Fabien Szymkiewicz

Carlos Minatchy

Philippe Reiffsteck

Keywords: Deep Foundations, Piles, Load Test, Instrumentation, Monitoring, Optical Fiber

The use of optical fiber for measuring strain and or temperature is increasing in civil engineering. Optical fiber is known to be relatively cheap, compared to the more traditional sensors commonly used, and can, depending on the chosen technology, give a complete distribution along the instrumented element, which can be very interesting in case of an element with variable dimensions along the length of said element. Therefore, for static pile loading tests, the use of optical fiber should present many advantages, compared to more conventional methods of strain measurements. The aim of this paper is to shed new light on the advantages and disadvantages of using optical fiber for the measurement of deformations along a foundation element. After a review of the state of the art of the most commonly used methods, the contribution of the optical fiber for the instrumentation of the foundation elements is evaluated, by comparing the results obtained by the use of the fiber with those obtained with more traditional methods, in some examples of full-scale load tests. Finally, some recommendations are given to maximize the awareness of the potential use of optical fiber in foundation monitoring.

INTRODUCTION

Static pile load test are usually carried either for design purposes, whether it be for control of this design or design by trial, or for constitution of a database then used to elaborate design methods. Therefore, they are critical tests for geotechnical structures and shall be thoroughly prepared.

As such, they are covered by complete and well detailed test standards [START_REF] Szymkiewicz | Static Pile Load Test: International Practice Review And Discussion About The European And Japanese Standards[END_REF]. Depending on the purpose of the test, the pile may or may not be instrumented. Most instrumentation installed inside piles measures the strain in the pile. Based on the geometry of the pile and the apparent modulus of the material of the pile, force can be derived from strain. Therefore, force-depth diagrams can be drawn for the instrumented piles (Fig. 1). Shaft resistance and base resistance can then be dissociated, and unit shaft friction can be assessed for each soil layer encountered, depending on the number of measurement points available along the length of the pile. t-z curves, linking the development of the unit shaft friction of a segment of pile to the displacement of this segment, can then be drawn, as well as q b -z curves. For lateral load tests, the principle is the same, except that it is p-y curves that are determined.

A number of different methods exist for the measurement of these strains, each with its advantages and disadvantages. Among these methods is the optical fiber: since approximately a decade, a number of geotechnical projects have included these technologies for monitoring, including piles [START_REF] Soga | The role of distributed sensing in understanding the engineering performance of geotechnical structures[END_REF]- [START_REF] Klar | Distributed strain measurement for pile foundations[END_REF]- [START_REF] Mohamad | Investigation of shaft friction mechanisms of bored piles through distributed optical fibre strain sensing[END_REF].

This paper aims to first outline the most common methods for the measurement of strains along the shaft of a pile, listing in the process the pros and cons of these methods. Then the use optical fiber for the testing of geotechnical structures, and particularly piles is outlined. The typical difficulties encountered and the benefits of the use of the optical fiber for geotechnical purposes are then highlighted, through practical examples. Conclusions are then drawn, and some innovative uses of optical fiber for the monitoring of foundations are proposed.

TRADITIONNAL METHODS FOR THE MEASUREMENT OF STRAINS ALONG A PILE

In this section, the three main methods for the measurement of strains along the shaft of a pile are described in details, and their technical, practical and economical strengths and weaknesses are listed.

Embedded Strain Gauges

This method is one of the oldest: it consists in placing, at defined depth, strain gauges that will be bonded to the steel cage reinforcing bars (one or more per depth) or pipe.

The advantages of this method are multiple: -this is a proven technology, with which many practitioners are familiar, -it is not very intrusive due to the small size of the gauges and the wires, which is a good point for small diameter piles, -it is quite affordable, -each sensor is independent from the others, which is important when encountering some damages, -each sensor can be compensated in temperature. However, there are a number of disadvantages when using this technology:

-It is time consuming to implement, as there are multiple steps to follow the whole process of preparation and bonding: first, flat rough surfaces should be ground on the rebars, then should be smoothed using finer sandpaper. Surface has then to be cleaned of any impurity using chemical products. The strain gauge may then be bonded to the surface (Fig. 2), using another chemical product, and two layers of protection should then be added on top of the gauge. If the gauges are not precabled, then an extra step will be the soldering of the cables to the gauge. Then, if necessary, soldering should be done for any extra cable length added, -It necessitates multiple tools and chemical products, some of them having to be kept refrigerated, and a clean environment, -Wires and gauges are very small, and may be damaged during the process of introducing the reinforcing cage inside the concrete (for CFA piles or equivalent), during the concreting phase (for bored piles), or during the driving phase (driven piles).

Fig. 2 Naked strain gauge bonded to the rebar

Embedded Vibrating Wire Strain Gauges

This method is very close to the first one: vibrating wire sensors are attached to the reinforcing bars of the steel cage at different depths, with one or more per depth.

The advantages of this method are as follow:

-the sensors are very robust (made of steel), -the sensors are not very intrusive (usual length is around 150 mm for a diameter of 25 mm, but can go up to 400 mm in length for particular applications), -the sensors need only to be attached to the rebars with adhesive rubber tape or plastic zip ties (Fig. 3a), -the sensors can be bought pre-cabled to the needed length, -cables are strong enough to resist the tension during the different phases of manipulation and realization of the piles.

