Marie Kerjean
email: marie.kerjean@inria.fr

Chiralities in topological vector spaces

Keywords: 2012 ACM Subject Classification Theory of computation Ñ Linear logic, Mathematics of computing Ñ Functional analysis, Theory of computation Ñ Categorical semantics Keywords and phrases Linear Logic, Functional Analysis, Category Theory Digital Object Identifier 10.4230/LIPIcs

come

Introduction

Linear Logic (LL) is the result of a decomposition of Intuitionistic Logic via an involutive linear negation. This linear negation takes its root in semantics: the linear negation of a formula is interpreted as the dual 1 of the vector space interpreting the formula. While LL's primary intuitions lie in algebra, the study of vectorial models [START_REF] Ehrhard | On Köthe sequence spaces and linear logic[END_REF][START_REF] Ehrhard | Finiteness spaces[END_REF] of it led to the introduction of Differential Linear Logic [START_REF] Ehrhard | Differential interaction nets[END_REF] (DiLL). This new proof system introduces the possibility to differentiate proofs and led to advances in the semantics of probabilistic and differentiable programming [START_REF] Ehrhard | The Computational Meaning of Probabilistic Coherence Spaces[END_REF][START_REF] Brunel | Backpropagation in the simply typed lambda-calculus with linear negation[END_REF].

Infinite dimensional vector spaces are necessary to interpret all proofs of DiLL. However these spaces are seldom isomorphic to their double dual. The class of all reflexive topological vector spaces, that is of spaces invariant via double-dual, moroever enjoys poor stability properties. More crucially, duality in topological vector spaces does not define a closure operator: simply considering E 2 does not produce a reflexive space. Thus historical models of Linear Logic traditionally interpret formulas via very specific vector spaces: vector spaces of sequences [START_REF] Ehrhard | On Köthe sequence spaces and linear logic[END_REF], vector spaces over discrete field [START_REF] Ehrhard | Finiteness spaces[END_REF]. How close is the differentitation at stakes in DiLL from the one of real analysis ? Denotational models of DiLL in real-analysis either don't interpret the involutivity of linear negation [START_REF] Blute | A convenient differential category[END_REF] or imply a certain discretisation for the interpretation of non-linear proofs [START_REF] Kerjean | Weak topologies for linear logic[END_REF][START_REF] Dabrowski | Models of linear logic based on the schwartz ε product[END_REF].

Polarization is a syntactical refinement of Linear logic arising for matters of proof-search [START_REF] Andreoli | Logic programming with focusing proofs in linear logic[END_REF][START_REF] Girard | A new constructive logic: Classical logic[END_REF]. By making vary the topology on the dual, this paper unveils polarized models behind preexisting models of DiLL and construct new ones. Meanwhile, it attaches topological notions to the concept of polarity in proof theory. We first revisit the poor stability properties of reflexive spaces by decomposing it in a polarized version model of MLL. We also revisit the notion of bornological spaces persistent in DiLL's denotational semantics [START_REF] Tasson | Sématiques et syntaxes vectorielles de la logique linéaire[END_REF][START_REF] Blute | A convenient differential category[END_REF] as XX:2 Chiralities in topological vector spaces an interpretation for positives. In a nutshell, we show that while (different variants of) complete spaces interpret negative connectives, barrelled spaces (as introduced by Bourbaki) or bornological spaces are the good interpretation for positive. While we acknowledge the poor computational value of these models -as only the multiplicative part of Linear Logic is properly interpreted here-we believe that our setting will extend to exponentials and non-linearity, as indicated in the perspective. Indeed, this paper unifies the duality at stakes in Linear Logic with the central notion of duality in functional analysis.

Smooth and polarized differential linear logic

Before diving into more details, we give a a few intuitions to the categorical semantics of (Differential) Linear Logic. We refer to the litterature for a detailed introduction [START_REF] Melliès | Categorical semantics of linear logic[END_REF][START_REF] Ehrhard | An introduction to differential linear logic: proof-nets, models and antiderivatives[END_REF].

Linear Logic is constructed on a fundamental duality between linear and non-linear proofs.

