Finally, D + V (φ(0))F [φ(0), φ(-τ )] ≤ τ bLM λ d+v 2 -aR -d-v λ d+v 2 ≤ ||φ|| d+v r (τ bLM -aR -d-v ),
and for all 0

≤ τ ≤ τ 0 = min{0.5/k, aR -d-v - bLM } for some ∈ (0, aR -d-v ) we have max θ∈[-2τ,0] V [φ(θ)] < γV [φ(0)] ⇒ D + V (φ(0))F [φ(0), φ(-τ )] ≤ -|φ(0)| d+v r
for all φ ∈ C [-2τ,0] that are solutions of (1) for t ≥ τ . Consequently [START_REF] Gu | Stability of Time-Delay Systems[END_REF], the trajectories of (1) are strictly decreasing to zero and stay bounded for t ≥ τ , but they also stay bounded for 0 ≤ t ≤ τ with ||x 0 || r ≤ ρ, which gives the desired result.

The results presented in this section open the door for a wealthy stability analysis of nonlinear time-delay systems.

B. Finite-time stability

Assume that the system (1) is finite-time stable for some τ > 0 and the settling-time functional is continuous and T 0 (0) = 0. According to such a definition there is x 0 ∈ C [-τ,0] such that T 0 (x 0 ) ≤ τ and at the instant T 0 (x 0 ) the righthand side is still dependent on initial conditions. Thus without additional assumptions on the right-hand side f (x τ ) and its dependence on x τ (0), or without skipping the continuity requirement of T 0 (x 0 ), an existence of finite-time stability phenomenon for time-delay systems is questionable [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF].

For the ordinary differential equations, finite-time stability is frequently related with homogeneity. It is well known that a homogeneous continuous-time system of negative degree is finite-time stable iff it has the origin locally attracting [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]. The recent results for finite-time stability analysis of timedelay systems [START_REF] Karafyllis | Finite-time global stabilization by means of time-varying distributed delay feedback[END_REF], [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF], [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF] are not based on homogeneity. The paper [START_REF] Karafyllis | Finite-time global stabilization by means of time-varying distributed delay feedback[END_REF] proposes design of a control, which implicitly contains some prediction mechanisms and time-varying gains in order to compensate the delay influence on the system dynamics and guarantee a kind of finite-time stability for the closed-loop system. The main result of [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF] is required to find a Lyapunov-Krasovskii functional V (φ) whose derivative is upper bounded by a certain negative function of the functional V (φ) itself (it is more restricted condition than in conventional Lyapunov-Krasovskii approach, where a function of |φ(0)| is required). Using the result of Proposition 2 on rescaling of solutions of homogeneous systems with non-zero degree, the following conditions of finite-time stability in time-delay systems can be established. Theorem 1. Let the system (1) be r-homogeneous with degree d < 0 and globally asymptotically stable at the origin IOD. Let 0 < T < +∞, where T is defined in one of the following ways for q ∈ (0, 1):

(a)

T = sup τ ∈(0,+∞) sup x0∈C [-τ,0] ,||x0||r=1 T x0 and T x0 is such that ||x t,x0 || r ≤ q||x 0 || r for all t ≥ T x0 ; (b) T = sup τ ∈(0,+∞) sup x0∈C [-τ,0] ,||x0||r≥1 T x0 and T x0 is such that |x(t, x 0 )| r ≤ q|x 0 (0)| r for all t ≥ T x0 .
Then (1) is finite-time stable at the origin IOD.

Proof. Since the system is globally asymptotically stable independently of the delay, then it is stable and according to definition of finite-time stability it is required to show finite-time convergence to the origin for all initial conditions and delays. Select some τ ∈ (0, +∞). Since the system is globally asymptotically stable, for any z 0 ∈ C [-τ,0] there is 0 ≤ t z0 < +∞ such that ||x(t + •, z 0 )|| r ≤ 1 for all t ≥ t z0 .

Next, the time of convergence to the origin equals t z0 + T 1,z0 , where T 1,z0 is the time of convergence from the sphere S r to the origin. It is necessary to show that T 1,x0 is finite for any x 0 ∈ S r .

First, consider the part (a). By conditions of the theorem, ||x t,x0 || r ≤ q||x 0 || r for any x 0 ∈ C [-τ,0] , ||x 0 || r = 1 for all t ≥ T ≥ T x0 and any τ ∈ (0, +∞). Take y 0 ∈ C [-τ,0] , ||y 0 || r = q k for some integer k ≥ 1. Then there exists x 0 ∈ C [-q kd τ,0] , ||x 0 || r = 1 such that y 0 (s) = Λ r (q k )x 0 (q kd s) for s ∈ [-τ, 0] and according to Proposition 2 y(t, y 0 ) = Λ r (q k )x(q kd t, x 0 ) for all t ≥ 0. Therefore, ||y t,y0 || r = q k ||x q kd t,x0 || r ≤ qq k ||x 0 || r = q||y 0 || r for all t ≥ T y0 = q -kd T x0 . Thus ||y t,y0 || r ≤ q||y 0 || r for any y 0 ∈ C [-τ,0] , ||y 0 || r = q k for all t ≥ q -kd T ≥ T y0 . Finally,

T1,x 0 = Tx 0 | ||x 0 ||r =1 + Ty 0 | ||y 0 ||r =q + Ty 0 | ||y 0 ||r =q 2 + . . . = T + q -d T + q -2d T + • • • = T k≥1 q -kd = T 1 -q -d < +∞.
This consideration can be repeated for any τ ∈ (0, +∞).

For the part (b) the proof follows exactly the same arguments (it is necessary to take y 0 ∈ C [-τ,0] , |y 0 (0)| r = q k for some integer k ≥ 1. Then there exists x 0 ∈ C [-q kd τ,0] , 1 ≤ ||x 0 || r such that y 0 (s) = Λ r (q k )x 0 (q kd s) for s ∈ [-τ, 0] and according to Proposition 2 y(t, y 0 ) = Λ r (q k )x(q kd t, x 0 ) for all t ≥ 0, and |y(t, y 0 )| r ≤ q|y 0 (0)| r for all t ≥ q -kd T ≥ T y0 ).

Lemma 1 and Corollary 2 can be used to check the global asymptotic stability IOD property that is required in Theorem 1. Uniformity of T with respect to delay value implies that finite-time convergence in (1) is possible under a "weak" dependence on delay. The main result of this theorem is another demonstration for complexity of existence of finitetime stability in the functional differential equations [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF]. Of course these results are obtained for homogeneous systems only, but they provide a good intuition about other cases also.