-The sensors are very quick to fix, even if the length of the cables (especially when working on long and / or heavily reinforced piles) can hinder the work and make the intervention longer, -each sensor is independent from the others, -each sensor can be compensated in temperature. On the other hand, this method presents some disadvantages:

-for piles heavily instrumented, the number and thickness of the cables can make this method intrusive, especially for small diameter piles (Fig 3b), -it can be expensive, when using a great number of sensor and instrumenting long piles, as the cost per sensor and per meter of cable is not negligible.

Retrievable extensometers

This method is a totally different one. It consists in a chain of sensors that will have to be inserted inside the pile after it realization. Technologies used in retrievable extensometers are usually strain gauges, vibrating wires strain gauges, LVDT or potential sensor based.

The principle is as follows: one or more reservations made of PVC or steel (which can be sonic tube, for example) are cast in the pile. Just before the load test begins, the retrievable extensometer, made of multiple segments whose lengths can vary to adjust to the geology on site, is inserted inside the pipe [START_REF] Ponts | La mesure des déformations à l'aide des extensomètres amovibles LPC : Méthodes d'essai LPC[END_REF]. When completely inserted and correctly positioned, it is locked into position, using either a pneumatic (with nitrogen being the gas expanding the anchors) or a mechanical system (Fig. 4).

The advantages of this method are as follows:

-it doesn't need the intervention of a team of external workers during the realization phase, -the material is not expendable and can be re-used for another pile, which can be economically interesting in case of repeated tests on the same site, -lengths of segments are adjustable, if needed, -it is particularly well adapted for the instrumentation of driven piles, as the sensors are not present during the critical phases and therefore cannot suffer from it. However, this method presents some major disadvantages:

-extra pipes should be positioned inside the reinforcing cage, which require extra work for the contractor before the realization of the pile. These pipes shall also be chosen in relation to the model of extensometer that will be implemented. Particular attention should be paid to the connection/welding of the segments of the pipe, to ensure that nothing obstructs the passage of the extensometer, -it is intrusive, especially for small diameter piles, -these extensometers are often limited to 8

to 12 segments, which can sometimes not be enough for long piles in a multi-layers ground, -nitrogen is needed for the use of most of these systems, -duration of use is limited in time, as there may be leaks of nitrogen due to worn-out valves and connections, -it is quite fragile and needs extra care when manipulated, -spare parts are expensive and not always easily found, -it needs constant maintenance and highly qualified technicians to maintain it, calibrate it and keep it ready to use, -on site preparation takes time, -during on site preparation phase, it necessitates a lot of space to be deployed safely. All these aspects make the use of a retrievable extensometer expensive, compared to the embedded methods.

EMBEDDED SENSORS VS RETRIEVABLE EXTENSOMETERS

While, for practical and economical reasons, the use of a retrievable extensometer is interesting when testing multiple piles of comparable size and/or driven piles, it becomes less competitive for projects with only a few piles, or with piles with different lengths, for the same reasons.

Furthermore, even if both methods aim is to measure strains along the pile, the measurement may differs : indeed, while the technologies of the sensors used for one or the other are often the same, the principle of measurement differs. The embedded sensors are measuring the strains over a length equal to their own length, while the retrievable extensometers measures strains between two anchors assumed to be fixed, and moving with the pile itself (Fig. 4) [START_REF] Ponts | La mesure des déformations à l'aide des extensomètres amovibles LPC : Méthodes d'essai LPC[END_REF], making the strain measurement 'global' compared to the 'local' measurements achieved with embedded sensors. Therefore, the retrievable extensometers integrate the strains over a longer sample. In consequence these measured strains are often larger than the ones achieved with embedded sensors, with a ratio dependent on the strain level (Fig. 5a). The resulting equivalent moduli derived from these measurements [START_REF] Lam | Critical assessment of pile modulus determination methods[END_REF] and used to convert strains into loads are in turn smaller.

In addition, the analysis of the results achieved are not as straight-forward as when using embedded sensors, as it is needed to interpolate results at the level of each anchor.

Nevertheless, results achieved with both measuring methods are quite comparable, as can be seen on Fig. 5b for the calculated displacement of the pile base, on Fig. 6a for the load distribution of a given load step and on Fig. 6b for the mobilization of shaft resistance during a static load test [START_REF] Szymkiewicz | Experimental study of heating-cooling cycles on the bearing capacity of CFA piles in sandy soils[END_REF], and the utilization of one method or the other should therefore be only linked to the context of the project, as it is not a question of accuracy.

CONTRIBUTION OF THE MEASURES OF STRAINS BY OPTICAL FIBER

Implementing optical fiber for the measurement of strains in piles was first done when long term monitoring of strains and temperatures was needed : therefore, at first, piles instrumented with these technologies were thermo-active piles.

Nowadays, with positive practical feedbacks on the use of this method available, it can be used for standard piles, for control or development purposes, as it allows measurements all along the fiber, with a resolution up to a few millimeters, depending on the technology used (Fig. 7) [START_REF] Kania | Application of Distributed Fibre Optic Cables in Piles[END_REF], and measurements achieved are in excellent concordance with measurements achieved with vibrating wire strain gauges (Fig. 8a).

