Exponential stabilization of a class of flexible microgrippers using dynamic boundary port Hamiltonian control.

Hector Ramirez, Yann Le Gorrec and Hans Zwart

Abstract-This paper deals with the control of a class of simplified models for flexible micro-grippers for DNA manipulation. The overall system is first modelled as a boundary controlled port Hamiltonian system made up as the interconnection of an infinite dimensional system (modelled as an undamped Timoshenko beam) representing the flexible arm of the gripper with two finite dimensional systems representing the DNA bundle and the suspension/actuator mechanism. The base of the arm is clamped on the suspension mechanism leading to under actuated system. The controller considered under strict dissipative port Hamiltonian format uses the velocity of the base of the tweezers arm as input and generates a force as output. The exponential stability of the closed loop system is derived by checking simple conditions on both the infinite and finite dimensional system.

I. INTRODUCTION

The manipulation of biological molecules by using micromechanical and optical devices such as: magnetic tweezers [START_REF] Amblard | A magnetic manipulator for studying local rheology and micromechanical properties of biological systems[END_REF], [START_REF] Gosse | Magnetic tweezers: micromanipulation and force measurement at the molecular level[END_REF], optical tweezers [START_REF] Simmons | Quantitative measurements of force and displacement using an optical trap[END_REF], AFM cantilevers [START_REF] Florin | Adhesion forces between individual ligand-receptor pairs[END_REF] and microfibers [START_REF] Ishijima | Sub-piconewton force fluctuations of actomyosin in vitro[END_REF], [START_REF] Cluzel | Dna: an extensible molecule[END_REF], are nowadays possible thanks to impressive technological progresses of the last years. The molecules manipulation is of particular interest for the characterization of basic properties of biological molecules such as DNA molecules [START_REF] Bustamante | Ten years of tension: singlemolecule dna mechanics[END_REF]. Silicon nanotweezers, (Figure 1). are used for !! !!! Fig. 1. Silicon nanotweezers this purpose: the principle is to trap DNA bundle between two arm tips using dielectrophoresis and then to characterize the DNA mechanical properties by using electrostatic actuation. The actuator is also used for monitoring enzymatic reactions on the DNA. The nanotweezers are so sensitive to

The contribution of Y. Le Gorrec and H. Ramirez has been done within the context of the NRA project HAMECMOPSYS with reference code ANR-11-BS03-0002. Further information is available at http://www.hamecmopsys.ens2m.fr/.

H. Zwart is with the Department of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands h.j.zwart@math.utwente.nl Y. Le Gorrec and H. Ramirez are with FEMTO-ST AS2M, 24 rue Alain Savary, 25 000 Besanc ¸on, France Yann.Le.Gorrec@ens2m.fr the stiffness variation of the DNA bundle that they become flexible and as consequence, current microfabrication processes tend to reduce the thickness of the beams in order to improve the sensitivity of the actuator. This leads to control problems that may be formulated in terms of partial differential equations (PDEs).

The framework of infinite dimensional port Hamiltonian systems has proven to be a powerful tool for the modelling and control of systems described by PDEs. Indeed, the definition of Boundary Controlled port Hamiltonian Systems (BCpHS) [START_REF] Van Der Schaft | Hamiltonian formulation of distributed-parameter systems with boundary energy flow[END_REF], [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF], [START_REF] Villegas | A port-Hamiltonian approach to distributed parameter systems[END_REF], [START_REF] Villegas | Exponential stability of a class of boundary control systems[END_REF], [START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinitedimensional Spaces[END_REF], have made it possible to develop constructive results analysis of well-posedness and stability of solutions.

In [START_REF] Ramirez | Exponential stability of boundary controlled port hamiltonian systems with dynamic feedback[END_REF] it has been shown that the interconnection of a fully actuated BCpHS with a linear finite dimensional controller renders the closed-loop system exponentially stable provided that the finite dimensional system is input strictly passive and exponentially stable. This result permits to elegantly, and quite easily, prove the exponential stability for a large class of linear controllers, in particular those arising from energy shaping methods using Casimir functions [START_REF] Macchelli | Modeling and control of the Timoshenko beam. the distributed port Hamiltonian approach[END_REF], [START_REF] Macchelli | Boundary energy shaping of linear distributed port-Hamiltonian systems[END_REF].

In this paper we use BCpHS to model and prove the exponential stability of a class of simplified models of underactuated silicon nanotweezer for DNA manipulation. More specifically the port Hamiltonian framework is used to construct a model of the manipulation process by interconnecting the different sub-systems: The infinite dimensional part consisting on the flexible arm of the gripper, modelled as an undamped Timoshenko beam; and the finite dimensional parts consisting on the DNA-bundle and the suspension/actuator mechanism. Preliminary results on a simplified version of the model have been presented in [START_REF] Ramirez | Exponential stability of a class of PDE's with dynamic boundary control[END_REF], [START_REF]Boundary port hamiltonian control of a class of nanotweezers[END_REF]. The stabilization of the process is performed by using a strict dissipative port Hamiltonian controller acting on the base of the tweezer through the transversal force only, the arm of the tweezers being clamped. We show that even when consider the extreme case of an undamped flexible arm, the closed-loop system is exponentially stable provided that the boundary controller is input strictly passive. The practical derivation of such controller (by using immersion/reduction method through Casimir invariants for example) is not detailed here. We focus in this paper on the analysis of the exponential stability of the underactuated closed loop system by checking simple conditions on both the infinite and finite dimensional systems and adapting the general result proposed in [START_REF] Ramirez | Exponential stability of boundary controlled port hamiltonian systems with dynamic feedback[END_REF].

The paper is organized as follows: The first section presents to port Hamiltonian models of the tweezer, the DNA and the suspension system as well as the model of the whole system after interconnection. The second section is devoted to exponential stability analysis. The paper ends with a conclusion and some perspectives.

II. PORT HAMILTONIAN MODELLING OF THE FLEXIBLE

MICROGRIPPER

In this paper we consider the simplified model of a flexible silicon based microgripper (Figure 2) used for DNA manipulation [START_REF] Boudaoud | Modeling and optimal force control of a nonlinear electrostatic microgripper[END_REF]. We assume that the tweezer is made up of largely flexible arm, that can be considered as a flexible beam with large deflection clamped to a transverse suspension system. The trapped DNA bundle is approximated by a spring/damper-mass-spring/damper system attached at the tip of the tweezer. The arm is actuated by using electrostatic forces generated by a comb drive actuator (not represented in Figure 2). Contrary to the model proposed in [START_REF] Ramirez | Exponential stability of a class of PDE's with dynamic boundary control[END_REF] (in which both force and torque at point a are used as control variables) we consider here the dynamics of the suspension system (the spring stiffness and damping coefficients are in a first instance considered constant). We assume that it is only possible to measure the transversal velocity at point a. whole system can be divided into three subsystems (Figure 3), namely the flexible arm, the DNA-bundle at the tip of the gripper and the actuated shuttle. Since we assume that the the flexible arm admits large deflections, we model it as a Timoshenko beam (infinite dimensional system). Furthermore, we consider the extreme case of an undamped Timoshenko beam, i.e, without dissipation. The choice of an undamped beam is motivated by control considerations. It is well known that dissipation adds robustness to a passive control scheme. Hence, by considering an undamped Timoshenko beam we are situating the control problem in a worst case scenario. We will show that the control is exponentially stable even when the natural damping is not considered.

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The DNA-bundle is modelled as a finite dimensional mechanical system. The subsystems are interconnected through their boundary power conjugated port variables. Let's note that only a part of the boundary port variables are used for the interconnection with the suspension system. The other part is considered equal to zero to account for the clamping of the beam. In Figure 3 the interconnection boundary port variables as well as the causality (depicted with arrows) are given. It is worth noting that even if the input of the beam at point a is a velocity (resp. angular velocity) (that is used for the interconnexion), the output of the port Hamiltonian controller is the force applied to the shuttle.

A. The Timoshenko beam

The damped and undamped Timoshenko beam has been widely studied as a distributed parameter port Hamiltonian system [START_REF] Macchelli | Modeling and control of the Timoshenko beam. the distributed port Hamiltonian approach[END_REF] and as BCS [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF] and the exponential stability of the system has been proved for static boundary feedback [START_REF] Villegas | A port-Hamiltonian approach to distributed parameter systems[END_REF], [START_REF] Villegas | Exponential stability of a class of boundary control systems[END_REF]. The BCS is defined as

∂ ∂t     x 1 x 2 x 3 x 4     =     0100 1000 0001 0010     P1 ∂ ∂z     Kx 1 1 ρ x 2 EIx 3 1 Iρ x 4     +     000-1 000 0 000 0 100 0     P0     Kx 1 1 ρ x 2 EIx 3 1 Iρ x 4     (1)
where the following state (energy) variables have been defined:

x 1 = ∂w ∂z (z, t) -φ(z, t) the shear displacement, x 2 = ρ(z) ∂w
∂t (z, t) the transverse momentum distribution, x 3 = ∂φ ∂z (z, t) the angular displacement, and x 4 = I ρ ∂φ ∂t (z, t) the angular momentum distribution, for z ∈ (a, b),t≥ 0, where w(t, z) is the transverse displacement of the beam and φ(t, z) is the rotation angle of a filament of the beam. The coefficients ρ(z), I ρ (z), E(z), I(z) and K(z) are the mass per unit length, the rotary moment of inertia of a cross section, Youngs modulus of elasticity, the moment of inertia of a cross section, and the shear modulus respectively. The matrices P 1 and P 0 defines the skew-symmetric differential operator of order 1 acting on the state space X = L 2 (a, b, R 4 ), J = P 1 ∂ ∂z + P 0 . The energy of the beam is expressed in terms of the energy variables,

E = 1 2 b a Kx 2 1 + 1 ρ x 2 2 + EIx 2 3 + 1 I ρ x 2 4 dz = 1 2 b a x(z) (Lx)(z)dz = 1 2 x 2 L where L = diag K, 1 ρ ,EI, 1

Iρ

. The boundary port variables are obtained by using integration by parts and factorization in order to define an extended Dirac structure including the boundary [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF]. They also can be directly parametrized from P 1 [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF], [START_REF] Villegas | A port-Hamiltonian approach to distributed parameter systems[END_REF] leading to:

f ∂,Lx e ∂,Lx =             (ρ -1 x 2 )(b) -(ρ -1 x 2 )(a) (Kx 1 )(b) -(Kx 1 )(a) (I -1 ρ x 4 )(b) -(I -1 ρ x 4 )(a) (EIx 3 )(b) -(EIx 3 )(a) (ρ -1 x 2 )(b)+(ρ -1 x 2 )(a) (Kx 1 )(b)+(Kx 1 )(a) (I -1 ρ x 4 )(b)+(I -1 ρ x 4 )(a) (EIx 3 )(b)+(EIx 3 )(a)             .
The control objective is to control the translational and angular position of the DNA-bundle. From the interconnection structure, the input and output of the beam are selected as follows:

u = v(b) ω(b) -v(a) -ω(a) , y = F (b) T (b) F (a) T (a) , (2)
which is achieved by defining It can by shown that with this choice of input and output the system (1) defines a an abstract boundary control system. Furthermore Ax = P 1 (∂/∂z)(Lx)+P 0 Lx with domain

u = W f ∂,Lx e ∂,Lx ,y = W f ∂,Lx e ∂,
D(A)= Lx ∈ H 1 (a, b; R n ) f∂,Lx(t) e∂,Lx(t) ∈ ker W
generates a contraction semigroup on X and the energy balance equation is defined as:

dE dt = u T y
In what follows the beam is clamped at point a. We then split the input/ouput in:

u =[u 1 ,u 2 ] ,y=[y 1 ,y 2 ] with u 1 = v(b) ω(b) -v(a) ,u 2 = ω(a) , y 1 = F (b) T (b) F (a) ,y 2 = T (a) , (3) Furthermore 
   u 1 u 2 y 1 y 2     = 1 √ 2 W W P 1 -P 1 II Lx(b) Lx(a)
and one can show (this property will be very important in the proof of the exponential stability) that even if u 2 is chosen equal to zero, there exists an ε such that:

u 1 2 + y 1 2 ≥ εLx(b) 2 (4)
Such result can be generalized to any u 2 corresponding to conditions at point a.