IV. APPLICATION TO STABILIZATION OF A CHAIN OF

INTEGRATORS

In [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], for a chain of integrators ξi = ξ i+1 , i = 1, . . . , n -1, ξn = u, where x = (ξ 1 , . . . , ξ n ) T ∈ R n is the state and u ∈ R is the control, the following stabilizing controller has been proposed:

u = - n i=1 k i |ξ i | αi sign(ξ i ),
where k i form a Hurwitz polynomial and α i ∈ R + . If the powers α i are selected in a way providing homogeneity of the closed-loop system and if all of them are smaller and sufficiently close to 1, then the system is finite-time stable [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]. For the case n = 2, α 1 = α 2-α , α 2 = α and an arbitrary value of α ∈ (0, 1) a solution has been given in [START_REF] Bernuau | Finite time output stabilization of the double integrator[END_REF]. Let us assume that the state is available for measurements with delays τ i ∈ (0, τ max ), 0 < τ max < +∞:

u (t) = - n i=1 k i |ξ i (t -τ i )| αi sign[ξ i (t -τ i )],
then applying the result of Lemma 2 (which can be easily extended to the case of multiple delays) to the closed-loop system we obtain that it is locally asymptotically stable in C [-τmax,0] if α i > 1 and sufficiently close to 1. Indeed, in this case the homogeneity degree is positive (Corollary 2 cannot be applied) and we can use the same technique as in [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] to prove global asymptotic stability of the delayfree control (for the case n = 2 the Lyapunov function V (x) = k1 α1+1 |ξ 1 | α1+1 + 1 2 ξ 2 2 can be used). These results are related with ones obtained in [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF] for a bounded global stabilizer of the chain of integrators, in both cases increasing of the delay leads to decreasing the set of attraction.

The results of computer simulation of this system for two different values of delays (for n = 2, k 1 = 1, k 2 = 2, α 1 = 1.5, α 2 = 1.2) is shown in Fig. 1. As we can conclude, fixing the initial conditions and increasing the value of delay the system becomes more oscillatory and finally unstable for a sufficiently big value of τ max .

V. CONCLUSIONS Extensions of the homogeneity theory are obtained for timedelay systems. It is shown that for any degree of homogeneity the solutions of a homogeneous system are interrelated subject to the delay rescaling. Next, this fact is utilized in order to show that local uniform asymptotic stability of homogeneous systems implies global one, and that for nonlinear homogeneous systems with non-zero degree, global asymptotic stability for a delay endorses this property for an arbitrary large delay. Thus analysis of stability in nonlinear time-delay systems differs significantly from the linear case, but in a positive sense: being stable for a delay such systems save this property for any delay value, which is an interesting advantage of nonlinear time-delay systems. Efficiency of the proposed approach is demonstrated on application to a finitetime stabilizing control from [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF].

  The authors are with Non-A team @ Inria, Parc Scientifique de la Haute Borne, 40 av. Halley, 59650 Villeneuve d'Ascq, France and CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve-d'Ascq, France. D. Efimov is with Department of Control Systems and Informatics, ITMO University, 49 Kronverkskiy av., 197101 Saint Petersburg, Russia. This work was supported in part by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).

  we denote by C [-τ,0] the Banach space of continuous functions φ : [-τ, 0] → R n with the uniform norm ||φ|| = sup -τ ≤ς≤0 |φ(ς)|, where | • | is the standard Euclidean norm); f : C [-τ,0] → R n ensures existence and uniqueness of solutions in forward time, f (0) = 0.

1 Figure 1 .

 11 Figure 1. Trajectories of stabilized delayed double integrator

If time is scaled t → λ d t then the argument of f : C [-τ,0] → R n in (1) is also scaled to f : C [-λ -d τ,0] → R n in (3).


for instance, Gustave Eiffel

Louis Renault

André Citroën

André Michelin

Marcel Schlumberger

Louis Blériot

William Lebaron

Bernard Yannou
email: bernard.yannou@ecp.fr

Jean-François Petiot

A View of Design (and JMD): The French Perspective A quick glance on history of science and engineering shows that the creation and invention of technical products has held a great appeal in France. A tradition of inventiveness, engineering and entrepreneurship has developed over the past several centuries; some famous mechanical and industrial pioneers

The French research in design is anchored in the mechanical discipline. In the late 1980's, a design community was progressively constituted through the desire to merge mechanics with computer science. An academic network, named PRIMECA (Pôle de Ressources Informatiques pour la MECAnique or Computer Science Resource Association for Mechanics), was created to promote the use and development of integrated design and manufacturing approaches in mechanics, and to encourage the training in Computer Aided Mechanical Design and Manufacturing. This network -still active nowadays -had a major influence on research in design. Several French researchers (including ourselves) are heirs of PRIMECA. This network probably was instrumental to the successful development in France of well-known CAD software, now in a leading position in the worldwide market. The success and the dynamism of this network also may contribute to a delay in the participation of the French design community to the Design Society as it emerged initially in Europe.

Since then, the scope of research in design has been widened, going from mechanical design to CAD and industrial engineering. As in other countries, design has now become open to the societal challenges of the 21 st century. The French design community now looks to integrate standard engineering with disciplines such as human factors, marketing, cognitive ergonomics, knowledge modeling, and artificial intelligence. New fields, such as systems engineering, sustainable design, customer-oriented design, or information management, have significantly emerged and are constituting progressively their corpus of knowledge. Some research in design can be seen as part of industrial engineering, keeps a 'tools-oriented' paradigm, and certain themes like PDM or PLM are being particularly well represented.

Another particularity of France lies in its system of engineer education beside universities, the Engineering School ("Grandes Ecoles") system. This system allows the establishment of strong connections with industry and a relative quick feedback of industrial needs on the curricula. It is common that Masters' thesis work (and also PhD with the system of the national CIFRE grants) takes place within companies. This system feeds research issues that are anchored in an industrial ground, and research in design is often pulled by the main French industries (i.e., automobile, aeronautics, energy). This fact will be reinforced by the recent decisions of the French government concerning the autonomy of Universities (e.g., a new right of the Universities to set initial professor salaries, and maybe soon to have merit raises and promotions to full professorships done locally by the University rather than centrally by the Ministry of Education), and the "Grand Emprunt" ("Big Loan") project for education and research fostering virtual labs between university faculties and companies.

With these strong industrial connections, design research is then generally more context dependent, tools oriented and puts a stronger value-creation requirements than in the US. This is why design processes, design knowledge and competences, design organization, design platforms, digital prototyping, design management, and integration of methodologies may be strongly emphasized by French design researchers. Rigorous approaches of "action-research" types derived from management sciences are often used to improve design methods and tools in an industrial context. A design research project is generally conducted in four stages:

1. A first observation round of design practices leads to design processes representation and to a diagnostic analysis.

2. Next, industrial improvement leads as well as a series of scientific issues are established, resulting often in several state-of-the-art assessments.