It features two conjunctions b and ˆ, two disjunctions `and ', as well as exponential connectives ? and ! on which structural rules are defined. The exponential ! encodes nonlinearity: in the call-by-name translation of Intuitionistic Logic to Linear Logic, traditional non-linear implications are translated as linear implications from the exponential: A ñ B "

!A

B. An involutive linear negation p´q K is defined inductively on formulas.

As such, a categorical model of Linear Logic is constituted of a linear-non-linear adjunction [START_REF] Melliès | Categorical semantics of linear logic[END_REF] between two categories. A monoïdal closed category pL, b, 1q interprets linear proofs and the multiplicative connectives, while a cartesian closed category pC, ˆ, 0q interprets nonlinear proofs. To interpret the involutive linear negation of LL, the category L must be ˚-autonomous. The exponential ! is a co-monad on L, coming from a strong monoïdal adjunction: ! :" E 1 ˝U and E 1 : C / / L $ U : L / / C. On top of that, interpreting DiLL necessitates an additive categorical structure on L and a natural transformation d : ! / / Id enabling the linearization of proof (hence their differentiation).

Topological vector spaces, to be defined precisely afterwards, are a generalization of normed or metric spaces necessary to higher-order functions. Smooth functions between topological vector spaces are those functions which can be infintely or everywhere differentiated. To handle composition or differentiation of smooth functions, the topology of their codomain must verify some completeness property 2 . However, this requirement for completeness mixes badly with reflexive spaces (those interpreting an involutive linear negation). Hence the difficulty to construct smooth models of DiLL.

Beyond the distinction between linear and non-linear proofs, polarization in LL [START_REF] Laurent | Etude de la polarisation en logique[END_REF] distinguishes between positive and negative formulas.

Negative Formulas: N, M :" a

| ?P | ˆP | N `M | K | N & M | J.
Positive Formulas:

P, Q :" a K | !N | ´N | P b Q | 0 | P ' Q | 1.
Semantically, polarization splits L in two categories P and N 3 . The developments of this paper all take place in the categorical setting developed by Mellies: chiralities [START_REF] Melliès | Categorical semantics of linear logic[END_REF] 4 are a decomposition of ˚-autonomous categories in two adjunctions. A strong monoïdal adjunction 2 As an example, Mackey-Completeness is a minimal completeness condition used by Frölicher, Kriegl and Michor [START_REF] Frölicher | Linear spaces and Differentiation theory[END_REF][START_REF] Kriegl | The convenient setting of global analysis[END_REF] to develop a theory of higher-order smooth functions 3 Beware that the name polarity is employed with its proof theory meaning: polarity describes a prooftheoretic behaviour of a formulas and their interpretation. In the theory of topological vector spaces, the polar of a set denotes the set of all linear forms which are bounded by 1 on this subsets. The two meaning of polarity are not unrelated in the light of Proposition 26, as in barrelled spaces the polar to a neighbourhood is bounded. 4 We warn the reader that chiralities have no obvious link with the orientation-related chiralities in physics p´q K L : P / / N op $ p´q K N : N op / / P interprets negations, accompanied with an adjunction interpreting shifts ˆ: P / / N $ ´: N / / P. This semantics enables an internal interpretation of polarized connectives -thus refining traditional interpretation in terms of dual pairs.

Organisation

We begin this paper by an introduction to topological vector spaces in section 2, leading to the introduction of two basic ˚-autonomous categories of vector spaces factorising through dual pairs. In section 3, we introduce chiralities as a categorical model of polarized MLL.

In section 4 we decompose the notion of reflexivity in a chirality of barrelled or weakly quasi-complete topological vector spaces -thus showing that chiralities are a relevant setting to the intricate theory of topological vector spaces. The last section 5 we give two chiralities based on bornological spaces, refining exisiting models of DiLL. The first one in section 5.2 refines the model based on convenient spaces [START_REF] Kerjean | Mackey-complete spaces and power series -a topological model of differential linear logic[END_REF][START_REF] Blute | A convenient differential category[END_REF], while the second in section 5.3

refines the models based on Schwartz ε tensor product. Most proofs are quite direct for the reader familiar with the theory of topological vector spaces. Those for which we didn't find a reference in the literature are given in appendix.

˚-autonomous categories of topological vector spaces

This preliminary section presents a rapid introduction to the various topologies on vector spaces and spaces of linear maps between them. We introduce in particular the weak and Mackey topology which both leads to ˚-autonomous DiLL, resulting respectively into a negative and positive interpretation of DiLL. § Definition 1. A Hausdorff and locally convex topological vector space is a vector space endowed with a Haussdorff topology making scalar multiplication and addition continuous, and such that every point has a basis of convex 0-neighbourhoods.