B. DNA-bundle model

The DNA-bundle is represented by the simple springdamper + load + spring-damper system of Figure 4 and thus admits a port Hamiltonian system representation. In Figure 4 k

E b (x c2 -x c1 ,x c2 ,M ẋc2 )= 1 2 k 1 (x c2 -x c1 ) 2 + k 2 x 2 c2 + M ẋ2 c2 (5)
and

J b = -J T b =   000 001 0 -10   ,R b = R T b =   k1 f1 00 000 00f 2   ≥ 0 g T b = 1 f1 01 000 S b = 1 f1 0 0 1 fθ > 0
where f θ is the rotational damping constant at the interconnection point.

III. THE CONTROLLER+SUSPENSION SYSTEM

The controller is considered under strict dissipative port Hamiltonian format, i.e. dynamic system of energy

E c3 (x c3 )= 1 2 x T c3 Q c3 x c3 defined as x c3 =(J c3 -R c3 ) ∂Ec3(xc3) ∂xc3 + B c3 u c3 y c3 = B T c3 ∂Ec(xc3) ∂xc3 + D c3 u c3 with J c3 = -J T c3 , R c3 = R T c3 ≥ 0, D c3 > 0.
The suspension system is modelled through a spring/damper system of potential energy:

E s (x a )= 1 2 kx 2 a
The interconnection of these two systems can be written under port Hamiltonian format:

va =(J a -R a ) dEa dva + g a u a y a = g T a dEa dva + S a u a with v a =( x a ,x c3 ) and E a (v a )=E s (x a )+E c3 (x c3
) the energy of the system and:

J a = 00 0 J c3 ,R a = 1 Dc3+f B T c3 Dc3+f B T c3 Dc3+f R c + Bc3B T c3 Dc3+f ≥ 0 g a = -1 Dc3+f - B T c3 Dc3+f S a = 1 D c3 + f > 0
IV. SYSTEMS INTERCONNECTION In this section we are interested in the interconnection of the infinite dimensional system with the finite dimensional ones. First of all let us gather the two finite dimensional systems as:

v =(J c -R c ) ∂Ec(v) ∂v + g c u c y c = g T c ∂Ec(v) ∂v + D c u c (
J c = -J T c = J b 0 0 J a ,R c = R T c = R b 0 0 R a ≥ 0 g c = g b 0 0 g a S c = S b 0 0 S a > 0
The total energy can be written on the form:

E c (v)= 1 2 v T Q c v with: Q c = diag k 1 ,k 2 , 1 M ,k,Q c3 > 0
The interconnection with the infinite dimensional system is done through the following relation:

u 1 y 1 = 1 0 r + 0 -1 10 u c y c ( 7
)
with the additional constraint:

u 2 = ω(a)=0 (8)
Remark 4.1: From the positivity of R c and the strict positivity of S c the dynamic system defined by ( 6) is exponentially stable and has all its eigenvalues in the left hand side of the complex plane.

V. S TABILITY ANALYSIS OF THE CLOSED LOOP SYSTEM

The exponential stability of the closed loop system is derived following two steps. The first one is concerned with the existence of solutions of the interconneted systems (finite+infinite dimensional systems). It aims at showing that the interconnected system defines a contraction semigroup. The second step consists in finding a Lyapunov function with exponential decay rate. The fact that the closed loop system defines a Boundary Control System with contraction semigroup is derived in Theorem 5.1.

Theorem 5.1: Let the state of the open-loop BCS satisfy

1 2 d dt x(t) 2 L = u(t)y(t).
Consider the LTI strictly passive finite dimensional system [START_REF] Cluzel | Dna: an extensible molecule[END_REF]

with storage function E c (t)= 1 2 v(t),Q c v(t) R m , Q c = Q c > 0 ∈ R m × R m .
Then the feedback interconnection [START_REF] Bustamante | Ten years of tension: singlemolecule dna mechanics[END_REF] of the BCS and the finite dimensional system is again a BCS on the extended state space

x ∈ X = X × V with inner product x 1 , x2 X = x 1 ,x 2 L + v 1 ,Q c v 2 V .
Furthermore, the operator A e defined by

A e x = JL 0 g c C J c -R c x v , Cx = I c W f ∂,Lx e ∂,Lx
with

I c = I 3 0 3,1 D(A e )= x v ∈ X V Lx ∈ H N (a, b; R n ),   f ∂,Lx e ∂,Lx v   ∈ ker WD , where WD = W + S c 0 00 W I 0 g T c
generates a contraction semigroup on X.

Proof: The proof is very similar to the one presented in [10, Theorem 5.8, pp:120] and in [START_REF] Ramirez | Exponential stability of a class of PDE's with dynamic boundary control[END_REF] but we consider here that the feedback is only applied in between u 1 and y 1 as stated in [START_REF] Bustamante | Ten years of tension: singlemolecule dna mechanics[END_REF], u 2 being freely chosen (equal to zero for the stability analysis). The proof is performed by showing that WD is full rank and by applying the Lumer-Phillips Theorem [12, Theorem 6.1.7, pp:69], which is divided in two parts: showing that A e is a dissipative operator (i.e. ReAx, x≤ 0) and that ran(I -A e )= X = X × V . The fact that WD is full rank is directly derived from the fact that W W and I 0 S c 0 01 0 0 ar full rank. From the strict positivity of D c it is easy to show that A e is dissipative using the Kalman-Yakubovich-Popov (KYP) Lemma [START_REF] Willems | Dissipative dynamical systems part I: General theory[END_REF], [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF]. The second part of the proof, ran(I -A e )= X, follows if the matrix (I -(J c -R c )) is non-singular. This is true as all the eigenvalues of the matrix A c are in the left half of the complex plane as it is stated in Remark 4.1.

We are now going to prove the exponential stability of the closed loop system by using the aforementioned contraction properties and Lyapunov arguments. The stability results from the strict dissipativity and the exponential stability of the finite dimensional system and the fact that under actuation is only considered on the a side of the beam.

The Lyapunov candidate that we propose to use is the total energy:

Ẽ = E + E c
To prove the result we first need some technical Lemma (the proofs are omitted here but can be found in [START_REF] Ramirez | Exponential stability of boundary controlled port hamiltonian systems with dynamic feedback[END_REF] as they not depend on the fact that the system is partially actuated). These Lemma are concerned with the energy of the finite dimensional system (6) and come from its strict dissipativity and exponential stability properties. Lemma 5.2, 5.3 and 5.4 give bounds on E c (τ ) and τ 0 E c (t)dt with respect to u c (t) 2 and y c (t) 2 .

Lemma 5.2: [START_REF] Ramirez | Exponential stability of boundary controlled port hamiltonian systems with dynamic feedback[END_REF] The energy of (6) satisfies for a positive time τ :

E c (τ ) ≤ κ 1 E c (0) + κ 3 τ 0 u c (t) 2 dt (9)
where κ 1 = κ 4 e -κ2τ with κ 2 ,κ 3 and κ 4 constants. Lemma 5.3: [START_REF] Ramirez | Exponential stability of boundary controlled port hamiltonian systems with dynamic feedback[END_REF] There exists positive constants ξ 1 ,ξ 2 and τ 0 such for all τ>τ 0 the energy of ( 6) satisfies

τ 0 E c (t)dt ≤ξ 1 τ 0 v (t)Q c R c Q c v(t)dt+ (10)
ξ 2 τ 0 u c (t) 2 dt
Lemma 5.4: [START_REF] Ramirez | Exponential stability of boundary controlled port hamiltonian systems with dynamic feedback[END_REF] For every δ 1 > 0 there exists a δ 2 > 0 such that for all τ>0 the energy of ( 6) satisfies the relation τ

0 δ 1 E c (t)+y c (t) 2 dt ≤ δ 2 τ 0 E c (t)+u c (t) 2 dt. (11)
We are now going to give some bounds on the total energy of the system in order to prove its exponential stability. For that we mainly need the property (4) i.e:

u 1 2 + y 1 2 ≥ εLx(b) 2 (12)
It leads to Lemma 5.5. Lemma 5.5: Consider a BCS as described in Theorem 5.1 with r(t)=0 ,u 2 =0 , for all t ≥ 0. Then, the energy of the system

Ẽ(t)= 1 2 x(t) 2 L + 1 2 v(t) T Q c v(t) satisfies for τ large enough Ẽ(τ ) ≤ c(τ ) τ 0 (Lx)(t, b) 2 R dt + 2c(τ ) c1 τ 0 E c (t)dt, Ẽ(τ ) ≤ c(τ ) τ 0 (Lx)(t, a) 2 R dt + 2c(τ ) c1 τ 0 E c (t)dt, (13)
where c is a positive constant that only depends on τ and c 1 a positive constant.

Proof: The proof is based on a multiplier technic and follows the proof of [13][11]. It mainly uses the contraction property of the semigroup and (4). These Lemma allow to prove the exponential stability of the closed loop system as stated in Theorem 5.6

Theorem 5.6: Consider the BCS defined by Theorem 5.1 with r(t)=0 ,u 2 =0 , for all t ≥ 0. Then the finite dimensional system (6) exponentially stabilizes the BCS.

Proof: The time derivative of the total energy is given by

Ė = -v Q c R c Q c v -u c S c u c ≤-v Q c R c Q c v -σu c u c , since S c ≥ σI = -v Q c R c Q c v -σ 1 u c u c -σ 2 u c u c = -v Q c R c Q c v -σ 1 u c 2 -σ 2 y 1 2 = -v Q c R c Q c v -σ 1 u c 2 -σ 2 y 1 2 + u 1 2 + σ 2 u 1 2
with 1 + 2 =1and where we have used that u c = -y 1 . Using (4)we have

Ė ≤-v Q c R c Q c v -σ 1 u c 2 -σ 2 Lx(t, b) 2 + σ 2 y c 2 . ( 14
)
Integrating this equation on t ∈ [0,τ] we have

Ẽ(τ ) -Ẽ(0) ≤- τ 0 v (t)Q c R c Q c v(t)dt+ - τ 0 σ 1 u c (t) 2 + σ 2 Lx(t, b) 2 -σ 2 y c (t) 2 dt.
Next choose τ sufficiently large such that Lemmas 5.3 and 5.5 hold. Using the latter lemma we have

Ẽ(τ ) -Ẽ(0) ≤- τ 0 v Q c R c Q c v + σ 1 u c 2 dt + σ 2 c(τ ) 2c(τ ) c 1 τ 0 E c (t)dt -Ẽ(τ ) + σ 2 τ 0 y c 2 dt.
Grouping terms we have that

Ẽ(τ ) 1+ σ 2 c(τ ) -Ẽ(0) ≤ - τ 0 v(t) Q c R c Q c v(t)dt -σ 1 τ 0 u c (t) 2 dt +σ 2 τ 0 2 c 1 E c (t)+y c (t) 2 dt .
Using Lemma 5.4 with δ 1 = 2 c1 we have

Ẽ(τ ) 1+ σ 2 c(τ ) -Ẽ(0) ≤- τ 0 v(t) Q c R c Q c v(t)dt + σ 2 δ 2 τ 0 E c (t)dt + σ( 2 δ 2 -1 ) τ 0 u c (t) 2 dt. (15)
Now, using Lemma 5.3 we obtain

Ẽ(τ ) 1+ σ 2 c(τ ) -Ẽ(0) ≤ (σ 2 δ 2 ξ 1 -1) τ 0 v(t) Q c R c Q c v(t)dt+ σ( 2 δ 2 (1 + ξ 2 ) -1 ) τ 0 u c (t) 2 dt.
Since 2 may be chosen to be arbitrarily small, i.e, 2 1 and since 1 =1-2 , we finally have that

Ẽ(τ ) ≤ c 2 Ẽ(0) (16)
with

c 2 = 1 (1+ σ 2 c(τ ) )
< 1 which proves the theorem.