The use of optical fiber for the measurement of strains presents some advantages:

-implementation, when choosing to simply link the fiber with duct tape to the rebars, can be very quick, -it is almost non-intrusive, as the fiber is generally only a 3 or 4 mm in diameter, -it is very appropriate for determining moments and p-y curves, as it provides a very high number of measurements along the pile, -it can be temperature compensated, -cost of the fiber in itself is very low and makes it possible to instrument most rebars. a b Fig. 8 Strain measurements comparison between optical fiber and vibrating wires (a) and Strain distribution for a given load step with an embedded fiber and a fiber sealed in pipe (b) Fig. 9 Strains measures along a bored pile: effect of tubing Optical fiber can also be inserted inside a pipe if available, and then sealed with grout, allowing for a possibility to monitor a pile if something were to go wrong with the primary instrumentation. Results from such instrumentation can be seen on Fig. 8b, where they are compared to results achieved with a fiber linked to a rebar : measurements are quite comparable and therefore this method can be used if needed. It is interesting to note that strains presented on Fig. 7b were also measured through this method, as the timing of the project did not allow for the pile to be instrumented before being realized. Furthermore, it also allows to detect layers that where not detected during the geotechnical investigation, and to observe behaviour that would not be seen when using a discrete methods of measuring strains.

Figure 9 shows the strains measured with an optical fiber and vibrating wire strain gauges, on a bored pile tubed down to 16m: at the end of the tube, strains measured by the optical fiber increases, due to a decreased stiffness of the pile, while the strain gauges, which are discretly distributed along the shaft, do not reflect this local behaviour [START_REF] Szymkiewicz | Feedback on static axial pile load tests for better planning and analysis[END_REF]. A variation in diameter is also always possible, when encountering a softer layer for exemple, and a continous measuring method is particularly adapted to detect these occurences [START_REF] De Battista | Distributed fiber optic sensors for monitoring reinforced concrete piles using Brillouin scattering[END_REF].

However, it is important to be aware of the difficulties / disadvantages of the optical fiber :

-while the cost of the fiber in itself is very low, the interrogator price is usually very high, -the interrogators are often not so rugged, and can be damaged during on site monitoring, due to dust presence mostly. Some interrogators are very vibration sensitive, -the fiber is also very fragile, and must be treated with extra care during set up, especially at the pile base, where the fiber is most exposed during concreting or insertion phases (Fig. 10a), and during construction phases such as lifting, when tensile stress due to the bending of the cage can exceed fiber tensile resistance (Fig. 10b), to avoid any break. -the fiber acting as a chain of sensors, a broken fiber will result in the total loss of information under the break. Furthermore, the steps of the realisation of a connection are numerous and tedious. The environment needs to be very clean, with close on site activities as low as possible. The apparatus needed are presented in Fig. 11.

Bad connections result in loss of signal that will in turn result in loss of precision, that will have to be added to the impact of the deployement method of the fiber and the pile.

It is also important to remark that, for cast in place piles, the concreting phase has an important impact on the initial response of the fiber, a seen on Fig. 12, which represent the variation of the frequency, along the length of a the fiber, before loading. The response of the fiber being chaotic, with huge variations in terms of measured strains happening over a very short length, the accuracy of the measures as well as the determined position of any particular point along the pile must be impacted by this concreting phase, depending on the fiber used, as well as the technology chosen and the developed algorithm. Brillouin signal of a fiber outside and inside a pile (frequency vs length)

CONCLUSIONS

The use of optical fiber for the measurement of strains during pile load tests or other related projects is now a method that is very reliable.

While measured values may differ from values achieved by more conventional technologies, this is only due to differences of measurement methods of the methods. Interpretation of the data shall give the same results as the other methods, whether it be displacements of the pile segment or load distributions along the pile, albeit with a higher resolution.

Nonetheless, it is still very rare to see this technology used for pile monitoring, the two main reasons for this being that:

-the set up, implementation and manipulation of the fiber should be done in a very controlled environment and with extra care, as it is very fragile.

-while the cost of the fiber is relatively low, the price of the interrogator, coupled with its fragility, leads to a certain reluctance to deploy them in dusty environments prone to hazards endangering their integrity and their proper functioning. It is also important to note that for cast-in place piles, the concreting phase must have an important impact on the fiber in itself, straining and stressing it locally very differently for one point to another. This can have a non-negligible impact on the results, especially if a high resolution and positioning accuracy if needed (for pile solicited under combined loading, for examples). It is therefore necessary to test reinforced fibers which can withstand this concreting phase better.

Nonetheless, further applications like the monitoring of piles at a young age for quality control purposes (by measuring of temperature inside the pile along the shaft to assess its possible change in diameter) may entice contractors to develop the use of this technology.

Fig. 1

 1 Fig. 1 Force distribution along a Continuous Flight Auger (CFA) pile instrumented with a retrievable extensometer

a b Fig. 3 Fig. 5

 35 Fig. 4Operating principle of a retrievable extensometer

  Fig. 6Comparison of load distribution for a given load step (a) and shaft resistance mobilization (b) for a CFA pile: results achieved with strain gauges and retrievable extensometer.

  Fig. 10 Fiber protection at the pile base (a) and cage lifting phase (b)

STATIC PILE LOAD TESTS: CONTRIBUTION OF THE MEASUREMENT OF STRAINS BY OPTICAL FIBER


Yutaka Yamamoto

Catherine Bonnet
email: catherine.bonnet@inria.fr.

Bézout Identity in Pseudoratoinal Transfer Functions

Keywords: Bézout identity, pseudorationality, distributions, Gel'fand representation, delay-differential systems AMS subject classification: 46F10, 46J15

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

This short note studies the issue of coprimeness for a certain class of systems.

We first consider the input/output relation described by

q • y = p • u (1)
where u and y denote the input and output, respectively, and q and p are given elements belonging to a ring R that acts on inputs and outputs and describes the type of input/output relations. The multiplication in (1) is interpreted according to the context it is considered.