3. New models are proposed.

4. These models are integrated into a design platform, a design methodology or guideline which is deployed, at least partly, in the industrial context, to globally assess the algorithmic efficiency as well as the organizational acceptance or effectiveness. For this last point, deployment protocols and organizational performance measurements must be also designed.

In summary, design research in France is considered as fully successful and completed if it can reach or has reached Stage 4 above. For a PhD research program in an industrial context, combining the identification of precise scientific issues and the corresponding state-of-the-art with a completed proof of concept in a company makes it difficult to draw sharp scientific conclusions and report them in published papers, which is the international rule.

In addition, it is worth noting that traditionally French design researchers have suffered from a language barrier with regard to international publicationsall written in English. This problem has been due in part to their inadequate overall knowledge of English, but also due to the way of writing: Use of linguistic precautions, use of metaphors, a wish to have a global vision of the problem, too much time spent in issue justification, emphasis on the limits or the weak points instead of facts and results. After all, French has been the language of diplomacy! In the past decade, with increased French participation to European projects, the acceptance of the English language prominence and the pressure for international publications, things are changing and French researchers are more and more accustomed to scientific English writing.

From these elements, an interesting question is: Why are there so few French researchers publishing in JMD?

The journal is well known by researchers in mechanism analysis and synthesis, and robotics, and is considered as a leading Journal in this field. They constitute the majority of the contributors to JMD, but they have now a more targeted journal, the Journal of Mechanisms and Robotics. In addition, Design Automation is less studied as such in France, for the reasons evoked above, namely that design deals with other domains. JMD requires careful analysis models, with solid mathematical formalisms and frequent use of statistics, design of experiments or applied mathematics. Again, the French researchers in design do not often make this formalization effort, being more results oriented.

But things are moving fast. The JMD scope and recent special issues have been more open to multidisciplinary issues. French researchers are also influenced by the American rigor and way of publishing. Globalization tends to act by reinforcing research quality, but preserving in the meantime historical differences and particular local nature. As a matter of fact, two upcoming events in design engineering, organized by the French design community, may be of interest to the JMD readership:

-The second CSDM conference, standing for Complex Systems Design and Management (see http://www.csdm2011.csdm.fr/), in Paris in December 7-9 2011 is co-sponsored by INCOSE and co-organized by Ecole Polytechnique, Ecole Centrale Paris and Supelec, three "Grandes Ecoles".

-The IDMME -Virtual Concept international conference (Integrated Design and Manufacturing in Mechanical Engineeringsee the 2010 event at http://www.idmmevirtualconcept.estia.fr/). This biennial conference (whose first event was held at Ecole Centrale de Nantes in 1996) emanates from the PRIMECA network and is now under the scientific partnership of ASME, Design Society, CIRP, and the Association Française de Mécanique (AFM). The next event will be a joint meeting with the Italian and Spanish design communities and will take place in 2013 in Spain (venue to be confirmed).

It is fair to say, that the French design community is quickly integrating with other design communities across the world, has much to learn but also much to offer. We expect that some of that will happen through JMD's publications.

The fist aviator to cross the France-England Channel, and the founder of Blériot Industries, becoming later Aérospatiale being further integrated into EADS company.

Considered as the inventor of steel-made skyscrapers and the first skyscraper designer: the Home Insurance Building of Chicago, 1885.

The TGV French train has the world speed records on commercial railways; 515 km/hr in May 1997 and 575 km/hr in April 2007.


Ariane Lambert-Mogiliansky

François Dubois
email: francois.dubois@math.u-psud.fr.

Our (represented) World: A Quantum-Like Object

Keywords: learning, Bohr complementarity, information, uncertainty AMS classification: 81Q99, 91C99

It has been suggested that observed cognitive limitations may be an expression of the quantum-like structure of the mind. In this paper we explore some implications of this hypothesis for learning i.e., for the construction of a representation of the world. For a quantum-like individual, there exists a multiplicity of mentally incompatible (Bohr complementary) but equally valid and complete representations (mental pictures) of the world. The process of learning i.e., of constructing a representation involves two kinds of operations on the mental picture. The acquisition of new data which is modelled as a preparation procedure and the processing of data which is modelled as an introspective measurement operation. This process is shown not converge to a single mental picture but can evolve forever. We define a concept of entropy to capture relative intrinsic uncertainty. The analysis suggests a new perspective on learning. First, it implies that we must turn to double objectification as in Quantum Mechanics: the cognitive process is the primary object of learning. Second, it suggests that a representation of the world arises as the result of creative interplay between the mind and the environment. There is a degree of freedom that modifies the objective of rational learning.

Introduction

It is a common place that human beings are not capable of holding very complex picture in mind (cf "small worlds" in Savage 1954). We consider reality focusing on one perspective at a time and show difficulties in combining perspectives as amply documented. This inability to seize reality in its full richness suggests that the process of developing an understanding of the world may not look like a puzzle that is assembled progressively. Instead, the human mind may exhibit structural limitations in terms of the incompatibility of perspectives in a way similar to properties in quantum mechanics. Ambiguous pictures of the kind provided in diag. 1 provides a suggestive illustration of this phenomenon.

Quantum Mechanics is a theory about the physical world. Its object is sub-atomic particles. The well-known "strange" features arise when an observer attempts to learn about a sub-atomic system by means of a measurement device. A fundamental concept in QM is Bohr complementarity: two properties can be incompatible in the sense that they cannot have a determinate value simultaneously. As a consequence it is impossible to experimentally access that physical world without interacting with it in a non-deterministic manner. Intrinsic indeterminacy fundamentally changes the way we have to think about the physical reality or at least about the physical reality that we can have some knowledge about. It introduce non-separability between the measurement process and the world. There is no (accessible) given "outside world" but a contextual world in which we play an active role.

For some time the quantum paradigm has been proposed to described among other things aspects of the human psyche including preferences, attitudes and judgements in decision theory and psychology (Lambert-Mogiliansky 2009, Busemeyer Bruza 2012, Khrenikov 2010 among others). In this paper we take a step forward in recognizing that knowledge is also an (psychic) object which in itself may exhibit non-classical features and we derive some implications of this hypothesis for learning.

The approach can be viewed as a reversal of standard QM in the following sense. A person considers a classical object e.g., a state agency, but her knowledge about it i.e., the mental picture in her mind can behave in a non-classical way. Since human beings makes decisions on the basis of perceived reality i.e., of mental pictures, our approach extends the possible practical relevance of the quantum paradigm beyond sub-atomic physics to much of macroscopic reality.