Remark 5.7: An undamped Timoshenko beam has been considered for the infinite dimensional part of the system. It should be noticed however that the stability proof also holds for a damped Timoshenko beam. Indeed, dissipation in the infinite dimensional model makes the inequalities in Lemma 5.5 strict inequalities.

VI. CONCLUSION

The model of a DNA-manipulation process has been formulated as a BCpHS. This has been achieved by modelling the sub-systems of the process as port-Hamiltonian systems and showing that the physically meaningful interconnection of the sub-models defines a BCpHS. More specifically, the flexible arm of the micro-gripper has been modelled as a flexible beam (undamped Timoshenko beam) that is clamped on one side, the DNA-bundle as a mass-spring-damper system and the suspension/actuator mechanism as a massspring system. In this model the flexible arm corresponds to a infinite dimensional system, so the complete DNA manipulation processes is the interconnection of an infinite dimensional system and two finite dimensional systems, where one of the input of the infinite dimensional system being assigned to zero. The stabilization has been achieved by interconnecting the suspension/actuator mechanism with a port-Hamiltonian controller (with realistic sensing and actuation variables). The complete process then remains a BCpHS but with respect a new energy function due to the contribution of the controller. We considered in this paper that the controller has been designed using a dissipative port Hamiltonian format. It has been assumed that no natural damping is present on the infinite dimensional part of the process, which makes the stability result more interesting, since natural damping (which is physically present) adds robustness to the closed-loop system.

Future work aims to include non-linear phenomena in the suspension/actuator mechanism and DNA-bundle.

Fig. 2 .

 2 Fig. 2. DNA manipulation through port Hamiltonian control

Fig. 3 .

 3 Fig. 3. Interconnection and control strategy

Fig. 4 .

 4 Fig. 4. DNA model resenting the DNA-bundle. Then we can write by using v b =( x c2 -x c1 ,x c2 ,M ẋc2 ) T , u b = F (b) T (b) T and y b = v(b) ω(b) T : vb =(J b -R b ) dEb dvb + g b u b y b = g T b

6 )

 6 with v =(v b ,v a ), u c =(u b ,u a )=(F (b),T(b),F(a)), y c = (y b ,y a )=(v(b),ω(b),v(a)), E c (v)=E b (v b )+E a (v a ) and


Rodolfo Cristofaro

Bruno De

Rodolfo De Cristofaro

Bruno Lecoutre
email: bruno.lecoutre@univ-rouen.fr.

A New Criterion for Prior Probabilities

Keywords: Data translated likelihood, Frequentist properties, Inductive logic, Jeffreys-rule, Likelihood principle, Objective Bayesian analysis, Principle of indifference, Stopping rule

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Howson and Urbach's book titled "Scientific Reasoning: The Bayesian Approach" (1996) is a carefully argued exposition and defense of the Bayesian view of scientific reasoning. In particular, according to Howson and Urbach, alternative methods of inference, especially those connected with significance testing and estimation, are really quite unsuccessful and, despite their influence among scientists, their pre-eminences are undeserved. What is more, subjectivity in the Bayesian approach is, first of all, minimal and, secondly, exactly right. The ideal of total objectivity is unattainable and alternative approaches to scientific reasoning, which pose as guardian of that ideal, in fact violate it at every turn; virtually no other method can be applied without a generous helping of personal judgment and arbitrary assumption.

We agree with the arguments advanced by the authors concerning inductive reasoning; arguments, which are in line with Rudolf Carnap's thinking on inductive logic. There is, however, a difference. It involves the analytical character of the solution to be given to the problem of statistical induction.

As Carnap (1962, p. 518) wondered, "why did statisticians spend so much effort in developing methods of inference independent of the probability axioms? It seems clear that the main reason was purely negative; it was the dissatisfaction with the principle of indifference (or insufficient reason). If we should find a degree of confirmation which does not lead to the unacceptable consequences of the principle of indifference, then the main reason for developing independent methods of estimation and testing would vanish. Then it would seem more natural to take the degree of confirmation as the basic concept for all of inductive statistics."

Today, are we able to say whether the unacceptable consequences of the principle of indifference have been avoided? It is hard to answer this question in the affirmative. Should the prior information be held to be irrelevant, a probability distribution should exist to be assigned to given hypotheses on which different individuals agree.

In other words, in order to control induction analytically, one would have to know how to assign the probabilities of the various hypotheses in case prior information is held to be irrelevant. Moreover, this should allow evaluating the subjective component due to other information (the information preceding the current experimentation).

Another point, which deserves to be investigated, has to do with the likelihood principle (LP). LP concerns foundations of statistical inference and it is often invoked in arguments about correct statistical reasoning. Let f (x | θ) be a conditional distribution for X given the unknown parameter θ ∈ Θ (the set of possible values of θ). According to the LP, in the inference about θ, after X = x is observed, all relevant experimental information is contained in the likelihood function for the observed x. As an operative implication of this principle, if different designs produce proportional likelihood functions, one should make an identical inference about a parameter θ from the data, x, irrespective of the particular design that yields x.

There are several counter-examples, and/or paradoxical consequences to the LP. What is more, some methods of conventional statistics are not consistent with the LP. For reference, see Severini (2000, pp. 79 ff.). The authors who do not accept these consequences reject the utilization of Bayes formula, and solve statistical inference by simply examining the likelihood function under a weak version of the LP. On this subject, one may see Cox and Hinkley (1974, p. 39).

If we accept the likelihood principle in its weak version, then the approach deriving from it is objective (or would at least allow one to assess the experimental information in an objective manner). Nevertheless, as we were able to show in a previous article (cf. de Cristofaro, 2004), the LP (both in its strong and weak version) is questionable. In fact, contrary to a widely held opinion, it is not a consequence of Bayes theorem.

To be honest, the differences, compared to the book by Howson and Urbach, do not involve the Bayesian approach -with which we basically agree -but the judgment about the choice of an objective prior distribution, and, more in general, the foundations of the so-called 'Objective Bayesian Analysis'.

2 On the irrelevance of stopping rule [START_REF] Howson | Scientific Reasoning: The Bayesian Approach[END_REF] do not speak explicitly of the likelihood principle, although they do deal with the issue when referring to the stopping rule (the rule that dictates when the trial should terminate).

The discussion is related to the example of an experiment that is designed to elicit a coin's physical probabilities of landing heads and tails. And the conclusion is as follows (p. 365): it does not matter whether the experimenter intended to stop after n tosses of the coin or after r heads appeared in a sample; the inference about θ [the probability of landing heads] is exactly the same in both cases.

The LP is based on an assumption held to be obvious and therefore not verified, which is the assignment of the prior probabilities about the parameter θ independently from the manner in which the trials were conducted: the socalled sampling rule, experiment, process of data generation, or -as we will call it -design, d. Quoting Lindley's words (2000), only the realized actual observation x is relevant: "This is the likelihood principle according to which values of x, other than that observed, play no role in inference." Nevertheless, the correct version of Bayes formula shows that, not only the data, but also d is relevant in inference.

Suppose that x is the observed value of a random variable X, whose probability distribution p(x | θ, e) depends on θ and e. Suppose also that θ itself has a probability distribution p(θ | e) conditional on e, called the prior distribution. Then, given x and e, the posterior distribution of θ, p(θ | x, e), is

p(θ | x, e) ∝ p(θ | e)p(x | θ, e). (1)
The statement of (1) is the expression of Bayes formula in accordance with the Carnap's philosophy.

Indeed, e comprises not only the beliefs of the experimenter before the experiment is performed, e * , but also the piece of information about d. That is, e = (e * , d).

In particular, x is determined by a particular design d with a given θ. Hence, p(x | θ, e) is not defined without a reference to d. Thus, the probability of x successes on n trials is different according to the supposed process of data generation: direct or inverse sampling, hypergeometric scheme, Markov process, and so on.

The conditional assertion about d is usually omitted when it is clear from the context which design has been chosen. But if two people assume the same prior with reference to different designs, it becomes important to spell out the designs each used. In fact, the correct expression of Bayes formula shows that the prior p(θ | e * , d) depends on d.

To conclude, the core of the LP is that, given x, d is irrelevant under the same likelihood. On the contrary, whatever the data may be, the evidence about d may affect the prior, and, consequently, the result of inference.

The omission of any reference to evidence has been the cause of unnecessary debates and unpleasant consequences. The very heart of the matter is the ancillary knowledge of how the data were collected. In fact, as we saw, d is a full part of e. It may play a basic role in inference.

In this regards, the often quoted sentence by Edwards, Lindman, and Savage (1963) should be revised: "The likelihood principle emphasized in Bayesian statistics implies, among other things, that the rules governing when data collection stops are irrelevant to data interpretation. It is entirely appropriate to collect data until a point has been proven or disproved, or until the data collector runs out of time, money, or patience."

This recommended practice appears as a pill hard to digest for experimenters willing to adopt Bayesian methods. Designs are open to unscrupulous manipulation if the experimenter is allowed to ignore the rules governing the data sampling and to choose the stopping point irrespectively of the prior.

In reality, Bayes formula (in its correct expression) does not allow us to choose at discretion the data, to ignore the rules governing their collection, or to stop the sampling irrespective of the prior.

We can find in Severini (2000, p. 79) a remark essentially correct: "information beyond that provided by the likelihood function is necessary for proper statistical inference". But the right answer to this remark is not the weak version of LP. It is the evidence about d and its effect on the prior.

Applying this idea, [START_REF] Bunouf | Bayesian priors in sequential binomial design[END_REF], 2008[START_REF] Bunouf | An objective Bayesian approach to multistage hypothesis testing[END_REF] developed Jeffreystype priors derived from likelihood augmented with the design information in multistage designs. They showed that the use of such priors corrects the posteriors from the stopping rule bias.

For a more detailed analysis regarding the foundations of LP, see de Cristofaro (2004).

The new principle of indifference

The principle of indifference is a rule for assigning probabilities under 'ignorance'. It was called "of indifference" by John M. Keynes, who was careful to note that it applies only when there is no knowledge indicating unequal probabilities. Anyway, we are not in a situation of total ignorance, since we know the design d that is going to generate the data. Thus, if we know that d is able to favor some hypothesis, then indifference principle does not apply.

Quoting Howson and Urbach (1996, p. 363), a fact is relevant to a set of hypotheses when knowing it makes a difference to one's appraisal of those hypotheses. In this connection, is it relevant for the assignment of an equal probability to each face of a die whether it is or not biased or whether its casting is or not fair? I think so. In the same way, apart from other information, in order to assign the same probability to every admissible hypothesis, the design d should be 'fair' or 'impartial', in the sense of ensuring the same support to all hypotheses.