With varied choices of the ring R, we can describe a variety of different classes of systems. For example, when R is the ring of polynomials R[s], it will give the class of finitedimensional linear systems with (1) interpreted in the Laplace transform domain. Likewise, R[z] corresponds to the class of finite-dimensional discrete-time linear systems with (1) considered in the sense of the z-transform.

When R = R[s, z] with s being the Laplace transform variable and z a finite-time delay, (1) can be interpreted as the class of delay-differential systems. Extensive studies have been conducted in this context in the literature; see, e.g., [START_REF] Rocha | Behavioral controllability of delay-differential systems[END_REF]; [START_REF] Glüsing-Lüerssen | A behavioral approach to delaydifferential systems[END_REF].

When R is taken to be the ring of stable rational functions R stable (s), it can be used to study stabilization with compensators over this ring [START_REF] Vidyasagar | Control System Synthesis[END_REF]). More generally, when we take R to be H ∞ , (1) can describe a certain class of distributed parameter systems as well. This class of systems is often adequate for studying stabilization/stabilizability over the compensators constructed in the ring H ∞ . This class has also been studied in depth in the literature. The authors have also conducted some research concerning a coprimeness condition over H ∞ [START_REF] Bonnet | Coprimeness of fractional representations[END_REF]).

In all these classes, the notion of coprimeness has played crucial roles depending on the context where (1) is interpreted. For example, for R = R[s] the coprimeness corresponds to the minimality of the representation (i.e., realization) naturally associated to (1); likewise for R [z]. For R = R stable (s) or H ∞ , the coprimeness yields stabilizability with compensators constructed over these rings.

In all these studies, a strong notion of coprimeness, i.e., the Bézout identity (or Bézout condition)

px + qy = 1 (2)
plays a critical role in deriving desired results, i.e., minimality or stabilizability, in the respective contexts.

For the study of distributed parameter systems, the first author has introduced the class of pseudorational impulse responses or transfer functions, and developed realization theory, various spectral analysis, and coprimeness conditions [START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF][START_REF] Yamamoto | Reachability of a class of infinitedimensional linear systems: an external approach with applications to general neutral systems[END_REF].

In this note, we will give a brief overview for this class of systems, the background and motivation for the study of this class, and then proceed to the study of Bézout identity over this class of systems.

The present article intends to give a new attempt to derive a condition for the Bézout identity (2) in the class of systems described by pseudorational impulse responses. We will provide a Banach algebra approach for a small class of impulse responses described by the space of measures, and then try to generalize it to the general class of distributions. To this end, we need to give some background materials, explain their motivations, and show how this approach is important for some distributed parameter systems. While we have once given a brief review of this background material [START_REF] Yamamoto | Pseudorational transfer functions-a survey of a class of infinite-dimensional systems[END_REF], it is important to further motivate more deeply into the need of developing actions induced by distributions and minimal representations there. This will lead naturally into the minimality of the state space representation and also controllability of behaviors over such a ring, e.g., the ring E (R) or E (R -) of distributions of compact support (see below for the definitions). The Bézout identity over E (R) has been studied in [START_REF] Yamamoto | Behavioral controllability and coprimeness for a class of infinite-dimensional systems[END_REF][START_REF] Yamamoto | Behavioral controllability and coprimeness for pseudorational transfer functions[END_REF].

The paper is organized as follows: In Section 2, we review some basics for the study of pseudorational impulse responses.

We will motivate and give the definition of pseudorationality [START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF]) and state the basic problem here. In Section 3, we show how the Bézout identity translates to an invertibility condition in a quotient algebra. In Section 4, we confine our attention to the algebra M(R -) consisting of measures with compact support contained (-∞, 0]. This restriction leads us to the study of a Gel'fand algebra. In Section 5, we attempt to generalize the result to the general case in E (R -), and also show that how a simple generalization fails. While a general result is available, a straightforward generalization is shown to be not possible because there exists a pair in M(R -) that satisfies a Bézout identity over E (R -) but is never coprime over M(R -). In the final section we will disscuss how this example suggests a further approach in this direction.

PSEUDORATIONALITY AND ITS BACKGROUND

Back in the early 1980s, the first author was trying to find a suitable class of transfer functions that accommodates at least delay-differential systems, and also enables us to study infinitedimensional systems with algebraic structures. The ring R[s, z] is one such example, but it has a fatal defect in that it is not closed under pole-zero cancellations. For example, 1e -s s has a common zero at s = 0, but when we cancel this common zero between the numerator and the denominator, the resulting entire (holomorphic on the whole complex plane) function

1 - s 2 + s 2 3! -• • •
is no longer expressible as the ratio of two-variable polynomials in s and z. This defect becomes critical when we want to establish a framework that allows us to freely form a minimal (canonical, i.e., reachable and observable) realization. This naturally raises the following question:

What is the smallest algebra that includes the differentiation and delay operator?

From here on, let us turn our discussion into the time domain with convolution that gives a multiplication structure. Then, the differentiation is expressed by the derivative δ of the Dirac delta, and time-delay operator is expressed as the convolution with δ a with a > 0. Likewise, the time advance operator of length is δ a with a < 0.

We now invoke an analogy with realization theory of the discrete-time finite-dimensional systems. Note here that the ring we consider there is R[z] where z is the time-advance operator, not the delay operator. This formalism is particularly suited for realization theory.