We abbreviate by lcs the term locally convex and Hausdorff topological vector space.

We denote by Topvec the category of lcs and linear continuous maps between them. The topology of a topological vector space E is thus described by the set V E p0q of all its 0neighbourhoods. From now on we work with locally convex Hausdorff topological vector spaces on R and denote them by lcs. Working with these object, we will be confronted to two definitions of equality: § Definition 2. 1. Two lcs E and F might have the same algebraic structure. The existence of a linear isomorphism between E and F will be denoted: E " F.

Two linearly isomorphic lcs E and F might have the same topological structure. The existence of a linear homeomorphism between E and F is stronger than algebraic equality

will be denoted:

E » F.
Functional analysis is basically the study of spaces of (linear) functions as objects of the same class as their codomains. To construct a topology on a space of linear function, one must decide of a bornology, that is of the class of sets on which convergence must be uniform. § Definition 3. The space of all linear continuous functions between lcs E and F is denoted

LpE, F q. The dual of a lcs E is denoted E 1 :" LpE, Rq. § Definition 4.
Several bornologies (that is, total collections of sets closed by finite union, arbitrary intersection and inclusion) can be defined on a lcs E. The following ones will be used in this article:

1. σpEq, the bornology of all finite subsets of E.

2. βpEq, the bornology of all T E sets absorbed by any 0-neighbourhood of E.

µpEq, the bornology of all absolutely convex compact sets in E σ

, that is of all the weakly compact absolutely convex sets.

Any bornology α on E defines a topology on LpE, F q, referred to as the topology of uniform convergence on α. It is generated by following sub-basis of 0-neighbourhoods:

W B,U " t | pBq Ă U qu
for B P α and U P V E p0q. We will denote by L α pE, F q the vector space LpE, F q endowed with this topology. All of the bornologies σ, µ, β make L α pE, F q and thus E 1 α a lcs.

Thus any bornology α defines in particular a topological dual p´q 1 α . The duals

E 1 σ , E 1 µ , E 1
β are called respectively the weak, Mackey and strong dual. § Definition 5. Any lcs E can be seen as a space of linear forms, through the following continuous linear injection:

ev E : " E / / pE 1 α q 1 x Þ Ñ δ x : pf / / f pxqq

Thus the topologies constructed above on spaces of linear forms can be defined on any lcs E,

for which the dual E 1 has been computed first hand. In particular, a lcs will be said:

1. Mackey when it is endowed with the topology induced by pE 1 µpEq q 1 µpE 1 q , 2. weak when it is endowed with the topology induced by pE 1 σpEq q 1 σpE 1 q , 3. barrelled when it is endowed with the topology induced by pE 1 βpEq q 1 βpE 1 q . Barrelled spaces are in particular Mackey [18,11.1].

We will denote respectively by E µ and E σ the lcs E endowed with the Mackey and the weak topology described above. § Remark 6. The weak topology is a very particular topology with a discrete flavour. On the contrary, examples of Mackey spaces are easy to find: as soon as a space is metrisable, it is Mackey. The basic example of metrisable spaces are the finite dimensional vector spaces or the normed spaces. For an example of a spaces with is metrisable and not normed, consider the space of smooth functions C 8 pR n , Rq endowed with the topology on uniform convergence of the iterated derivative on compact subsets of R n . Examples of barrelled lcs then include complete metrisable spaces, and as such Banach spaces. Example of non-metrisable spaces include spaces of distributions as C 8 pR n , Rq 1 β § Definition 7. E is said to be semi-reflexive when E " pE 1 β q 1 , and reflexive when

E » pE 1 β q 1 β .

As a corollary, a lcs E is reflexive if and only if it is barrelled and semi-reflexive.

Reflexive spaces are stable by product or direct sums. Thus using the strong dual as interpretation for the negation of linear logic gives us very little chance to construct a model of DiLL without strongly restricting the kind of vector spaces one handles. On the contrary, any space is invariant under double weak or Mackey dual.