That being stated, the new Principle of Indifference is as follows (cf. de Cristofaro, 2008):

Given the set of all admissible hypotheses H, let h denote any one element of the partition of H and let d denote the projected design, then we are allowed to assign the same probability to every h if (i) prior information is considered to be irrelevant, and (ii) d is impartial.

In statistics, in order to ensure the impartiality of d towards the parameter θ, it is sufficient that the superior extreme ordinate of the posterior p(θ | x, d), for a possible x, is situated on the same level of any other curve of the posterior obtainable from d (superior profile criterion).

In plain language, if d is genuinely impartial to θ, then all possible values of θ should be equally supported whatever the data may be. Likewise, the superior extreme ordinate of the posterior curves graphed for all possible data, x, should be constant.

Let (θ | t) be the standardized likelihood of θ given a possible observation of the sufficient statistic t about θ (whose integral with respect to θ ∈ Θ is 1). Apart from a proportionality constant that does not depend on t or θ, a means whereby profile criterion can be put to the test consists in plotting the function

h(θ) = sup t (θ | t), (2)
in order to see whether it is or not constant. Of course, the assumption of a uniform prior for θ is justified in the affirmative.

For instance, the Binomial mean θ is far from being impartial. In case of n = 24 trials and x successes, we have:

h(θ) = sup x=0,...,24 25! x!(24 -x)! θ x (1 -θ) 24-x . ( 3
)
This function is U-shaped, and, therefore, the assumption of a uniform density prior for θ is not justified. (cf. the standardized likelihood curves in Figure 1).

A similar criterion about the choice of an objective prior was introduced by Fisher in the year 1922 and it was worked out by [START_REF] Box | Bayesian Inference in Statistical Analysis[END_REF] for reference to Fisher, see p. 35). According to these authors, the prior distribution for a parameter, let us say θ, is assumed to be locally uniform if different sets of data translate the likelihood curve on the θ-axis, leaving it unchanged in shape and spread (that is, the data only serve to change the location of the likelihood). On the other hand, if θ is not data translated in this sense, then Box and Tiao suggest expressing the parameter in terms of a new metric φ = φ(θ), so that the corresponding likelihood is 'data translated'.

As we can see from the examples and figures shown by Box and Tiao (1973, pp. 27-39) if the parameter θ is data translated (and the design d is impartial to θ), then the superior extreme ordinate of every possible curve of the standardized likelihood for θ is situated on the same level of any other curve obtainable from d.

For illustration, suppose x = (x 1 , ..., x n ) is a random sample from a Normal distribution N (µ, σ 2 ), where σ is supposed known. The likelihood function of µ is

(µ | x) ∝ exp - n 2σ 2 (m -µ) 2 , ( 4
)
where m is a possible determination of the average of observations. This function is represented by a Normal curve with its maximum value that remains constant for all possible determinations of m. In particular, when m = µ, (4) is proportional to a constant. Thus, the design is impartial to µ, and the density prior for µ can be assumed locally uniform. Now, it could happen that the quantity of interest was not µ but its reciprocal γ = µ -1 . In this case, the posterior or standardized likelihood for γ is

(γ | σ, m) ∝ γ -2 exp - n 2σ 2 (m -γ -1 ) 2 , (5)
which is a curve with its maximum ordinate proportional to γ -2 . In particular, when m = γ -1 , ( 5) is proportional to γ -2 .

On the other hand,

p(γ | σ) = p(µ | σ) dµ dγ = p(µ | σ)µ 2 ∝ γ -2 . (6)
As we saw with reference to (5), the density prior for γ (besides being proportional to the Jacobian of the transformation from µ to γ) is proportional to the superior profile of the posterior (or standardized likelihood) curves for γ, with reference to all possible determinations of m from the intended design. Namely,

p(γ | σ) ∝ sup m (γ | σ, m) = γ -2 . ( 7
)
Notice that the reference to 'standardized likelihood' is due to [START_REF] Box | Bayesian Inference in Statistical Analysis[END_REF], where we can find some unremarked illustrations about the property of the prior we have just mentioned (cf. pp. 27, 30, 35, 39).

More in general, if we do a one-to-one transformation of the parameter θ concerning an impartial design in terms of a new metric φ = φ(θ), then the prior

p(φ) ∝ sup φ p(φ | φ), (8)
where p(φ | φ) is the posterior for φ, given a possible observation of the sufficient statistic φ about φ.

We can see the argument from another viewpoint: if (8) holds, then exists a transformation θ = θ(φ) that makes the design impartial with respect to θ.

The profile criterion, we suggested with reference to the uniform distribution, is a particular case of a more general rule, given by ( 8).

According to this rule, the prior probability for the parameter φ conditional to d, p(φ | d), is proportional to the superior profile of corresponding posterior curves, p(φ | x, d), considered for all possible x obtainable from d. A rule we can apply to any prior.

As an example, the density prior for Binomial mean θ suggested by Jeffreys is

p(θ) ∝ [θ(1 -θ)] -1/2 . ( 9
)
This prior is approximately proportional to the superior profile of standardized likelihood curves for given n

(θ | x) = 1 B(x + 1, n -x + 1) θ x (1 -θ) n-x , (10)
where B(α, β) is the complete Beta function. Note that (θ | x) is the density of a Beta distribution of parameters x + 1 and n -x + 1, which is the posterior distribution for θ associated with an uniform prior.

In the case of n = 24 trials, Figure 1 shows the curves of standardized likelihood for x = 1, 3, 7, 12, 17, 21, 23 successes, together with the prior of (9). As we can see, ( 9) is approximately (with a very good approximation) proportional to the corresponding maximum of the curves shown in Figure 1. A similar approximation holds between ( 9) and sup

x 1 B(x + 1 2 , n -x + 1 2 ) θ x-1 2 (1 -θ) n-x-1 2 . ( 11
)
in which we can recognize the density of a Beta distribution of parameters x + 1 2 and n -x + 1 2 , which is the posterior distribution for θ associated with the Jeffreys prior β( 12 , 1 2 ).

Because of this approximation, the transformation φ = sin -1 √ θ does not make the Binomial experiment exactly impartial. As we can see in Figure 2, the superior profile of the standardized likelihood curves for φ is not constant, although it is nearly so. If necessary, a prior may be assumed which is approximately uniform. But, strictly speaking, the design is not impartial. Notice that, although the superior profile of standardized likelihood for φ is nearly constant, the corresponding curves are rather far from being data translated. That suggests that the right criterion for the assumption of a uniform prior is based on the profile of likelihood rather than 'data translated likelihood'.

Given the general rule, defined by ( 8), we could build approximate prior distributions, by assuming each time the prior for θ:

g r (θ) ∝ sup t g r-1 (θ) (θ | t), (12)
where r = 1, 2, ..., s, s is fixed in such a way that

g s (θ) ≈ g s-1 (θ), (13)
and

g o (θ) = (θ | t) (θ | t) . ( 14
)
Of course, the iteration could go up to the asymptotic prior distribution, if it exists.

We note in passing that in the above-mentioned procedure the prior for r = 1 may approximate very well the prior for r = 2. That is, the approximation may be rather good already for r = 1.

A further good approximation for the prior concerning the parameter θ could be provided by the superior profile of the sampling distribution of the t given θ, with reference to all possible determinations of t from the intended design. detail, in a future paper.

The profile criterion could be extended to the case of two or more parameters.

For instance, we can consider the choice of the prior with reference to a random sampling, in samples of size n, from a Normal distribution N (µ, σ 2 ), where µ and σ are both unknown. The likelihood of (µ, σ) is

(µ, σ | x) ∝ σ -n exp - n 2σ 2 s 2 + (m -µ) 2 , (15)
where m and s are the maximum likelihood estimators about µ and σ, respectively. According to Box and Tiao (1973, p. 49), this likelihood is data translated (and the design impartial) in terms of (µ, log σ). Then, following the profile criterion, the density prior for (µ, σ) is proportional to the Jacobian of the transformation from (µ, log σ) to (µ, σ). That is, in accordance with the modified Jeffreys-rule, p(µ, σ) ∝ σ -1 .

Jeffreys has made fundamental contributions to statistics in order to obtain analytical prior distributions. Anyway, we believe that the basic idea for choice and evaluation of an objective prior is profile criterion.

In the end, it is clear that the thesis, supported by Howson and Urbach (1996, p. 429), according to which no theory can adequately represent 'ignorance' between alternatives, has to be revised in the light of the new principle of indifference.

4 An illustration of the profile criterion

Profile criterion prior for Binomial sampling

For a Binomial sample of size n with x successes, the standardized likelihood (θ | x) has been given in [START_REF] De Cristofaro | A new formulation of the principle of indifference[END_REF]. In order to determine the profile criterion density defined by π(θ) ∝ sup x (θ | x), let us consider the ratio

R(y) = (θ | x + 1) (θ | x) = n -x x + 1 θ 1 -θ . (16)
For a given x, this ratio is equal to one for θ = (x + 1)/(n + 1), is greater than one for θ < (x + 1)/(n + 1) and is smaller than one for θ > (x + 1)/(n + 1). Consequently we get

π(θ) ∝ θ | X(θ, n) , (17)
where

X(θ, n) = j for j n + 1 ≤ θ ≤ j + 1 n + 1 (0 ≤ j ≤ n). (18)
For θ in each interval [j/(n+1), (j+1)/(n+1)] (0 ≤ j ≤ n), π(θ) is proportional to the density of the Beta distribution β(j+1, n-j+1). The fact that R(j) = 1 when θ = (j + 1)/(n + 1) ensures the continuity of the density. The constant of normalization 1/ 1 0 π(θ)d(θ) can be easily computed from incomplete Beta functions. Moreover, recurrence relations can be derived to get more efficient computer algorithms. The prior for n = 5 is plotted in Figure 3 and can be compared with the Jeffreys prior. The constant of normalization is 1/3.018. It can be seen in Figure 3 that the transformed profile criterion prior for φ = sin -1 √ θ is approximately uniform on the interval [0.17,1.40], that is for θ between about 0.03 and 0.97.

Numerical applications

For a Binomial sample of size n, it follows that the corresponding posterior density π(θ | x) within each interval defined by j (0 ≤ j ≤ n) is proportional to the density of the Beta distribution β(x + j + 1, 2n -x -j + 1), with a coefficient of proportionality that depends on j and is equal to

B(x + j + 1, 2n -x -j + 1) B(j + 1, n -j + 1) . ( 19
)
Here again the constant of normalization can be easily computed from incomplete Beta functions and recurrence relations can be derived, assuring the feasibility of the procedure. For instance, for n = 5 the posterior associated with x = 2 and x = 0 are plotted in Figure 4.

For x = 2 the posterior is very closed to the posterior Beta distribution β(2.5, 3.5) associated with the Jeffreys prior. These distributions have respective medians 0.4070 and 0.4068. We get for instance the respective 95% equal tails credible intervals [0.0930,0.7894] and [0.0944,0.7906].

For the extreme case x = 0 the posterior introduces a correction to the Jeffreys posterior β(0.5, 5.5). These two distributions have respective medians 0.070 and 0.042. We get for instance the respective 95% equal tails credible intervals [0.0025,0.4046] and [0.00009,0.3794].