The most important element of our theory is Bohr-complementarity of mental perspectives of one and the same object. They parallel that of properties for sub-atomic particles e.g., spin along different angles. We propose that Bohr complementarity of perspectives captures the cognitive limitations that are responsible for our difficulties to synthese information along different perspectives into one single coherent picture. The difficulties arise from the impossibility for incompatible perspectives to simultaneously have determinate value. Just as in quantum physics, the system (here our mental picture) makes discrete jumps when attempting (by means of introspection) to find a determinate value along a series of incompatible perspectives. We model the cognitive process of learning as involving two essentially different operations. The first operation corresponds to receiving information framed according to some perspective. We model this as a preparation procedure. The second operation entails processing information or updating the mental picture in terms of one's preferred perspective. It is modeled as an introspective measurement operation. This learning process is not Bayesian as soon as we allow for Bohr complementarity of perspectives. The issue of learning by non-classically minded agents has been approached in Danilov and Lambert Mogiliansky [7, 8] and more recently in Busemeyer and Bruza [4].

1.

The basic model

The parallel with QM is as follows: the system is the represented world, it is a psychic object. At any point in time the individual's knowledge about the (represented) world is captured by the current mental picture or state. We denote it |ϕ ∈ H where H is finite dimensional Hilbert space of mental pictures of the (relevant) world. The state can be characterized from different perspectives. A perspective is modelled as an operator similar to an observable. It operates on the mental picture to generate an outcome, a possible (set of ) picture(s) belonging to that perspective. The present analysis addresses incomplete knowledge due to either incomplete information or quantum indeterminacy or both.

1-1.

The notion of measurement Before entering into the analysis let us remind of some basic notions and how they relate to our issue. Generally, a measurement is an interaction between a system and some measurement device, which yields some result, the outcome of the measurement that we can observe and record. [START_REF] Bohr | Atomic Theory and the Description of Nature[END_REF] Two measurements are compatible if they, roughly speaking, can perspective, we find out that it is not consistent with out earlier held beliefs. [START_REF] Bohr | Atomic Theory and the Description of Nature[END_REF] For instance in the case of the Stern-Gerlach experimental setup, we let the electron travel through a non-homogeneous magnetic field and observe deviation either up or down. In another example we let people play the Prisoner Dilemma (or the UG) and observe their choice. In the present context we see part of the cognitive process (see below) as a measurement of the mental picture.



Ariane Lambert-Mogiliansky and François Dubois be performed simultaneously or more precisely, if the performance of one measurement does not affect the result of the other. Suppose that the first measurement gave outcome o; then we perform the second measurement and the first one anew. In case we are dealing with compatible measurements we obtain outcome o with certainty. If all measurements are compatible we can substitute them with a single finest (complete) measurement, which is also first-kind. [START_REF] Busemeyer | Quantum Models of Cognition and Decision[END_REF] Performing that measurement we learn everything about the system. Such a system is classical. The existence of incompatible measurements is a distinctive feature of non-classical systems. It is closely related to the impact of measurements on the state and to the existence of "dispersed" states. A state is dispersion-free when the outcome of every possible measurement is uniquely determined, there is no reason for the state to change. If all pure states are dispersion-free then measurements do not impact on pure states and therefore all measurements are compatible. On the contrary, if a state is dispersed then by necessity it will be modified by an appropriate measurement. On the other hand, the change in a pure state is the reason for incompatibility of measurements. The initial outcome of a first measurement is not repeated because the system has been modified by a second measurement (see Danilov and Lambert-Mogiliansky [7, 8]).

In this paper a measurement is an introspective operation that acts on the agent's mental picture. The agent's asks himself a question: what are the implications of new information for my understanding of a specific matter? and an answer, the outcome of the mental process is brought to consciousness e.g., my current understanding in terms of a recommendation for action. We propose that some cognitive limitations documented by practitioners (give references) can be suitably modelled as the result of incompatible mental operations that act on the mental picture to modify it.

1-2.

Pure and mixed mental states A pure state is a maximal information state. It may be the eigenstate of some perspective so it gives maximal and precise knowledge of the world along some perspective. Or it may be a superposed state i.e., not the eigenstate of any perspective (observable). Such a state also contains maximal knowledge but is characterized by some "dispersion entropy" (see definition below). No measurement can be performed on a pure state and provide new information without losing some information. [START_REF] Busemeyer | Quantum Dynamics of Human Decision-Making[END_REF] In contrast a mixed state is a familiar probabilistic combination of states which reflects incomplete knowledge. Measurements can be performed on a mixed state to reach a maximal information pure state. In this section we briefly discuss how a mixture of mental states can emerge.

At a first a mixed state seems to be a most natural mental object. So for instance the "world" is a friend's mood. You may be able to assign a probability to her willingness to go out with you. Although that situation appears to be most ordinary, we need to make clear how such a mixed state may arise in our context. In the world of QL-minded agents there exist three possible scenario:

-the state has been prepared to a mixed state i.e., information provided in terms of a probability distribution over possible states [START_REF] Busemeyer | Dynamic and consequential consistency of choices between paths of decision trees[END_REF] ; -an introspective complete measurement is initiated but not completed. The initial superposition has decohered into a probabilistic mixture of the possible states but not collapsed into one of the eigenstates; [START_REF] Danilov | Measurable Systems and Behavioral Sciences[END_REF] -a superposed state has spontaneously decohered;

In the present paper we shall not consider spontaneous decoherence as we do not explicitly consider time evolution. We focus on probabilistic mixture induced by the environment through well-identified new information or by means of an introspective mental processes.

1-3.

Intentionality: a preferred basis We shall assume that an individual endeavours to construct a representation i.e., learn about the world because he has some concern in mind e.g., he has to select an action. This assumption is consistent with psychological and neurobiological evidence that human cognitive processes (including perception) are structured by some form of intentionality.

A perspective is defined as an "observable" that is an operator R that applies on the state or mental picture. It is characterized by its eigenvalues {r 1 , ...r n } interpreted as the possible eigenpictures belonging perspective R . A perspective is a "complete set of commuting observables" so that a eigenstate in any perspective is a maximal information state. Consequently and importantly any two perspectives R = P yield alternative descriptions of the same "world". A pure mental picture can written as a superposition of the possible eigenpictures in any perspective of the (relevant) world:

|ϕ = i λ i |r i , j λ 2 i = 1, λ i ∈ R
which can also writes as a density operator

ρ = |ϕ ϕ|
When the initial mental picture is not a pure state but a statistical mixture of pure states, it can only be represented with a density operator ρ:

ρ = i τ i |ϕ i ϕ i | , τ i ≥ 0 , j τ i = 1
where |ϕ i ϕ i | is a classical "ket-bra" notation to represent the orthogonal projector on the vector |ϕ i . The density operator allows to express a situation where the agent's mental state is characterized by both intrinsic uncertainty and incomplete information.