In analogy with this, we consider the convolution actions induced by the finite sum of elements of δ and δ a i as follows:

x(t) → ∑ i α i δ (k i ) δ a i x = ∑ i α i x (k i ) (t -a i ) (3)
where α i ∈ R and δ (k i ) denotes the k i -th order derivative of δ .

However, the finite sum (3) does not constitute a closed space convenient for analytical handling. We thus want to take a suitable completion of the elements of form (3) under a suitable topology.

One natural choice of such a topology is that of Schwartz distributions [START_REF] Schwartz | Théorie des Distribution[END_REF]). We thus introduce the following notions.

Let D denotes the space of distributions on R. Let E (R) be its subspace consisting of those having compact support. E (R -) is also its subspace with support contained in the negative half line (-∞, 0]. D + denotes the subspace of D consisting of elements having support bounded on the left. Distributions such as Dirac's delta δ a placed at a ∈ R, its derivative δ a are examples of elements in E (R). If a ≤ 0, then they belong to E (R -). If we take the closure of elements of type (3) in D , it is E (R). If we confine (3) to those with nonpositive a i , it will be E (R -). (See [START_REF] Schwartz | Théorie des Distribution[END_REF][START_REF] Schwartz | Méthodes Mathématiques pour les Sciences Physiques[END_REF] for more detail.)

We thus consider fraction representations over E (R -) (relying on the analogy with R[z] and

E (R -)). Definition 2.1. An impulse response function G (supp G ⊂ [0, ∞)
) is said to be pseudorational [START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF]) if there exist q, p ∈ E (R -) such that

(1) G = q -1 * p where the inverse is taken with respect to convolution and belongs to D + ; (2) ord q -1 =ord q, where ord q denotes the order of a distribution q (Schwartz (1966)).2

If this condition is satisfied, we call (p, q) a pseudorational pair. The Laplace transform q-1 p is called a pseudorational transfer function.

The delay-differential equation:

ẋ(t) = x(t -1) + u(t) y(t) = x(t),
admits the representation y = (δ -1δ ) -1 * δ -1 * u, and hence it is pseudorational.

The main problem that concerns us here is the following: Problem Given a pseudorational pair (p, q) ∈ E (R -) × E (R -), characterize a condition under which p and q satisfy the Bézout identity: p * x + q * y = δ (4) for some x, y ∈ E (R -).

If we consider E (R) instead of E (R -), it gives a necessary and sufficient condition for the behavior defined over D (Yamamoto ( 2016)). Actually, the Bézout condition over E (R) is in close relationship with that in E (R -) (Yamamoto ( 2016)).

COPRIMENESS IN E (R -)

We first translate (4) to a divisibility condition by considering the principal ideal (q) = q * E (R -) generated by q in E (R -). Note first that ( 4) is easily seen to be equivalent to p * φ = δ mod q.

(5) for some φ ∈ E (R -). In other words,

[p] * [φ ] = [δ ] (6) in E (R -)/(q). This means that the equivalence class [p] is invertible in the quotient algebra E (R -)/(q).
Condition (6) by itself is not so easy to handle because of the intricate topology of E (R -). However, because q has compact support, the following remarkable property holds: Proposition 3.1. Take any

T > 0 such that supp q ⊂ (-T, 0]. Then E (R -)/(q) ∼ = E ([-T, 0])/(q) (7)
Proof Let π be the projection operator

π : D → D (0,∞) : ψ → ψ|(0, ∞) (8)
where D (0,∞) is the space of distributions with support contained in (0, ∞). Given ψ ∈ D , define the following operator π q as π q : E (R -) → E (R -) : ψ → q * π(q -1 * ψ).

(9) Now for a distribution ψ ∈ D + , define (ψ) as (ψ) := inf{t ∈ supp ψ} (10) where supp ψ denotes the support of ψ.

Take any x ∈ E (R -) along with π q x. We claim that π q x belongs to E (R -) (hence ( 9) is well defined as a map from E (R -) into itself) and that x ∼ = π q x mod q. We have q -1 * (x-π q x) = q -1 * x-q -1 * q * π(q -1 x) = q -1 * x-π(q -1 * x).

The last term φ := q -1 * xπ(q -1 * x) belongs to E (R -) because q -1 * xπq -1 * x must be zero on (0, ∞). That is to say,

xπ q x = q * φ ∈ q * E (R -) = (q). This also shows that π q x = xq * φ ∈ E (R -). In other words, [x] = [π q x] in E (R -)/(q). Moreover, since (π(q -1 * x)) ≥ 0 and (q) ≥ -T , the support of π q x = q * π(q -1 * x) must be contained in [-T, 0] by Lemma A.1. That is, for every x ∈ E (R -), there always exists an element π q x such that supp π q x ⊂ [-T, 0], and x ∼ = π q x mod q. This proves (7).

2 Remark 3.2. Proposition 3.1 claims that as far as a pseudorational impulse response is concerned, we can confine our attention to those inputs with support contained in [-T, 0] with -T < (q). This result is not so surprising if we pay proper attention to the compact-support property of q. Since q has bounded support, its maximum length should determine the maximum length of memory needed to reconstruct the state or future outputs. This can be easily guessed once we resort to the analogy with realization theory for discrete-time linear systems: The degree of the denominator polynomial q(z) determines the dimension of the state in the standard reachable realization, and the degree here exactly corresponds to the length of the support of q here. The projection scheme used above is an analogy to the finite-dimensional theory developed by [START_REF] Fuhrmann | Algebraic system theory: an analyst's point of view[END_REF].