Frequentist properties

One of the most common approaches to the evaluation of an objective (or neutral) prior distribution -at first developed by [START_REF] Welch | On formulae for confidence points based on integrals of weighted likelihoods[END_REF] -is to see whether it yields posterior credible sets that have good frequentist coverage properties. Actually, the Jeffreys credible interval has remarkable frequentist properties. Its coverage probability is very close to the nominal level, even for small-size samples, and it can be favorably compared to most frequentist intervals [START_REF] Brown | Interval estimation for a binomial proportion (with discussion)[END_REF][START_REF] Cai | One-sided confidence intervals in discrete distributions[END_REF]. So, [START_REF] Cai | One-sided confidence intervals in discrete distributions[END_REF] concluded: "The results show that the Jeffreys and second-order corrected intervals provide significant improvements over both the Wald and score intervals. These two alternative intervals nearly completely eliminate the systematic bias in the coverage probability. The one-sided Jeffreys and second-order corrected intervals can be resolutely recommended." (p.81).

It can be shown that the correction introduced by the profile criterion prior provides again better frequentist properties than the Jeffreys interval. Let us consider for illustration the same example as Cai (2005, pp. 69-70): the coverage probability of the 99% upper limit confidence interval for a binomial proportion with n = 30. Figure 5 plots the coverage probabilities of the profile criterion interval as a function of θ. In order to compare the performance of the intervals, let us introduce the following "optimal" coverage probabilities. For any value θ, let us define P + , the smaller possible coverage probabilities superior to 99%, and P -, the larger possible coverage probabilities inferior to 99%. P + and P -are the coverage probabilities of the two 99% Bayesian posterior limits respectively associated with the priors β(1, 0) and β(0, 1) (Lecoutre, 2008, p. 790). Note that the first limit is the 99% upper limit of the Wald interval. For any θ, an optimal coverage probability must be either P + or P -. We will conventionally consider as optimal the closer value to 99%. For the Jeffreys and profile criterion intervals, Figure plots the differences between the coverage probability and the optimal coverage probability, showing that the profile criterion leads to superior performance.

Figure 5. Coverage probability of the 99% upper limit (profile criterion prior) for a binomial proportion with n = 30 (top curve) and differences between the coverage probability (Jeffreys and profile criterion intervals) and the optimal coverage probability.

Note that, even for very extreme proportions, the coverage probability remains exceptionally good. So, for all values θ form 0.001 to 0.999 by step of 0.001, the "less good" coverage probabilities are respectively 0.9702 (θ = 0.995) for the Jeffreys prior and 0.9417 (θ = 0.998) for the "profile criterion". By contrast, for the second order corrected interval, we can get the unacceptable coverage probabilities 0.0825, 0.0566 and 0.0291 for θ = 0.997, 0.998 and 0.999.

Conclusion

The new principle of indifference allows us to be consistent with probability axioms besides achieving the long sought-after objective of science: an induction that can be analytically controlled in all its constituent elements. In this way, we actually answer Hume's challenge, and, in the Carnap's words, we can take the degree of confirmation as the basic concept for all of inductive statistics.

This is not to deny the importance of the subjective theory of probability.

In effect, the ideal of total objectivity is unattainable. Yet, the result of the inference can be notified in objective (or analytical) form, not only when prior knowledge is held to be irrelevant, but also in the other cases, making the subjective component of information (that determined the induction) explicit.

Quoting Fisher's words (1955), "we have the duty of formulating, of summarizing, and of communicating our conclusions, in intelligible form, in recognition of the right of other free minds to utilize them in making their own decisions".

Similarly, according to [START_REF] Carnap | Logical Foundation of Probability[END_REF], inductive logic alone does not and cannot determine the best hypothesis on a given evidence, if the best hypothesis means that which good scientists would prefer. It tell them to what degree the hypothesis considered is supported by the observations.

In other words, the most important task of probability calculus in statistics is to provide objective measures of evidence produced by experiments or other sample surveys, with an inference entirely probabilistic.

Anyway, much work still remains to be done in order to implement procedures and build objective prior distributions. Later, it should be opportune to review all the methods of inference, so concluding the induction with a probability distribution, opportunely analyzed for a description (possible exhaustive) of the inductive process.

In conclusion, a necessary and sufficient condition for consistency of scientific inference is not only the agreement with the rules of probability calculus, but also the analytical solution that the new principle of indifference is able to give the problem of statistical induction. In passing, this was the essential requisite of Carnap's work, or, rather, his own goal.

Figure 1 .

 1 Figure 1. Standardized likelihood curves for Binomial mean θ and Jeffreys prior distribution.

Figure 2 .

 2 Figure 2. Standardized likelihood curves for φ = sin -1 √ θ.

Figure 3 .

 3 Figure 3. Profile criterion prior (thick line) and Jeffreys prior β( 1 2 , 1 2 ) (thin line) for a Binomial sample of size n = 5. The top curve is the transformed profile criterion prior for φ = sin -1 √ θ.

Figure 4 .

 4 Figure 4. Posterior distributions associated with the profile criterion (thick line) and Jeffreys priors (thin line) for a Binomial sample of size n = 5 with x = 2 (top figure) and x = 0 successes (bottom figure).

This reasoning leads us to the following definition: If, under d, all hypotheses are equally supported whatever the data may be, we shall say that d is impartial.


P Ryckelynck

L Smoch

On the functional equation αu + C ⋆ (χu) = f

Keywords: Mathematics Subject Classification (2010). 39A06, 39A70, 39A12. Keywords. Functional equations, matricial polynomial functions, matrix sequences

We study in this paper the functional equation

denotes the discrete convolution of an almost zero matricial mapping C with discrete support together with the product of u and the characteristic function χ of a fixed segment. This equation combines some aspects of recurrence equations and/or delayed functional equations, so that we may construct a matricial based framework to solve it. We investigate existence, unicity and determination of the solution to this equation. In order to do this, we use some new results about linear independency of monomial words in matrix algebras.

Introduction

Let [t 0 , t f ] be some interval of time and χ its characteristic function, and ε > 0 a fixed time delay. Let (c k ) k∈Z be an almost zero sequence of matrices in C d×d where d denotes the "physical" dimension. We define next C : R → C d×d by

C(t) = { c -t/ε if t ∈ εZ 0 otherwise
. We denote by I d the identity matrix of size d and we fix throughout a matrix α ∈ C d×d . We have the discrete convolution product over εZ:

(u ⋆ v)(t) = ∑ s∈εZ u(s)v(t -s), t ∈ R
where u and v are two mappings defined in R with matrix values of format

(d × d, d × 1) or (1 × d, d × 1).
This paper is devoted to the study of the following functional equation

αu + C ⋆ (χu) = f (1)
which may be rewritten equivalently as

αu(t) + ∑ k∈Z c k u(t + kε)χ(t + kε) = f (t), (2)
where u, f : R → C d , u being unknown. The sum in the left hand-side of (2) will be denoted throughout the paper by u = C ⋆ (χu), where the operator may occur in the context of discrete calculus of variations (see [START_REF] Ryckelynck | Discrete Calculus of Variations for quadratic Lagrangians[END_REF][START_REF] Ryckelynck | Quadratic choreographies[END_REF]). Equation ( 1) may be thought as a discrete version of the differential equation αu(t) + u ′ (t) = f (t) where u ′ has been replaced by . An analogous functional equation appears in the field of time-scale calculus and has led to many works, see for instance [START_REF] Agarwal | Dynamic equations on time scales: a survey[END_REF][START_REF] Hilger | Differential and Difference Calculus -Unified![END_REF] and the literature therein. However, we do not require here that the operator behaves as a derivative.

We define N as the largest integer such that c N ̸ = 0 or c -N ̸ = 0. For example, when N = 1, equation ( 2) rewrites as

f (t)-αu(t) =                          0 if t < t 0 -ε c 1 u(t + ε) if t 0 -ε ≤ t < t 0 c 1 u(t + ε) + c 0 u(t) if t 0 ≤ t < t 0 + ε c 1 u(t + ε) + c 0 u(t) + c -1 u(t -ε) if t 0 + ε ≤ t ≤ t f -ε c 0 u(t) + c -1 u(t -ε) if t f -ε < t ≤ t f c -1 u(t -ε) if t f < t ≤ t f + ε 0 if t f + ε < t .
(3) In general, the equations (2) may be thought as a mixture between recurrence equations and delayed functional equations.

We focus in this paper on the following two problems. What are the analytical properties of functions of the shape u and is the functional equation [START_REF] Agarwal | Dynamic equations on time scales: a survey[END_REF] well-posed?

The paper is organized as follows. In Section 2 we provide some analytical properties of the functions u. Section 3 presents some technical lemmas for matricial linear recurrences. Section 4 is concerned with the non-vanishing property for entries and determinants of matricial polynomial functions. It ensures that the linear compatibility equations extracted when solving [START_REF] Agarwal | Dynamic equations on time scales: a survey[END_REF] are Cramer for the specific case N = 1. We prove in Sections 5 and 6 that (1) is well-posed if α is invertible, with proper accuracies, according to the respective cases N = 1 and arbitrary N . The case α = 0 is also investigated, and stands for the characterization of the range of the operator . At last, Section 7 gives some concluding remarks and perspectives.

Analytical properties of the functions u

In order to comprehend equation [START_REF] Agarwal | Dynamic equations on time scales: a survey[END_REF] and present some robust methods for solving it, we provide first some features of the operator . Proof. The main ingredient of the proof is the explicit formula for u(t) provided in [START_REF] Ryckelynck | Discrete Calculus of Variations for quadratic Lagrangians[END_REF]:

                                                                             0 if t < t 0 -N ε c N u(t + N ε) if t 0 -N ε ≤ t < t 0 -N ε + ε c N -1 u(t + N ε -ε) + c N u(t + N ε) if t 0 -N ε + ε ≤ t < t 0 -N ε + 2ε . . . . . . c N -p u(t + N ε -pε) + . . . + c N u(t + N ε) if p ∈ {0, . . . , 2N -1} and t 0 + (p -N )ε ≤ t < t 0 + (p + 1 -N )ε . . . . . . c -N u(t -N ε) + . . . + c N u(t + N ε) if t 0 + N ε ≤ t ≤ t f -N ε . . . . . . c -N u(t -N ε) + . . . + c p-N u(t -N ε + pε) if p ∈ {2N -1, . . . , 0} and
t f + (N -p -1)ε < t ≤ t f + (N -p)ε . . . . . . c -N u(t -N ε) + c 1-N u(t -N ε + ε) if t f + N ε -2ε < t ≤ t f + N ε -ε c -N u(t -N ε) if t f + N ε -ε < t ≤ t f + N ε 0 if t f + N ε < t (4)
Assertion 1. is an obvious consequence of formula (4). Next, let u : R → C d , then formula [START_REF] Ryckelynck | Discrete Calculus of Variations for quadratic Lagrangians[END_REF] shows that the (4N + 2) points t 0 + kε and t f + kε, |k| ≤ N may be points of discontinuity of u. Moreover, if t δ is any point of discontinuity of u, the values t δ + kε, |k| ≤ N , give rise in general to points of discontinuity of u. Lastly, since there does not exist any other point of discontinuity of u, assertion 2. is proved. Let us prove assertion 3., i.e. the conditional injectivity of . We first notice help to (4) that if u vanishes in [t 0 , t f ], then all the terms occuring in (4) are equal to 0 so that u is zero everywhere. Conversely, suppose that det(c N ) ̸ = 0. If u ∈ ker( ), second row of formula [START_REF] Ryckelynck | Discrete Calculus of Variations for quadratic Lagrangians[END_REF] shows that u = 0 in [t 0 , t 0 + ε[. Next, by using this result, the third row of (4) shows that u = 0 in [t 0 + ε, t 0 + 2ε[ and so on. Then we may prove easily that u = 0 in [t

0 , t 0 + ε[∪[t 0 + ε, t 0 + 2ε[∪ . . . ∪]t f -ε, t f ].
We may proceed similarly when det(c -N ) ̸ = 0 by starting from the last but one row of (4). As a consequence, if u, v : R → C d , we get

u = v ⇔ u = v in [t 0 , t f ]
which proves the third assertion. We proceed in the very same way to prove that measurability or integrability of u on [t 0 , t f ] is equivalent to the same property for u on R.