An action problem is defined as a correspondence from D : H → A where H is the space of mental pictures of the world and A the set of actions. For each (mentally represented) state of the world, it defines which actions the agent wants to undertake.

Hypothesis 1 There exists a perspective R * such that D is a coarsening of R * . [START_REF] Danilov | Decision-making under nonclassical uncertainty[END_REF] Hypothesis 1 implies that if the agent receives (maximal) information framed in perspective R * he will know exactly what to do. We also say that R * is fully congruent with D. Generally, D has a lower dimensionality than H, i.e., distinct eigenpictures can call for the same action.

1-4.

The learning process In our context, it is in place to distinguish between two essentially different operations in the learning process [START_REF] Damasio | L'Erreur de Descartes[END_REF] . We shall present them as two successive phases.

The first phase of the mental process is an interaction between the mind (mental picture) and the outside world. The individual acquires new information expressed within some perspective. This is a very important point: information is not "neutral", it is always framed i.e., it comes together with a perspective (model) of the world. Because it is framed, to integrate a piece of information, the individual may have to (temporarily) switch to the perspective in which it is formulated. Either the information belongs to the initial perspective R or to some alternative perspective P, P = R. In both cases information acquisition corresponds to a fully deterministic evolution of the mental picture. It operates on the initial state(picture) so as to project it onto the one picture encoding the information content provided. In the terminology of QM, phase 1 resembles a process of preparation10 : ρ → ρ where ρ is the new density matrix. Let

ρ = i θ i |r i r i |, i θ i = 1. If the information is consistent with R, ρ = i θ i |r i r i |
, where θ i are calculated according to the familiar Bayes rule. We have a classical case of belief updating. The interesting case is when P = R, the mental picture is then "forced"11 into the perspective P and ρ in basis |p j writes:

ρ = i γ i |p i p i | , γ i = 1,
Depending on the provided information content the density matrix may be a pure state or mixed state (if the information content is probabilistic). A first central point to emphasize is that when P R = RP i.e., we are dealing with incompatible (alternative) perspectives, information embodied in ρ the initial state is lost in the process. A preparation in P modifies the mental picture into a specific mental picture distinct from the initial one and the result is the same whatever the initial state in R. Again this would not have been the case if the provided information was in terms of the same perspective as the initial state R. Of course this is a simplification but it it aims at capturing the following. The loss of information reflects the fact that the two pieces of informations (ρ and ρ ) do not form a consistent picture for the individual. It is as if they are expressed in two different languages that do not translate into each other unambiguously and the individual cannot but think (form a mental picture) in either one. In reality, the process is of course much more complex. In particular, as earlier mentioned the individual may keep some memory.

Phase 2: Updating The second phase is a purely mental operation which does not appeal to the outside world. It is an introspective measurement operation. It measures the mental state ρ with respect to the preferred perspective R * . This operation may be conceptually decomposed into two steps in a way originally proposed by von Neuman (1932). The first step corresponds to the formulation of the mental picture in terms of a convex combination (probabilistic mixture) of possible outcomes in the preferred perspective R * . This corresponds to making the measurement but not registering the result. The state is projected on the preferred basis and decohered. The second step entails a decision -one (or a subset of possible) result is singled out. It corresponds to registering the result of the measurement. We shall refer to these two steps as respectively analysis and decision.

a. Analysis Generally, the concerns that determine intentionality may not require a complete measurement. So both the analytical step and the decision step may be coarse (as opposed to fine or complete measurement). In that case the outcome will entail the preservation of some quantum coherence so the state obtained after the decision is also a superposition. We next describe the process for a complete measurement which fully decoheres the state into a mixture of eigenpictures of R * before selecting one of them. The state ρ is expressed in terms of the projectors associated with the eigenvectors of R * :

ρ = k,l i γ i r * k ||p i p i ||r * l |r * k r * l |
The ρ contains off the diagonal elements which do not correspond to any eigenstate of R * , they cannot be observed. The analysis step associated with a complete measurement corresponds to projecting the state into the R * perspective, i.e., decoherence takes place  Ariane Lambert-Mogiliansky and François Dubois so only diagonal elements are left: ρ → ρ * = j |r * j r * j | ρ |r * j r * j | and we write ρ * :

ρ * = k θ * k |r * k r * k | with diagonal coefficients θ * j = i γ i | p i | r * j | 2 , j θ * j = 1.
This new density operator is the result of "analysis" in the sense that the possible results have been identified with their respective probability for occurrence. Some information from the preparation stage is "transmitted" modulo the correlation coefficients | r * i | p j | which are the probabilistic weights attached to the eigenstates in ρ * . Some information is "lost" with the disappearance of the non-diagonal elements. The correlation coefficients linking two representations are features of the mind; they may but need not reflect features of the world that is represented. This is a fundamental distinction between quantum cognition and quantum mechanics. In Quantum Mechanics the correlation coefficients pertains to observables representing properties of a physical system. The subject matter of the present investigation is not the world but the represented world which is a mental construct. In particular, our model applies to the representation of classical objects. The coefficient captures the quantum-like structure of the mind actualized in cognitive activity.

b. Decision

The decision step is a non-deterministic process. When the updating phase corresponds to a complete measurement, the resulting density operator represents an eigenpicture of the preferred perspective [START_REF] Espagnat | À la recherche du réel -Le regard d'un physicien[END_REF] . Decision selects one of the |r * j r * j | with the corresponding probability θ * j ρ * → ρ * * = |r * j r * j |. At this point, the individual has a maximally informed mental picture in R * and all information in P from ρ has been "lost". Recall that it affected the mental process through the probability for obtaining any of the possible results. We see that after completing the cognitive process corresponding to interpreting the information in his own mental model, the agent's actual state of information is a maximal information state which cannot be easily traced back to the initial ρ or even to intermediary ρ and ρ * . We next develop a simple numerical example.

Example

In this section we illustrate the learning process defined above in numerical example. Consider a two dimensional Hilbert space and a 2X2 density matrix in the canonical preferred basis R * . Let ρ represent the initial state ρ = .25|r *

1 r * 1 | + 0.75|r * 2 r * 2 |
(1) ρ = .25 0 0 .75 .

Phase 1: Information acquisition [START_REF] Espagnat | À la recherche du réel -Le regard d'un physicien[END_REF] The state resulting from a coarse measurement is a pure state that is a superposition of eigenpictures. After the analysis step, the matrix ρ given in (2) becomes

(3) ρ = .75 0 0 .25 .