GEL'FAND ALGEBRA STRUCTURE OF THE SPACE OF MEASURES

We have seen that the existence of the Bézout condition reduces to the invertibility of [p] in the quotient algebra E (R -)/(q). It is also seen that this space E (R -)/(q) is isomorphic to E ([-T, 0])/(q) for some T > 0 so that its structure is quite simplified. However, the space E (R -)/(q) is still not that easy to tackle due to a rather complex topological structure of E (R -)/(q).

We now choose to confine ourselves to the subspace M(R -) that is the subspace of E (R -) consisting of measures, i.e., those with elements of order 0. As shown in Proposition 3.1, M(R -)/(q) ∼ = M([-T, 0])/(q) for some T > 0. (Proposition 3.1 claims this fact for E (R -), but the proof remains essentially the same. Note that ord q -1 =ord q = 0 by condition (2) of Definition 2.1, so that q -1 is also a measure.) We here observe that the space M([-T, 0])/(q) has a remarkable advantage over E ([-T, 0])/(q) in that it can be regarded as a Banach space with respect to the strong dual topology as the dual space of the space of continuous functions C[-T, 0]. Furthermore, it inherits a natural algebra structure induced from M(R -) (with respect to convolution) with the unity element [δ ]. In other words, it is a Gel'fand algebra [START_REF] Gel'fand | Commutative Normed Rings[END_REF]; [START_REF] Berberian | Lectures in Functional Analysis and Operator Theory[END_REF]).

A Gel'fand algebra is known to have a remarkable property in that the invertibility of an element can be well tested by characterizing the space of its maximal ideals [START_REF] Berberian | Lectures in Functional Analysis and Operator Theory[END_REF]; [START_REF] Gel'fand | Commutative Normed Rings[END_REF]). This fact is best suited to study the invertibility condition (6).

Let us now make the following Assumption:

Assumption 1 There exists σ ∈ R such that p(s) and q(s) do not vanish on {s| Re s ≥ σ }. Remark 4.1. Pseudorationality assumes the existence of q -1 ∈ D + , hence the above condition is automatically satisfied by Lemma A.4. If p -1 ∈ D + is assumed, it also satisfies Assumption 1. If Assumption 1 is satisfied we may assume that σ can be taken to be zero, without loss of generality. For if necessary, we can always shift the complex variable as s → sσ , and this clearly does not affect the coprimeness relationship.

The following theorem was first given in Yamamoto ( 2007a), but we here give a more complete proof for the sake of completeness. Theorem 4.2. Let p, q ∈ M(R -), and satisfy Assumption 1. Suppose that there exists c > 0, a ∈ R such that |p(s)| + |q(s)| ≥ c > 0 (11) for every s ∈ C -= {s ∈ C| Re s ≤ 0}. Then the (p, q) is a Bézout pair, i.e., satisfies the Bézout identity (4).

For the proof, we need some preliminaries. The question here is to find a condition under which [p] is invertible in M([-T, 0])/(q). By Gel'fand representation theory [START_REF] Berberian | Lectures in Functional Analysis and Operator Theory[END_REF][START_REF] Gel'fand | Commutative Normed Rings[END_REF]), an element [p] is invertible if and only if it belongs to no maximal ideals.

Consider the Laplace transform of elements in M(R -). It is easy to see that this is a subalgebra of H ∞ (C -). Then, as in [START_REF] Hoffman | Banach Spaces of Analytic Functions[END_REF], we see that the correspondence ψ → ψ(s) considered for s ∈ C -gives the Gel'fand representation.

What is then a maximal ideal in M(R -)? Take any λ ∈ C -, and consider the point evaluation

φ λ : f → f (λ ).
(12) It is easy to see that φ λ is a complex homomorphism (i.e., homomorphism from M(R -) to C), and hence ker φ λ is a maximal ideal of M(R -). Observe however that this does not necessarily yield a maximal ideal in M(R -)/(q), because in order to be an ideal in this space, this ideal should contain (q). In other words, q should vanish there. If M is given by

M λ = { f | f (λ ) = 0}
, then this means that λ should be a zero of q for M λ ⊃ (q). Now let λ 1 , λ 2 , . . . , λ n , . . . (13) be the set of zeros of q. Then we have maximal ideals M λ 1 , M λ 2 , . . . , M λ n , . . . of M(R -)/(q). But these are not all. There are other maximal ideals that are centered at "infinity".

To see this, let us first start with the following proposition: Proposition 4.3. Let f ∈ M(R -), and suppose that φ ( f ) = 0 for some complex homomorphism, i.e., f belongs to a maximal ideal ker φ . Suppose also that φ does not agree with any of M λ n as given above. Then there exists a sequence µ n such that

• µ n → ∞ and • f (µ n ) → 0 as n → ∞
Proof Suppose there exists no such µ n . Then there exists δ > 0 and R > 0 such that | f (s)| ≥ δ for |s| ≥ R. In view of the Hadamard factorization (A.3) of q, it follows that either (1) f (s) has infinitely many zeros, or (2) f (s) has only finitely many zeros.

The first case is clearly impossible by | f (s)| ≥ δ . Hence f has only finitely many zeros. But this yields f (s) = e αs P(s) where P is a polynomial. Note that α ≥ 0 because the inverse Laplace transform of f is a measure in M(R -). Since α = 0 just corresponds to a constant, we assume α = 0, so that α > 0. But then e αs can have infinitely many zeros along the imaginary axis, and this contradicts | f (s)| ≥ δ for |s| ≥ R. Hence f must be a polynomial. But this is again impossible unless f is a (nonzero) constant because the inverse Laplace transform of f must be a measure. Therefore f must be a constant. But this yields φ (1) = 0, which clearly means that φ annihilates the whole space, and this contradicts the fact that φ is a nontrivial complex homomorphism (or ker φ is a maximal ideal).