Preliminary lemmas for matricial linear recurrences

As we shall see in Sections 5 and 6, solving (1) queries to deal with matricial nonstationary recurrences of the shape

w n+1 = M n w n + g n , n ∈ N, (5)
where M n is a s × s matrix and w n , g n are vector sequences in C s , s being a fixed integer. Since non-commutative products occur in the explicit formula, let us denote by

m ⨿ k=0 M k = M m M m-1 . . . M 0
this left-side product.

Lemma 3.1. The solution to the recurrence ( 5) is given by

w n = ( n-1 ⨿ k=0 M k ) w 0 + n-1 ∑ k=0 ( n-1 ⨿ ℓ=k+1 M ℓ ) g k . ( 6
)
Proof. The cases n = 0 and n = 1 are true and use the conventions of the empty sum and empty (left-)product. The proof for n > 1 is straightforward by using induction. The particular case when M is independent on n is of some importance and gives rise to the solution to stationary matricial recurrence of the shape (5) as follows:

w n = M n w 0 + n-1 ∑ k=0 M n-k-1 g k . ( 7
)
In the following two lemmas, we deal with sequences of non-commutative matricial polynomials.

Lemma 3.2. Let d ≥ 2, β, γ ∈ M d,d (C
) and (δ ′ n ) and (δ ′′ n ) be the sequences of matrices defined by the recurrences

δ ′ n+1 = βδ ′ n + γδ ′ n-1 , δ ′′ n+1 = δ ′′ n β + δ ′′ n-1 γ, n ≥ 1 (8) with δ ′ 0 = δ ′′ 0 = 0 and δ ′ 1 = δ ′′ 1 = I d .
Then, for all n ≥ 2, we have

δ ′ n = δ ′′ n = ∑ β m1 γ m2 β m3 γ m4 . . . ( 9
)
where the sum is extended over all the multiplets (m 1 , m 2 , . . . , m n ) ∈ N n with

m j ≥ 1 if 2 ≤ j ≤ n -1 and [ n 2 ] ∑ i=1 (m 2i-1 + 2m 2i ) = n -1. ( 10
)
The number of monomials in the formula (9) is the Fibonacci number F n .

Proof. First, let us mention that formula (9) is true for n = 0 and n = 1 by using the empty sum convention and the initial conditions. Let us prove now the formula (9) for the sequence (δ ′ n ). Suppose that the result (9) holds for n and n -1 and that δ ′ n-1 and δ ′ n are of lengths F n-1 and F n respectively. Then, help to (9) and (8) we see that δ ′ n+1 is the sum of the words excerpted from the sum δ ′ n multiplied on the left by β and those from the sum δ ′ n-1 multiplied on the left by γ. So we have proved by induction that δ ′ n has the shape

δ ′ n = ∑ e (n,m1,m2,...) β m1 γ m2 β m3 γ m4 . . .
where the coefficients e (n,m1,m2,...) are convenient positive integers independent of β and γ. Let us note that if β = γ = I d then, obviously, δ ′ n = F n × I d so, the number of monomial words in the sum (9) is F n . Let us prove now that all the coefficients e (n,m1,m2,...) are in fact equal to 1. Indeed, we see that all the F n-1 words coming from δ ′ n-1 are distinct from the F n words arising from δ ′ n since the first ones begin with β while the second ones start with γ. All the words δ ′ n+1 occuring in (9) being distinct, their cardinality is F n-1 + F n = F n+1 . So far, we have proved the result (9) for δ ′ n and the proof for δ ′′ n is the same. In the following, we set

δ n = δ ′ n = δ ′′ n , ∀n ∈ N. Lemma 3.3. Let d ≥ 2, β, γ ∈ M d,d (C), (g n ) n ∈ (C d ) N
, and let (w n ) n ∈ (C d ) ∈ N the vector sequence satisfying for all n ≥ 1 the following two-order matricial recurrence equation:

w n+1 = βw n + γw n-1 + g n . ( 11
)
Then for all n ∈ N ⋆ we have

w n = δ n w 1 + δ n-1 γw 0 + n-1 ∑ k=1 δ n-k g k . ( 12
) Proof. Let us define M = ( β γ I d 0
)

. We may easily prove by induction that

M n = ( δ n+1 δ n γ δ n δ n-1 γ
) , for all n ∈ N ⋆ , by using the recurrence (8) for the sequence (δ n ). Next, (11) may be rewritten as

( w n+1 w n ) = M ( w n w n-1 ) + ( g n 0 ) . By setting w ′ n = ( w n w n-1 ) , g ′ n = ( g n 0 )
and by using a slightly adapted version of formula (7), we get 12) is obtained by reading the first row of the previous formula.

w n = M n-1 w 1 + ∑ n-1 k=1 M n-1-k g k . Now, the formula (

Linear independency of words in matrix algebras

We shall obtain a result which ensures that the foregoing linear compatiblity equations which arise when solving the functional equation ( 1) are Cramer for almost all choice of matrices α

, c -1 , c 0 , c 1 ∈ C d×d if N = 1.
Let us call a matricial polynomial function of two matricial indeterminates (ζ, ξ) an expression of the following shape

ψ(ζ, ξ) = ∑ κ m1,m2,... ζ m1 ξ m2 ζ m3 ξ m4 . . . ( 13
)
with coefficients in C. As an example, we may cite the sequence of matrices occuring in Lemma 3.2. Given such a function, for all (p, q) ∈ N 2 , we denote by K pq the sum of all the coefficients κ m1,m2,... such that m 1 + m 3 + . . . = p and m 2 + m 4 + . . . = q. Similarly, for all (p, q, r, s) ∈ N 4 , we introduce the sum Kp,q,r,s of all the coefficients m 1 × m 2 × . . . × κ m1,m2,... such that m 1 +m 3 +. . . = p, m 2 +m 4 +. . . = q, and where r and s count the occurences of ζ and ξ respectively in the monomial of (13) associated to the multiplets (m 1 , m 2 , . . .). We must have necessarily |r -s| ≤ 1. We shall say that the function ( 13) is generic when at least one sum K pq and one sum Kp,q,r,s are nonzero. For instance, the matrices δ n occuring in (9) are the specializations at (β, γ) of generic polynomials, since the coefficients are either 1 or 0.

Let us give now the main result of this section.

Theorem 4.1. Suppose we are given l generic matricial polynomial functions ψ (k) (ζ, ξ), and let us set ψ

(k) i,j (ζ, ξ) = (ψ (k) (ζ, ξ)) ij . Then, the set of couples (ζ, ξ) such that ∀i, j, k, ψ (k) i,j (ζ, ξ) ̸ = 0, with 1 ≤ i, j ≤ d, 1 ≤ k ≤ l, and ∀k, det(ψ (k) (ζ, ξ)) ̸ = 0, with 1 ≤ k ≤ l consists in an open dense subset of (C d×d ) 2 .
Proof. In order to prove this result, we first show that if ψ(ζ, ξ) is the generic matricial polynomial function defined by (13), then all its entries ψ i,j (ζ, ξ) are nonzero polynomials w.r.t. their 2d 2 indeterminates. The proof goes as follows. Let us specialize ψ at matrices of the shape

ζ =   x z 0 1,d-2 0 x 0 1,d-2 0 d-2,1 0 d-2,1 I d-2   and ξ =   y t 0 1,d-2 0 y 0 1,d-2 0 d-2,1 0 d-2,1 I d-2   for convenient x, y, z, t ∈ C. We easily get ψ(ζ, ξ) =   ψ(x, y) ψ(x, y, z, t) 0 1,d-2 0 ψ(x, y) 0 1,d-2 0 d-2,1 0 d-2,1 ψ(1, 1)I d-2   ( 14
)
where ψ is the function (13) evaluated at complex numbers and is equal to ∑ K p,q x p y q while ψ is the polynomial ∑ Kp,q,r,s x p-r y q-s z r t s . Due to the hypotheses, the sets of couples (x, y) or quadruplets (x, y, z, t) such that ψ(x, y) ̸ = 0 or ψ(x, y, z, t) ̸ = 0 respectively, are open everywhere dense subsets in C 2 and in C 4 respectively. Now, in order to deal with the zero-like entries of ( 14), we use the following property of ψ. We have

ψ(g -1 ζg, g -1 ξg) = g -1 ψ(ζ, ξ)g,
where g is any invertible matrix in C d×d . Especially, if we choose g as a permutation matrix µ σ , we obtain

ψ ij (µ -1 σ ζµ σ , µ -1 σ ξµ σ ) = ψ σ(i),σ(j) (ζ, ξ).
The symmetric group S d acts transitively on the set of couples (i, i) with 1 ≤ i ≤ d and transitively on the set of couples (i, j) with 1 ≤ i, j ≤ d and i ̸ = j. Thus, since ψ 1,1 and ψ

)) = ( ∑ p,q K p,q ρ p σ q ) d = ψ(ρ, σ) d which is nonzero in C[ρ, σ] since there exists at leat one coefficient K pq ̸ = 0.
Collecting all these informations, we may work with open dense sets in C 2d 2 instead of dealing with nonzero polynomials. Let us consider for all k ∈ {1, . . . , l} and i, j ∈ {1, . . . , d} the closed algebraic varieties of codimension 1

F (k) ij = {(ζ, ξ) ∈ (C d×d ) 2 /ψ (k) ij (ζ 11 , . . . , ξ dd ) = 0} and F (k) det = {(ζ, ξ) ∈ (C d×d ) 2 / det(ψ (k) (ζ 11 , . . . , ξ dd )) = 0}.
Since each hypersurface is meager, the union (

∪ ijk F (k) ij ) ∪ (∪ k F (k)
det ) is a closed meager subset of (C d×d ) 2 . Thus, its complementary is an everywhere open dense subset of (C d×d ) 2 . This ends the proof of the result. is an open dense subset of (C d×d ) 2 . In the very special case when the matrices β, γ are multiples of I d , i.e. β = βI d and γ = γI d with β and γ in C, we easily find help to (8) that

δ n = 1 √ β2 +4γ (( β+ √ β2 +4γ 2 ) n - ( β- √ β2 +4γ 2 ) n ) I d .
The set of couples ( β, γ) such that

β+ √ β2 +4γ β- √ β2 +4γ
is not of the shape exp

( 2ikπ n ) is open dense in C 2 and

Solving explicitly the functional equation when N = 1

Let us show now how to solve the functional equation ( 1) when N = 1 by using the preceding results. We restrict ourselves to the quite opposite cases α invertible or α = 0. Let us define M ≥ 1 as the integer part of

t f -t0 ε . Theorem 5.1. If N = 1, for all quadruplets (c -1 , c 0 , c 1 , α) in an open dense subset of (C d×d ) 4 , the equation αu + C ⋆ (χu) = f
admits one and only one solution.