Only the diagonal elements (representing observable states) are left. If we compare the matrices ρ and ρ , we observe that the probabilities associated with the eigenpictures of the preferred perspective have been inverted so if we complete the measurement, there is now 75% chance that the agent ends up being convinced of r 1 instead of 25% as before he acquired new information. In order to better understand why this is a reflection of cognitive constraints we need to introduce the concept of dispersion entropy and explore further the example which we do below.

Learning under irreducible uncertainty

2-1. Dispersion entropy -a measure of doubt In this section we propose a concept of entropy that captures the amount of (relative) uncertainty that characterizes the mental picture of an agent. The first important point is that this uncertainty is not "absolute" but relative to a specific perspective e.g., corresponding to some objective that requires action. To make the point most clear we shall therefore confine ourselves to pure states i.e., states of maximal information. In those states, the corresponding mental picture is a superposition of eigenpictures i.e., the only uncertainty that is left is due to the quantum-like structure of the mind.

Shannon entropy which measures uncertainty associated with a classical probability is defined for a discrete probability distribution {p j } according to S = - As does Shannon entropy, von Neumann entropy measures the amount of uncertainty due to incomplete information while we are interested in measuring intrinsic uncertainty. We therefore talk about the entropy of dispersion rather than of probability distribution. [START_REF] Epstein | Non-Bayesian updating: A theoretical framework[END_REF] The relative entropy of dispersion is defined:

Definition The dispersion entropy of a pure state ρ = |ψ ψ| with |ψ = λ i |ϕ i , λ 2 i = 1 relative to a specific perspective A with eigenvectors {|a i } is

S (ρ, A) . = - α j log α j where α j = i λ 2 i | a j | ϕ i | 2 , α i = 1.
A few important remarks: 1. Dispersion entropy is a relative concept, it varies with the perspective; 2. Dispersion entropy is equivalent with the von Neumann entropy of the mental picture ρ = α i |a i decohered by the measurement of A but before the decision (see analysis in Phase 2 of section 2.2 ).

3. S (ρ, A) = 0 when the mental picture |ϕ is an eigenpicture of A and is positive otherwise.

In contrast with the standard concept of entropy we cannot say that the dispersion entropy of a pure state measures how much information we might gain by actually performing the measurement -because we are in a zero von Neumann entropy state. The S (ρ, A) of pure state ρ captures the non epistemic impact of the operation of measurement of A on the mental picture . [START_REF] Gioia | Symbols, Script and Sense-making: creating meaning in the organizational experience[END_REF] In the context of quantum cognition we would like to link dispersion entropy to a psychological state of "doubt".15 S (ρ, A) is then a measure of the extent of "inner uncertainty". It can be resolved by an introspective measurement. [START_REF] Gilboa | Maxmin Utility with non-unique priors[END_REF] An introspective measurement does not only take the logical consequence of information (the projection that decoheres the state), it also includes a resolution (what we called "decision"). [START_REF] Gilboa | Rationality of Beleifs or: why savage's axioms are neither necessary nore sufficient for rationality[END_REF] Another way to put it is that a (complete) introspective measurement operates on a pure mental picture to resolve a state of doubt or lack of conviction.

We next develop a simple argument showing how additional information can increase dispersion entropy relative to the preferred basis. For the purpose of the argument, we assume that the agent's mental state is initially an eigenpicture of his preferred perspective |ϕ = |r * 2 . This state reflects information that is perfectly congruent with

R * but is framed in R : |ϕ = λ i |r i (= |r * 2 )
. The agent is not aware of r * 2 unless he updates "his beliefs" which corresponds to the (measurement) operation of R * . If he did, he would obtain R * |ϕ = |r * 2 , with probability 1. He would then be fully determined i.e., convinced with respect to the decision problem. There would not be any (relevant) uncertainty left with γ 2 = 1 we have S (ρ, R * ) = 0. But assume that instead he acquires new information relative to another but related aspects of the issue. We denote this perspective P. This corresponds to steps 1 of the cognitive process above. The mental state |ϕ is prepared into |ϕ equal to some |p i i.e., information is maximal in perspective P. "Enriched" with this new information, the agent now updates his mental picture with respect to his preferred perspective (step 2):

|ϕ = |p 2 = j γ j |r * j → any |r * j with probability γ 2 j = 1
We see that while he previously potentially "knew" r * 2 (but was unaware of it) he is now in a state of hesitation where he believes that there is only a probability of γ 2 2 < 1 that the "true" eigenpicture is r * 2 . [START_REF] Hanany | Undating prefrences with multiple priors[END_REF] The acquisition of new information has triggered the loss of "conviction" reflecting previously held information which is in effect lost leaving the agent in a state of intrinsic uncertainty captured by a strictly positive dispersion entropy:

S (ρ , R * ) = - j γ 2 j log γ 2 j > 0.
where ρ = j γ 2 j |r * j r * j |. Proposition. Assume that the agent's mental state is one eigenpicture of his preferred perspective, acquiring additional information in terms of a perspective that is not compatible with the preferred one affects the mental state so as to increase dispersion entropy reflecting doubt or lost conviction. This is an important result. From the point of view of "inner conviction" additional information is unambiguously detrimental when the starting point is an eigenpicture of the preferred perspective as illustrated in the argument above: the new information necessarily induces uncertainty with respect to the agent's action relevant perspective. More generally i.e., starting from an arbitrary eigenpicture (possibly mixed and with non zero relative dispersion entropy) as in the example above, new information may either decrease or increase dispersion entropy. In the next section we discuss possible implications for rational learning.

Remark 1 In our case r * 2 is true because it reflects an information congruent with R * and |p i is also true. The agent is not mistaken. He simply cannot hold in one single picture both informations. The introspective operation when processing information p 2 involves the whole (cognitively limited) mind and therefore upsets earlier held beliefs.. We illustrate the result in Proposition 1 in figure 2 where the dimensionality of a perspective is 2. The broad line corresponds to the preparation stage and the thin lines to the measurement. Using the numerical example above, we illustrate the impact of new information on dispersion entropy. The starkest results appears when starting out with ρ = |r * 2 r * 2 | and assume that the state was generated as the result of the acquisition of maximal information in perspective R * and thus associated with zero dispersion entropy relative to R * . After acquiring information the state of knowledge/conviction is ρ

= |p 1 p 1 | = 2+ √ 2 4 |r * 1 r * 1 | + √ 2 4 |r * 1 r * 2 | + |r * 2 r * 1 | + 2- √ 2 4 |r * 2 r * 2 |. Then ρ = 2+ √ 2 4 |r * 1 r * 1 | + 2- √ 2 4 |r * 2 r * 2 |. The new state is associated with the following dispersion entropy S(ρ , R * ) = -2+ √ 2 4 log 2+ √ 2 4 - 2- √ 2 4 log 2- √ 2 4 = 0.4165 > S(ρ, R * ) = 0.
So now we readily see that new information created a state of doubt when there should not have been any -the agent has maximal information with respect to the perspective relevant to his decision. The doubt does not arise as the result of the information content associated with |p 1 p 1 | but exclusively as the result on the cognitive constraint expressed by the incompatibility between the two perspectives corresponding to R * and P respectively.