2

In particular, this holds also for q. Then if M is a maximal ideal of M(R -)/(q), then π -1 (M) is clearly a maximal ideal of M(R -), and this should contain (q).

We are now ready to prove Theorem 4.2.

Proof of Theorem 4.2 Suppose (11) holds, but p belongs to a maximal ideal in M(R -)/(q). If p belongs to one of M λ n , then this would clearly contradict (11). Hence assume that p vanishes at no λ n , n = 1, 2, . . .. Then by Proposition 4.3, there exists µ n such that µ n → ∞ and p(µ n ) → 0. Since this maximal ideal should contain q, q should also vanish there, and hence a suitable subsequence of q(µ n ) should go to 0. This clearly contradicts (11).

2

Here are some examples:

Example 4.4. The pair (e s/2 -1, e s -1) is not a Bezout pair.

The pair possesses infinitely many common zeros.

Example 4.5. The pair (e s , e s/2 -1) is a Bezout pair. It is easy to check (11). This can also be directly verified by e s -(e s/2 -1)(e s/2 + 1) = 1. Remark 4.6. Condition ( 11) is the same as that in the celebrated Corona theorem by Carleson for H ∞ [START_REF] Duren | Theory of H p Spaces[END_REF]; [START_REF] Garnett | Bounded Analytic Functions[END_REF]). One should of course be careful not to confuse the present result with the Corona theorem, because such conditions crucially depend on the choice of a ring. The proof here is good deal simpler than that of the monstrous Corona theorem [START_REF] Duren | Theory of H p Spaces[END_REF]; [START_REF] Garnett | Bounded Analytic Functions[END_REF]). This is because the algebra M(R -)/(q) is much "smaller" than H ∞ , and the way it yields "cancellation at infinity" is quite much restricted by the discrete zeros {λ n } whereas in the case of the Corona theorem, there are almost arbitrary ways in which such sequences go to infinity.

EXTENSION TO E (R -)

Let us first make the following assumption:

Assumption 2: The algebraic multiplicity of each zero λ n of q(s) is globally bounded.

The Gel'fand algebra structure and the characterization of the maximal ideal space in M([-T, 0])/(q) is fairly appealing, and not so difficult as the case for H ∞ . It is thus quite tempting to try to generalize the above result to the general case of E (R -) or E (R).

We first note the following Theorem 5.1 obtained in Yamamoto (2007a[START_REF] Yamamoto | Behavioral controllability and coprimeness for pseudorational transfer functions[END_REF]. Theorem 5.1. Let q -1 * p be pseudorational, and suppose that there exists a nonnegative integer m such that |λ m n p(λ n )| ≥ c > 0, n = 1, 2, . . . (14) Then the pair (p, q) satisfies the Bézout identity (4) for some φ , ψ ∈ E (R -).

The proof given in Yamamoto (2007a[START_REF] Yamamoto | Behavioral controllability and coprimeness for pseudorational transfer functions[END_REF] is fairly complicated and highly technical. It does involve some elaborate analysis of complex analytic functions of exponential type, and some deep facts of their growth orders.

It is thus tempting to try to give a proof by using Theorem 4.2, extending the result for M(R -) to E (R -).

Let us first prepare some pertinent facts on the structure of E (R -). Since every element of E (R -) has compact support, it is of finite order [START_REF] Schwartz | Théorie des Distribution[END_REF]). That is, for every ψ ∈ E (R -), there exists r ≥ 0 such that ψ = (d/dt) r ψ 0 (15) for some ψ 0 ∈ M(R -) and r ≥ 0. This readily implies

E (R -) = ∪ ∞ r=0 (d/dt)M(R -).
(16) In other words, the algebra E (R -) is derived as the differentiated union of measures.

We now suppose that we are given a pseudorational pair (p, q) belonging to E (R -). Since E (R -) is the nested union of differentiated measures, we may hope that we can reduce the coprimeness problem of E (R -) into that of M(R -). A procedure like the Euclid division algorithm can be a hint for this.

Suppose for the moment that p is of order 0 and q is of order 1. Suppose also that q(s) has one real zero, say, λ . Then the inverse Laplace transform of q(s)/(sλ ) should be of order zero because the division by sλ should act as an integration. Therefore, both p and L -1 [ q(s)/(sλ )] should be of order zero, i.e., measure.

Then it is naturally expected that the coprimeness of (p, q) should reduce to that of (p, L -1 [ q(s)/(sλ )]).

In fact, if (p, q 0 * q 1 ) is coprime in a ring R (p, q 1 ) is coprime and vice versa. So it is natural to expect that the Bézout condition of (p, q) is translated to that of (p, L -1 [ q(s)/(sλ )]) where the latter belong to the space of measures M(R -), where Theorem 4.2 is available.

However, this seemingly reasonable idea unfortunately does not work. The following counterexample shows why. Example 5.2. Consider the pair (δ -1δ , δ -1 ). This pair is clearly pseudorational. The element δ -1δ has order 1, and δ -1 has order 0, i.e., measure. They admit Laplace transforms se s -1 and e s , respectively. They satisfy the Bézout identity

(se s -1) • (-1) + s • e s = 1, (17) or
(δ -1 -δ ) * (-δ ) + δ * δ -1 = δ , (18)
and hence the pair is coprime over E (R -).