Proof. We may assume by density that c 1 is invertible. If M = 1, the result follows easily from the inspection of (3). The main part of the proof consists in solving (3) and is valid for all α. Outside [t 0 -ε, t f + ε], one has f (t) = αu(t) which allows to determine u if α is invertible. Next, we focus on the interval [t 0 + ε, t f -ε]. By using (2), we get the recurrence equation:

u(τ +(n+1)ε) = c -1 1 (-(α+c 0 )u(τ +nε)-c -1 u(τ +(n-1)ε)+f (τ +nε)), ( 16
)
∀τ ∈ [t 0 , t 0 + ε[ and n ≥ 1. So we see that ( 16) is a recurrence of the shape (11). Using Lemma 3.3 with

β = -c -1 1 (α + c 0 ), γ = -c -1 1 c -1 , ( 17
)
and w n = u(τ + nε), g n = c -1 1 f (τ + nε) for τ fixed, we get

u(τ + nε) = δ n u(τ + ε) + δ n-1 γu(τ ) + n-1 ∑ k=1 δ n-k c -1 1 f (τ + kε), ( 18
)
for all instant τ +nε ≤ t f . Since τ ∈ [t 0 , t 0 +ε[, we have in any case n ≤ M -1.

More explicitly, we shall use (18) in the following ranges of indices:

τ ∈ [t 0 , t f -M ε[⇒ max{n/τ + nε ≤ t f } = M , (19) τ ∈ [t f -M ε, t 0 + ε[⇒ max{n/τ + nε ≤ t f } = M -1. ( 20
)
We find it convenient to denote by ℓ = ℓ(τ ) the maximal value of n for which we may use (18), given by ( 19) or (20). Although the formula (18) is true for n = 1, this result is tautological in this case. For n ≥ 2, (18) expresses the solution to the functional equation ( 1) in the interval [t 0 + 2ε, t f ], as a linear combination of the restrictions u(τ ) and u(τ + ε) of u to the two intervals [t 0 , t 0 + ε[ and [t 0 + ε, t 0 + 2ε[. The coefficients of this linear relationship are independent on time and constitute the sequence of matrices (δ n ) which depend only on β and γ. Now, let us determine the two additional unknown functions u(τ ) and u(τ + ε) by solving (2) near the boundaries t 0 and t f . In order to do this, let us write explicitly the four remaining equations:

if t ∈ [t 0 -ε, t 0 [, f (t) = αu(t) + c 1 u(t + ε), (21) if t ∈ [t 0 , t 0 + ε[, f (t) = (α + c 0 )u(t) + c 1 u(t + ε), ( 22
) if t ∈]t f -ε, t f ], f (t) = (α + c 0 )u(t) + c -1 u(t -ε), ( 23
) if t ∈]t f , t f + ε], f (t) = αu(t) + c -1 u(t -ε). ( 24
)
We note that the restriction of u to 22) and (25) give

u(t) = c -1 1 f (t -ε) -c -1 1 αφ(t -ε). ( 25
) If t ∈ [t 0 + ε, t 0 + 2ε[, (
u(t) = c -1 1 f (t -ε) -c -1 1 (α + c 0 )c -1 1 f (t -2ε) + c -1 1 (α + c 0 )c -1 1 αφ(t -2ε
). (26) The formulas (25) and (26) give the two additional restrictions of u needed in the expansion (18). We note that these restrictions, at the time being, depend on the auxiliary restriction φ of u. Now it remains to solve (23) and (24). First, the two functions occuring in the r.h.s. of the equation ( 23) are already known and may be expressed through (18). The equation ( 23) may be thought as a constraint either on φ or on the l.h.s. f of (1). We may observe that φ(τ -ε) occurs two times in u(t) and two times in u(t -ε). So, the coefficient we are looking for is equal to

θ = ((α + c 0 )δ ℓ + c -1 δ ℓ-1 )coeff of φ in u(τ + ε)] +((α + c 0 )δ ℓ-1 + c -1 δ ℓ-2 )γ[coeff of φ in u(τ )],
where ℓ = M or ℓ = M -1, depending on the location of τ w.r.t. t f -M ε. Direct inspection of these coefficients from (25) and (26) yields the explicit expression of the coefficient of φ in (23):

θ = ((α + c 0 )δ ℓ + c -1 δ ℓ-1 )(c -1 1 (α + c 0 )c -1 1 α) +((α + c 0 )δ ℓ-1 + c -1 δ ℓ-2 )γ(-c -1 1 α) (27) = [((α + c 0 )δ ℓ + c -1 δ ℓ-1 )c -1 1 (α + c 0 ) -((α + c 0 )δ ℓ-1 + c -1 δ ℓ-2 )γ]c -1 1 α.
Now, in order to simplify the matrix θ, we use the formulas (17) to eliminate α + c 0 and c -1 and we get

θ = c 1 ((βδ ℓ + γδ ℓ-1 )β + (βδ ℓ-1 + γδ ℓ-2 )γ)c -1 1 α.
By using three times the recurrence (8), we obtain

θ = c 1 δ ℓ+2 c -1 1 α.
At the time being, no assumption has been made about α. From now on, we consider the case when α is invertible and chosen in such a way that det(δ ℓ+2 ) ̸ = 0 (see (17) and Remark 4.1). Therefore, det(θ) ̸ = 0. Hence, (23) may be solved w.r.t. φ(τ -ε). As a by-product of the previous results, φ(τ -ε) is a linear combination of the values of f on the set {τ + kε |k| ≤ N }. It remains to solve equation ( 24) on ]t f , t f + ε]. We get simply u(t) = α -1 (f (t)c -1 u(t-ε)) for all t ∈]t f , t f +ε], where the r.h.s. has already been determined. Then we have proved that, for a generic quadruplet (c -1 , c 0 , c 1 , α) and for all f : R → C d , there exists one and only one solution to the functional equation [START_REF] Agarwal | Dynamic equations on time scales: a survey[END_REF].

Next, let us consider the case when α is zero, which stands for the characterization of Im( ).

Theorem 5.2. For all triplets (c -1 , c 0 , c 1 ) in an open dense subset of (C d×d ) 3 and for all g : [t 0 , t f ] → C d , there exists one and only one extension f of g to R such that Supp(f ) ⊂ [t 0 -ε, t f + ε] and the functional equation f = u admits one and only one solution in [t 0 , t f ].

Proof. We may assume by density that c 1 and c -1 are invertible. Let be given a function g : [t 0 , t f ] → C d and let us construct an appropriate extension f : R → C d of g as follows. We set

f (t) =                0 if t < t 0 -ε p(t) if t 0 -ε ≤ t < t 0 g(t) if t 0 ≤ t ≤ t f q(t) if t f < t ≤ t f + ε 0 if t > t f + ε
for some auxiliary functions p and q with values in C d . The proof consists in showing that f lies in the range of if and only if p and q are uniquely determined by g. We keep the notations of the previous proof. When α = 0, the formulas ( 16) to (26) still hold, we have θ = 0 and the auxiliary function φ does not occur anymore. Let us solve the two equations ( 23) and (24) w.r.t. the two restrictions p and q of f . To write these equations, we introduce the operators

∆ ℓ f (τ ) = ℓ-1 ∑ k=1 δ ℓ-k c -1 1 f (τ + kε).
Let us notice that these operators may be thought as discrete integral operators. Plugging formula (18), with the maximal value ℓ allowed for n, in the equation ( 23), we get for all t ∈]t f -ε, t f ]

f (t) = c 0 (δ ℓ u(τ + ε) + δ ℓ-1 γu(τ ) + ∆ ℓ f (τ ))+ c -1 (δ ℓ-1 u(τ + ε) + δ ℓ-2 γu(τ ) + ∆ ℓ-1 f (τ )),
t and τ being connected as previously by the requirements that τ ∈ [t 0 , t 0 +ε[ and t-τ ε ∈ N. Remembering the values of β, γ given by ( 17) and the recurrence (8), we get:

f (t) = -c 1 (δ ℓ+1 u(τ + ε) + δ ℓ γu(τ )) + c 0 ∆ ℓ f (τ ) + c -1 ∆ ℓ-1 f (τ ). ( 28
)
Since we may respectively rewrite (25) and (26) as

u(τ ) = c -1 1 p(τ ) and u(τ + ε) = c -1 1 f (τ ) -c -1 1 c 0 c -1 1 p(τ ),
we see that the coefficient of p = f (τ -ε) in ( 28) is given by:

(δ ℓ+1 c -1 1 c 0 -δ ℓ γ)c -1 1 = -δ ℓ+2 c -1 1 .
We assume the triplet (c -1 , c 0 , c 1 ) lies in the open dense subset where det(δ ℓ+2 ) ̸ = 0 and thus, we have shown that the restriction p exists and is unique. This being done, the equation (23) yields

f (t + ε) = q(t) = c -1 (δ ℓ u(τ + ε) + δ ℓ-1 γu(τ ) + ∆ ℓ f (τ )),
t and τ being connected as before, which is well-determined. As a consequence, the formula (18) gives rise to one unique function u on [t 0 , t f ] such that u = f and this ends the proof.

A matricial-based framework when N is arbitrary

Let us use a matricial-based framework for solving the functional equation for arbitrary N . In this section we assume throughout that c N and α are invertible, mainly for the ease of exposition. We shall say that the (2N + 2)tuple (c -N , . . . , c 0 , . . . , c N , α) is generic in (C d×d ) 2N +2 provided the foregoing system (39) is Cramer. This system (39) consists in N equations w.r.t. N unknowns, generalizing the auxiliary function φ occuring in Section 5. Its coefficients are matricial polynomial functions w.r.t. c -1 N (c 0 + α) and the 2N -1 matricial indeterminates c -1 N c k , k ∈ {-N, . . . , N -1} and k ̸ = 0, which in a sense generalize (13).

Theorem 6.1. If the (2N + 2)-tuple (c -N , . . . , c 0 , . . . , c N , α) is generic, the functional equation ( 1) admits one and only one solution.

Similarly, we shall say that the (2N + 1)-tuple (c -N , . . . , c 0 , . . . , c N ) is generic in (C d×d ) 2N +1 provided the foregoing matrix Θ defined by (40) is invertible. This matrix is associated to a linear system of N equations w.r.t. N unknowns p 1 , . . . p N , which are the restrictions of f to parts of [t 0 -N ε, t 0 [ in order that f lies in the range of the operator . Theorem 6.2. If the (2N + 1)-tuple (c -N , . . . , c 0 , . . . , c N ) is generic, for all g : [t 0 , t f ] → C d there exists one and only one extension

f : R → C d such that Supp(f ) ⊂ [t 0 -N ε, t f + N ε]
B(t) =      c -1 N f (t) 0 . . . 0      ∈ C 2dN , U(t) =          u(t + (N -1)ε) . . . u(t) . . . u(t -N ε)          ∈ C 2dN , ( 29)
A(t) =          -c -1 N c N -1 χ(t + (N -1)ε) . . . -c -1 N (α + c 0 χ(t)) . . . -c -1 N c -N χ(t -N ε) I d 0 . . . . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 I d 0          . ( 30
)
The matrix A(t) ∈ C 2dN ×2dN is locally constant w.r.t t. Especially, when t ∈ [t 0 + N ε, t f -N ε], all the characteristic functions of the entries of A(t) are equal to 1 and we shall agree to denote by A the constant value of A(t) in this segment. Let us mention that the matrix A depends only on the 2N + 2 matrices c i and α and is nothing but the block companion matrix of the functional equation (1).