Remark 2 It must be emphasized if we relax slightly the assumption that |ϕ = |r * 2 (i.e., allowing for initial incomplete information), the result in the Proposition can be obtained with a classical model of a different flavor. [START_REF]Choices, values and frames[END_REF] Assume the agent has priors that puts nearly all the weight (but not all) on r * 2 . Next, he receives information p 2 . If according to the agent's structural model, that information is highly correlated with r * 1 , he updates his beliefs and is now less certain about what to do than he was before he got the information. His new beliefs exhibit a distribution with higher entropy. However, he has not lost information as in the quantum case. Instead the interpretation is that his priors were wrong so new information brought him closer to the truth.

In view of the remark above a legitimate question is: are we only talking about interpretation or do quantum cognitive limitations really have a more profound bearing on learning? The next section addresses this question.

2-2.

Rational Learning under Irreducible Uncertainty Learning whether it is by classically minded agent or QL-minded ones is always about the construction of a mental representation of the world. In our context it is important to distinguish between two properties of a representation. On the one hand the mental picture may be more or less informationally complete. And on the other hand, the mental picture may be more or less connected with the outside world it is supposed to represent. In a classical context this distinction also exists and information from the outside world is used to complement and test the mental representation. If the picture is incomplete, new information will bring it closer to the outside truth. If the model is wrong attempts to update it with new information will give rise to inconsistencies. It should be stressed that no satisfactory approach exists as to how a classically minded agent proceeds in face of data revealing failures of the underlying model. There is no obvious incremental path of modifications (the notion of distance is ambiguous at best) and certainly no prescription for discrete jumps between alternative models.

For a quantum-like agent, the situation is similar except that there is not one single "true" model (mental picture) but a multiplicity of Bohr complementary (correlated) "true" models. Discrete jumps between representations are parts of the cognitive process by force of complementarity. With quantum cognition the link between the epistemic state (the mental picture) of the represented world and the world in itself is ambiguous. In fact the term "epistemic state" does not seem proper in face of intrinsic indeterminacy as the multiplicity of zero entropy states suggests. As repeatedly emphasized by cognitive scientists we cannot address knowledge without simultaneously addressing human cognitive activity. As we shall see this takes a precise meaning in our context.

2-2-1.

Bayesian and QL-learning Bayesian learning operates within a Boolean algebra. The objective is to learn the parameters of the model of the system. [START_REF] Khrennikov | Ubiquitous Quantum Structure -From Psychology to Finance[END_REF] Provided the priors are not inconsistent with the true model, we know (Schwartz 1965) that starting from any such priors Bayesian updating converges to the true state.

With quantum learning, we are in a Hilbert space, that is there exists a variety of resolutions of the system i.e., a variety of perspectives, of valid theories of the system. Therefore, the state of knowledge of a quantum minded individual does not converge with new information but can evolve forever. Consider again the example above. After the agent performed the introspective measurement R * , if the agent is being asked about his understanding in P, he will perform a new measurement of the eigenpicture resulting from the first measurement and ρ will not be recovered. Performing those measurements alternately, he will keep oscillating without converging i.e., without being able to settle for a definite value in both P and R * simultaneously. [START_REF] Lambert-Mogiliansky | Quantum indeterminacy in Dynamic decision-making: Self-control through identity management[END_REF] Of course this example is simplistic and the agent might simply remember ρ . But in more sophisticated context, we expect the modification of the mental picture to be effective.

In Quantum Mechanics, irreducible uncertainty gave rise to a rich philosophical debate about the nature of reality (see e.g., D'Espaganat 1979). In our context when the irreducible uncertainty is a consequence of the quantum structure of the mind, new questions arise. The mental picture can also "drift away" from a "truthful representation" of the outside world. [START_REF] Lambert-Mogiliansky | Type indeterminacy -A Model of the KT(Khaneman Tversky)-man[END_REF] We next briefly discuss what rational learning may mean in such a situation.

2-2-2.

The object of QL-learning and "Knowledge about the world" In quantum mechanics when physicists were confronted with the impossibility of first hand objectification, they resorted to second hand objectification. That is basically the Hilbert space model of quantum mechanics. It captures how measurements impact on the system and how measurements are related among each other. The state vector is the central object that encapsulates all information and it is the basis for making predictions. Similarly, the situation we are facing in quantum cognition suggests that the primary object of scientific learning must be the human mind i.e., the structure of mental perspectives, how they affect the mental representation of the world and how they are related to each other. The current mental picture is the state of the representation, it encapsulates the information known to the individuals and it is the basis for making prediction and further operations. This structure is assumed to exhibit invariance reflecting psychological regularities. [START_REF] Lambert-Mogiliansky | Transparency in public life ; a quantum cognition perspective[END_REF] Hence we propose that the primary object of QL-learning is the human cognitive process -the non-boolean algebra of mental perspectives. What knowledge about the outside world concern, we have seen that for QL-minded agents knowledge about the world is a mental picture that reflects both inputs from the outside world (information) and input from the inside world (decision in introspective measurement). At any point [START_REF] Lambert-Mogiliansky | Quantum indeterminacy in Dynamic decision-making: Self-control through identity management[END_REF] The general result is a transposition into cognition of the basic feature of quantum mechanics namely that it is not possible for complementary properties to have a determinate value simultaneously. [START_REF] Lambert-Mogiliansky | Type indeterminacy -A Model of the KT(Khaneman Tversky)-man[END_REF] The role of the acquisition of new information from an incompatible perspective is neither a complement nore a test of the truthfulness of the current picture. But it is an operation that forces a (temporary) reconnection between the mental picture and the outside world. Only information within the perspective of the mental picture is a true test in a way similar to learning in the classical world with classically minded agents. [START_REF] Lambert-Mogiliansky | Transparency in public life ; a quantum cognition perspective[END_REF] At least we do expect invariance at the individual level.