The former element se s -1 has one positive zero, say α. This means that se s -1 sα (or its inverse Laplace transform) has order 0 because division by sα entails in integration of δ -1δ once, whereby yielding an element of order zero, i.e., a measure.

In other words, the pair (or the respective inverse Laplace transforms)

se s -1 s -α , e s (19)
belongs to M(R -), and they are coprime over E (R -). However, this does not guarantee that this pair admits a coprime factorization over M(R -) in the sense of Theorem 4.2.

To see this, observe that se s -1 admits infinitely many zeros λ n such that Re λ n → -∞. (This can easily be seen by noting that it is the characteristic function of the retarded delaydifferential equation ẋ = x(t -1) + u.) Indeed, λ n e λ n = 1 admits infinitely many solutions such that e λ n = 1/λ n , n = 1, 2, . . .. This also implies that p(λ n ) → 0 as n → ∞. That is, it contradicts condition (11) of Theorem 4.2, and cannot be a Bézout pair in M(R -).

In other words, the pair can admit a Bézout identity over E (R -) with x, y ∈ E (R -), but it cannot satisfy a Bézout condition over the algebra of M(R -) because the latter algebra is much smaller than E (R -) and does not give as much freedom as that induced by E (R -). This can be more directly seen by noting the identity

s • e s + se s -1 s -α • (α -s) = 1.
This looks trivial and not any different from ( 17). The difference here is that the multiplying factor αs that makes the pair (19) satisfy the Bézout identity does not belong to (the Laplace transform of) M(R -). To cover this situation, we do need Theorem 5.1, which cannot be, unfortunately, covered as a natural variant of Theorem 4.2.

In fact, at the zeros λ n of q, e λ n = 1/λ n holds, so that the p(λ n ) clearly satisfy condition ( 14) for m = 1. This condition can also be rewritten as |s p(s)| + |s q(s)| ≥ c > 0, ∀s ∈ C -.

(20)

CONCLUDING REMARKS

We have seen that the space M(R -) of measures admits a Gel'fand algebra structure, and it yields a concrete Corona-like condition (11) for the Bézout identity for a pseudorational pair (p, q). We have also pursued to derive the general condition ( 14) for the Bézout identity over E (R -), but also seen that a straightforward reduction idea does not work. The modified generalized Corona-like condition (20) may, however, suggest that there could still be a possibility of generalizing (11) to a more general context in E (R -).

Roughly speaking, the order of a distribution α is the least integer r such that α = (d/dt) r β for some measure β .

1 This author was supported in part by the Japan Society for the Promotion of Science under Grants-in-Aid for Scientific Research No. 19H02161. The author also wishes to thank DIGITEO and Laboratoire des Signaux et Systemes (L2S, UMR CNRS), CNRS-CentraleSupelec-University Paris-Sud and Inria Saclay for their financial support while part of this research was conducted.

Appendix A. PRELIMINARY MATERIALS For (α) defined by (10), the following lemma holds:

Proof The proof is immediate from the local version of Titchmarsh's theorem on convolution [START_REF] Donoghue | Distributions and Fourier Transforms[END_REF]). Actually, if α, β = 0, then α * β = 0. Hence (α * β ) ≥ (α) + (β ) readily follows. However, it can also be deduced from the local version of Titchmarsh's theorem that if α and β are nonzero in some neighborhoods of (α) and (β ), respectively, then α * β is nonzero in a neighborhood of (α) + (β ). This proves (A.1). 2

We now need some basic properties for the Laplace transform of elements in E (R -): Theorem A.2. (Paley-Wiener, [START_REF] Schwartz | Théorie des Distribution[END_REF])). A complex analytic function f (s) is the Laplace transform of a distribution φ ∈ E (R -) if and only if f (s) is an entire function that satisfies the following growth estimate for some C > 0, a > 0 and integer m ≥ 0: if and only if it satisfies the estimate

2) for some C > 0, a > 0 and integer m ≥ 0. In this case, the support of φ is contained in [-a, 0] This implies the following important lemma: Lemma A.3. Let f (s) is an entire function that satisfies the above Paley-Wiener estimate (A.2). Suppose λ ∈ C is a zero of f (s). Then f (s)/(sλ ) also satisfies (A.2) (with different C, a, m, of course), and hence it is the Laplace transform of a distribution φ ∈ E (R -).

Proof Draw a unit circle around λ . Outside of this circle, f (s)/(sλ ) should satisfy the same estimate of type (A.2). On the other hand, inside this circle, f (s)/(sλ ) is a continuous function, and hence bounded. Combining these two facts easily yields the conclusion. 2

For a distribution φ ∈ E (R -), that φ (s) is an entire function of exponential type as in (A.2) yields the fact that it allows the socalled Hadamard factorization as the infinite-product consisting of its zeros [START_REF] Boas | Entire Functions[END_REF]):

The following lemma is of relevance to Assumption 1: Lemma A.4. [START_REF] Schwartz | Méthodes Mathématiques pour les Sciences Physiques[END_REF]). An analytic function φ (s) is the Laplace transform of a distribution f ∈ D + if and only if there exists c ∈ R such that |φ (s)| is bounded by a polynomial in s.

Suppose that f ∈ D + is invertible in D + . Then 1/ f (s) satisfies a polynomial estimate for some half plane {s| Re s > c}. This means that there exist no zeros of f (s) for Re s > c.