When t < t 0 -N ε or t > t f +N ε, we have already seen that the solution to the equation ( 1) is given by u

(t) = α -1 f (t). Now, if t ∈ [t 0 -N ε, t f -N ε],
(2) may be written as

u(t + N ε) = c -1 N [f (t) -(c N -1 χ(t + (N -1)ε)u(t + (N -1)ε) + . . . +(α + c 0 χ(t))u(t) + . . . + c -N χ(t -N ε)u(t -N ε))]. ( 31
)
When t ∈ [t 0 -N ε, t 0 [, this equation expresses u(t + N ε) explicitly as a combination of the N auxiliary functions φ k (τ ) = u(τ -kε), 1 ≤ k ≤ N , which are at this level unknown. Obviously, the functions φ k (τ ), 1 ≤ k ≤ N , stand for the last N components of U(τ ). By using definitions (29) and (30), recurrence (31) may be rewritten as

U(t + ε) = A(t)U(t) + B(t), ∀t ∈ [t 0 -N ε, t f -N ε].
The preceding recurrence and Lemma 3.1 yield the following formula

U(t + nε) = C n (t)U(t) + F n (t) (32) for t 0 -N ε ≤ t ≤ t + (n -1)ε ≤ t f -N ε, where C n (t) =            n-1 ⨿ k=0 A(t + kε) if t + (n -1)ε < t 0 + N ε, A n-N ( N -1 ⨿ k=0 A(t + kε) ) otherwise (33)
and

F n (t) = n-1 ∑ k=0 ( n-1 ⨿ ℓ=k+1 A(t + ℓε) ) B(t + kε). ( 34
)
Focusing on the first row of (32), i.e. the first component u(t + (n + N -1)ε) of U(t + nε), we emphasize on the fact that the unknown function u(t) may be expressed as a linear combination of the delayed functions f (t + kε), k ∈ Z through F n (t), as well as the auxiliary functions φ k (τ ), 1 ≤ k ≤ N , through U(t). So, as a first conclusion of these calculations, u(t) is well-determined in the whole interval ] -∞, t f ], as a combination of the auxiliary functions

φ k . When t ∈ [t f -N ε, t f +N ε],
we may deduce from (1) and ( 4) the two systems of N functional equations which are analogous to (23) and (24), i.e.

             f (t) = αu(t) + c -N u(t -N ε) + . . . + c N -1 u(t + (N -1)ε) in ]t f -N ε, t f -(N -1)ε] f (t) = αu(t) + c -N u(t -N ε) + . . . + c N -2 u(t + (N -2)ε) in ]t f -(N -1)ε, t f -(N -2)ε] . . . f (t) = αu(t) + c -N u(t -N ε) + . . . + c 0 u(t) in ]t f -ε, t f ] (35)
and

             f (t) = αu(t) + c -N u(t -N ε) + . . . + c -1 u(t -ε) in ]t f , t f + ε] f (t) = αu(t) + c -N u(t -N ε) + . . . + c -2 u(t -2ε) in ]t f + ε, t f + 2ε] . . . f (t) = αu(t) + c -N u(t -N ε) in ]t f + (N -1)ε, t f + N ε] . ( 36
)
Let us consider the case when α is invertible. We convert (35) into a linear system for the N auxiliary unknown functions φ k . To do this, we note first that (32), with n = M and t = τ -N ε, writes as

U(τ + (M -N )ε) = C M (τ -N ε)U(τ -N ε) + F M (τ -N ε). ( 37
)
The first N components of U(τ -N ε) which occur in the right-hand side of the previous equality are the auxiliary functions while the last N components are values of the function α -1 f at instants τ -

(N + k)ε, 1 ≤ k ≤ N .
Next, by shifting adequately t in the system of equations (35), we get for all

t ∈ ]t f -N ε, t f -(N -1)ε] the following equality      f (t) f (t + ε) . . . f (t + (N -1)ε)      = D    u(t + (N -1)ε) . . . u(t -N ε)    (38)
where

D =          cN-
(α + c0) c-1 . . . c-N 0 . . . . . . 0          ∈ C N d×(2N d)
is a rectangular block Hankel matrix. By setting t = τ + (M -N )ε in (38) and using (37), we easily get

DC M (τ -N ε)U(τ -N ε) =    f (τ + (M -N )ε) . . . f (τ + (M -1)ε)    -DF M (τ -N ε). (39) By partitioning the (N d) × (2N d) matrix DC M (τ -N ε) as (Θ 1 Θ 2 ), where Θ 1 , Θ 2 ∈ C N d×N d
, we see that the previous linear system consists in N vectorial equations in C d depending on the N auxiliary functions φ k and is Cramer provided that Θ 1 is invertible. Under the assumption of genericity, the previous condition is satisfied and thus, there exists one and only one solution to (39). The auxiliary functions being determined, as a rule, the function u is well defined and unique on ] -∞, t f ]. Since α is also invertible by assumption, we may compute the various extensions of u on [t f , t f + N ε] using the system (36). This ends the proof of Theorem 6.1. The proof of the second theorem proceeds in the same way. Let p k be the unknown restriction of f to [t 0 -kε, t 0 -(k -1)ε[, for all k ∈ {1, . . . , N }. Similarly let us define q k as the unknown restriction of f to ]t f +(k -1)ε, t f + kε], for all k ∈ {1, . . . , N }. When α is zero and det(c N ) ̸ = 0, the system (4) shows that u(t) may be arbitrarily chosen outside [t 0 , t f ]. Indeed, the auxiliary functions φ k do not occur in the calculation of u. However, u is welldetermined in the whole interval [t 0 , t f ] provided g is given in this interval and the p k , 1 ≤ k ≤ N , are already determined. In order that u satisfies the systems (35) and (36), the restrictions p k and q k , 1 ≤ k ≤ N , must be chosen adequately. Let us focus on (35) or, equivalently, on (39). Indeed, both systems still hold when α = 0. However, our approach is different from previously since the unknowns which are the p k 's are located exclusively in the term -DF M (τ -N ε). Indeed, the above discussion shows that U(τ -N ε) does not depend on the restrictions p k nor the matrix (f (τ +(M -k)ε)) 1≤k≤n . Let us determine the coefficient [Θ] ij ∈ C d×d of p j (τ ) in the i-th equation in the system (39). To do this, we use the convention that for some matrix Θ ∈ C (N d)×(N d) , the symbol [Θ] ij denotes the (d × d)-block located at the (i, j)-entry of Θ. Then, help to formulas (29) and (34), we see that

[Θ] ij = -   D M -1 ⨿ k=N -j+1 A(τ + (k -N )ε)   i1 c -1 N . ( 40
)
We assume that the multiplet (c -N , . . . , c N ) is generic, which amounts to say that Θ is invertible. Therefore, (39) is Cramer and admits one and only one solution (p 1 , . . . , p N ). Next, by using the same transformation which led to (38), the system (36) may be rewritten as

   f (t) . . . f (t + (N -1)ε)    =    q1(t) . . . qN (t)    =      c-1 c-2 . . . c-N c-2 . . . c-N 0 . . . . . . . . . . . . c-N 0 . . . 0         u(t -ε) . . . u(t -N ε)   ,
for all t ∈]t f , t f + ε]. Since the matrix (u(t -kε)) 1≤k≤N is known, we obtain as a conclusion that the restrictions (p k , q k ), 1 ≤ k ≤ N , are uniquely determined by g, which ends the proof of Theorem 6.2.

Final comments

Before concluding this paper, it is natural to ask if there exist other ways to handle the functional equation αu + C ⋆ (χu) = f . Since (1) is a convolution equation, one might want to use the discrete Laplace (or Z-) transform to solve it. Indeed, if the Z-transform is defined by Lu(p) = ∑ t∈εZ exp(-pt)u(t) then one has the formula L(u ⋆ v)(p) = (Lu)(Lv) for all functions u and v on the real line, with compact support and values in suitable algebras. So, when α = 0, the image of (1) is L(C)L(χu) = L(f ) and, thus, if L(C) is invertible, we get

L(χu)(p) =

(∑ k∈Z c k exp(pkε)

) -1 (∑ k∈Z exp(-pkε)f (kε)

) .

Unfortunately, the determination of the original u has been transferred to a difficult problem of moments. Moreover, this approach does not apply to the general case α ̸ = 0. Another way to proceed would be to apply the Picard's method. Indeed, one sees that (1) is equivalent to

u = α -1 f -α -1 u = α -1 f -α -1 (α -1 f -α -1 u) = . . . = α -1 f + ∑ ∞ k=1 (-1
) k α -k-1 k f . Thus if ∥α -1 ∥ is small enough, one may hope that is a contraction of a suitable Banach or Fréchet function space F. The main difficulty in this approach is the requirement that the vector space F is complete, and it is related to the increasing number of points of discontinuity of k f , as k increases. By comparison, the approach developed in this paper is less complicated and more robust in the sense that the solution is given explicitly. Indeed, the central parts of the proofs of the four previous theorems highlight not only the well-posedness of (1) but also an effective computational way to construct interval by interval the solution u.

In scope of future work, we suggest the three following problems. First, the formulas (32) and (33) imply that, in the very special case when f is constant, u(t) may be expressed through exponential-monomial functions inside [t 0 + N ε, t f -N ε]. So, in this case, the search for pseudo-periodic solutions of (1) is linked to the difficult elimination problem of requiring that the spectrum Sp(A) of A is included in the unit circle of C. Second, we conjecture the following variant of Theorem 4.1. If the coefficients κ m1,m2,... occuring in formula (13) are rational or even algebraic over Q and, in contrast, the entries ζ i,j and ξ i,j generate a subfield of C of transcendance degree equal to 2d 2 , then all the quantities ψ (k) i,j (ζ, ξ) and det(ψ (k) )(ζ, ξ) do not vanish. Lastly, we conjecture also that for all N ≥ 2, the genericity assumption on the multiplets in Theorems 6.1 and 6.2 hold on open dense subsets of appropriate products of matricial algebras. This conjecture seems natural in light of the two theorems presented in Section 5.

Remark 4 . 1 .

 41 As an example, for all integer m, the set of couples (β, γ) ∈ (C d×d ) 2 such that det(β) ̸ = 0, det(γ) ̸ = 0, βγ ̸ = γβ and det(δ n ) ̸ = 0, ∀n ∈ {0, . . . , m}, (15)

  and the functional equation f = u admits one and only one solution in [t 0 , t f ]. Proof. (Of both theorems.) Let u : R → C d be a solution to equation (1), related to A, B and U defined over R as follows:

  Theorem 2.1. Let ν ∈ N ∪ {∞}. 1. The operator maps the space of the functions u : R → C d of piecewise-C ν regularity, to the space of compactly supported functions v : R → C d of piecewise-C ν regularity, such that Supp(v) ⊂ [t 0 -N ε, t f + N ε]. 2. If u : R → C d has p points of discontinuity on R, then u has at most (4N + 2) + (2N + 1)p points of discontinuity on R. 3. If c N or c -N is invertible then the kernel of consists of the functions which vanish on [t 0 , t f ]. 4. If c N or c -N is invertible then, for all u : R → C d , u is measurable, (respectively integrable, of piecewise-C ν regularity) on R if and only if u is measurable (respectively integrable, of piecewise-C ν regularity) on [t 0 , t f ].

  , for those couples, we have det(δ n ) ̸ = 0.Remark 4.2. The set of generic polynomials is an open dense subset in the vector space of matricial polynomial functions of given degree. Remark 4.3. Let us mention how to proceed if we deal with non-generic polynomials. We may use in this case the so-called defect Theorem of P.M.

Cohn, see

[START_REF] Lothaire | Algebraic Combinatorics On Words[END_REF] Theorem 9.6.1, p. 283

]. If K is a commutative field and Ω is an alphabet of non commuting variables, we may denote by K⟨⟨Ω⟩⟩ the algebra of non commutative formal series in these variables. Then the defect theorem states that if φ 1 , φ 2 are elements of K⟨⟨Ω⟩⟩ without constant term, satisfying a non-trivial relation Φ(φ 1 , φ 2 ) = 0 for some non commutative series Φ in two variables then φ 1 , φ 2 commute. This theorem allows to exhibit at least one nonzero polynomial entry to ψ(ζ, ξ), but the conclusion is not as so accurate than in Theorem 4.1.