Our (represented) World: A Quantum-Like Object of time the representation we have of the world depends on the path of information acquisition and introspective measurements performed. By choosing a specific path, the individual chooses -let be in a non-deterministic manner -his "world". He is the creator of the world he lives in. We know from psychologists that people systematically underscore their responsibility in failure (relative to bad luck) and overscore their impact on success. It has been suggested that this can be fully rational to optimize motivation and selfimage. [START_REF] Mura | Projective Expected Utility[END_REF] The quantum cognitive approach provides a framework for effectively acting in this manner, it internalizes the (emotional) constraints that justify lying to oneself into cognition itself. Since QL-agents do have some freedom in the determination of the world they live in -it only seems rational to create a world in which we are more productive and feel good. Rational learning about the world transforms into a choice problem the outcome of which affects our well-being in a way similar to other choices. As all creative processes, it is constrained -the cognitive process is structured. Moreover one should not end up with a mental picture that is fully disconnected with the outside world because that undermines the value of our actions which are based on that mental picture. But we are invited to recognize a significant freedom. We thus propose that rational construction of a representation of the world entails selecting what information to acquire and which introspective measurements to perform so as to produce a mental picture that maximizes our well-being in the actual world. For instance when having very specific practical concerns adequacy between the mental picture and the outside world may be particularly valuable. In such a case a strong conviction in a less relevant perspective obtained after a series of introspective measurements may benefit from being upset by new information. Future research is called upon to characterize rational learning by QL-agent in terms of optimal decision-making with respect to information acquisition and introspective measurements.

Concluding remarks

This paper is a first exploration of learning by cognitively limited agents where the limitations are modeled appealing to quantum like characteristics of the mind. A main motivation is that the quantum approach has shown successful in explaining behavioral anomalies in decision-making while it also seems able to capture concerns expressed by practitioners dealing with information transmission and communication. In particular, the fact that i. people reason about reality within the frame of some perspective ("mental script") or as we call it, a mental picture, ii, the multiplicity of potential representations of reality generates specific problems for learning. We model cognitive limitations in terms of the multiplicity of Bohr complementary mental representations of one and the same reality. We find that with intrinsic uncertainty, additional information induce loss of information that can be detrimental.

Our analysis calls for a new approach to learning. First, it puts the cognitive process at the center: that is the logical structure behind the geometry of perspectives. Second, it unveils a form of creative freedom in the construction of a mental representation that invites a fully different approach. Namely learning about the world becomes a part of decision-making whose objective is to maximize well-being.

2

 2

Figure 1 .

 1 Figure 1. What do you see ?

j

  p j log p j .  It has been generalized by von Neumann to apply to quantum states and in particular to density operators. Let ρ = λ i |ϕ i ϕ i |: S (ρ) = -tr (ρ log ρ) or equivalently for S (ρ) = -λ i log λ i .

Figure 2 .

 2 Figure 2. Detrimental information



  Our (represented) World: A Quantum-Like ObjectConsider an alternative perspective P with two eigenpictures {p 1 , p 2 } . The basis {|p 1 , |p 2 } corresponds to a rotation of {|r * 1 , |r * 2 } by an angle of π/8 . The agent receives incomplete information in P i.e., the true state is either p 1 or p 2 with respec-

	tive probability γ 1 = (2 +	√ 2)/4 and γ 2 = (2 -	√	2)/4. The density operator repre-
	senting the state that obtains as a result of the acquisition of the new information is ρ = (2 + √ 2)/4|p 1 p 1 | + (2 -√ 2)/4|p 2 p 2 |. Note that ρ does not depend on ρ, it only
	reflects the newly acquired information.
	Phase 2: Updating
	The density operator must now be written in terms of the |r * i r * i | projectors ρ =
	.75|r * 1 r * 1 | + 0.25|r * 2 r * 2 | + 0.25|r * 1 r * 2 | + .25|r * 2 r * 1 | or
	(2)	ρ =	.75 .25 .25 .25	.

Let us take an example. we are interested in learning about a state agency. We may learn everything about the recruitment system which informs about the qualifications of people in various positions. It is a personnel perspective that we can map back to a perspective guided by say a concern about the correctness of a decision. If we next learn about the allocation of responsibility and associated path of decisions, it activates other concepts (another network of neurones). We then try to map back into a

If we can perform a measurement twice in a row. If the outcomes of the two measurements always coincide, we say that the measurement is a first-kind measurement.

When the state already is a the eigenstate of the observable than no information is either lost or gained.

You know your friend received the results of a determinant exam but only the probability for the outcomes, if she passed she is in a mood to go out.

You have reflected over her mood and formulated the uncertainty without "solving it" for yourself.

A measurement M is coarser than M if every eigenset of M is contained in some eigenset of M , see Danilov and Lambert-Mogiliansky[START_REF] Danilov | Measurable Systems and Behavioral Sciences[END_REF][START_REF] Danilov | Decision-making under nonclassical uncertainty[END_REF] p. 334.

Describing the two phases as essentially different is a simplification because information acquisition is not "direct" either. It also involves mental processes: perception is not neutral and not always classical (cf quantum zeno effect). Nevertheless a major distinction between the 2 phases is that the first involves an input from the outside while the second does not.

In Physics a preparation procedure operates on a large number of systems to filter those whose properties are the desired ones. But one can also prepare a single system into a specific state by submitting it to a series of measuremnt until de desired result is obtained. The idea we retain is that the procedure secures that the system is in a specified state.

Clearly the individual could neglect the information and he would not be forced into anything. But if he wants to integrate it, he must exert some mental flexibility and switch to the perspective in which that information takes its meaning.

See sect 2.2 above.

Clearly the definition could be extended to include arbitrary state i.e., mixed state. It would then capture the combined uncertainty from incomplete information and intrinsic uncertainty and would remain a relative concept.

[START_REF] Heisenberg | Edition Seuil: Heisenberg[END_REF] The term doubt is used in it s psychological meaning i.e., not the cartesian rational doubt that defines the scientific

approach.[START_REF] Gilboa | Maxmin Utility with non-unique priors[END_REF] It could also be resolved by information acquisition in

perspective A.[START_REF] Gilboa | Rationality of Beleifs or: why savage's axioms are neither necessary nore sufficient for rationality[END_REF] Another way to put it is that an introspective measurement is a way to deal with "doubts" where doubt is not (only) a problem of information but an inner state of lack of conviction.

After the introspective process, he will end up believing r * 2 with some probability less than 1.

See Dubois and Lambert-Mogiliansky (2014)for a detailed argument.

Some consider also Bayesain updating with multiple priors (see e.g., Hanany and Klibanov[START_REF] Hanany | Undating prefrences with multiple priors[END_REF]). But there is no consensus as to how to proceed -in sharp contrast with Bayesian updating of single priors.

The idea is that motivation is not fully maleable and must be incited: if you feel bad about yourself youmay not find the energy to make bold moves.