When a monoidal category resists ˚-autonomy, the traditional solution is to consider pairs of objects of this category, and interpret negation as the switching of position inside a pair. This way, one can enforce the dual of constructions which do not preserve reflexivity -typically tensor products. Chu categories of vector spaces as defined by Barr [2] are a categorical axiomatization of the notion of dual pairs [START_REF] Mackey | On infinite dimensional linear spaces[END_REF]. § Definition 8 (Chu categories of vector spaces). Object of Chu are pairs of vector spaces pE 1 , E 2 q equipped with a symmetric non-degenerate linear form ă ¨, ¨ą: E ˆF / / R.

Morphisms of Chu are pairs of linear maps:

pf 1 , f 2 q : pE 1 , E 2 q / / pF 1 , F 2 q with f 1 : E 1 / / F 1 and f 2 : F 2 / / E 2 such that for every x P E 1 , y P F 2 one has ă f 1 pxq|y ą"ă x|f 2 pyq ą.
Chu is a ˚-autonomous category when endowed with the following constructions:

pE 1 , E 2 q K " pE 2 , E 1 q pE 1 , E 2 q b pF 1 , F 2 q " pE 1 b F 1 , LpE 2 , F 1 qq pE 1 , E 2 q pF 1 , F 2 q " pLpE 1 , F 1 q, E 1 b F 2 q § Theorem 9 (
The Mackey-Arens theorem). The weak topology on E is the coarsest locally convex topology on E which preserves the dual, while the Mackey topology is the finest. In particular:

pE σpE 1 q q 1 " E 1 " pE µpE 1 q q 1 .
Work by Barr [2] reinterprets this theorem in terms of dual pairs: the Mackey Topology induces a right adjoint to the functor D : E Þ Ñ pE, E 1 q from the category Topvec of lcs and continuous linear maps to the category of dual pairs, while the weak topology induces the left adjoint to this functor.

Mackey Chu Weak K K E Þ Ñ pE, E 1 q pE, F q Þ Ñ E µpF q F Þ Ñ pF, F 1 q pF, Eq Þ Ñ F wpEq
These adjunctions naturally result in ˚-autonomous categories over Weak and Mackey.

However these constructions are saturated: topologies on tensor products or hom-sets are defined from the dual and are in no way internal. We showed in previous work that Weak spaces provide a negative interpretation of DiLL [START_REF] Kerjean | Weak topologies for linear logic[END_REF], in the sense that negative connective preserve weak topologies (see proposition 17).

Likewise, as hinted by the above diagram, we will show that Mackey spaces provide a positive interpretation of MLL -which could be extended to LL also by formal power series.

To show this, we need to dive into topological tensor products. Here again, topologies on vector spaces introduce a variety of distinct notions of continuity. § Definition 10. Consider E, F and G three lcs. We denote:

1. BpE ˆF, Gq the vector space of all continuous bilinear functions from E ˆF (endowed with the product topology) to G.

HB α pE ˆF, Gq the vector space of all α-hypocontinuous bilinear functions from E ˆF

to G, where α P tσ, µ, βu. These are the bilinear maps h such that for any B E P αpEq and B F P αpF q, the families of linear functions ty P F Þ Ñ Hpx, yq|x P B E u and tx P E Þ Ñ hpx, yq|y P B F u are equicontinuous.

3.

BpE ˆF, Gq the vector space of all separately continuous bilinear functions from

E ˆF to G.
Continuity implies α-hypocontinuity, which in turns implies separate continuity. While separate continuity is too weak to be compatible with a fine topology on vector spaces, continuity is in general too strong to ensure the monoidal closedness of our models. Hypocontinuity turns out to be the good notion to work with, as in historical models of DiLL [START_REF] Ehrhard | On köthe sequence spaces and linear logic[END_REF].

For concision, we respectively denote as BpE ˆF q, HB α pE ˆF q and BpE ˆF q the spaces of scalar bilinear forms BpE ˆF, Rq, The projective tensor product is commutative and associative [START_REF] Jarchow | Locally convex spaces[END_REF][START_REF] Frölicher | Linear spaces and Differentiation theory[END_REF] on lcs and preserves this class of topological vector spaces. So does the weak tensor product E b σ F [19, 2.12].

HB
For wider bornologies, commutativity is immediate but associativity becomes more specific, as its asks to have a good knowledge of the bornology α on E b α F . This question is sometimes called as "Grothendieck' problème des topologies". § Proposition 12. For any lcs E, F and G we have a linear isomorphism LpE b α F, Gq " HB α pE ˆF, Gq. In particular, pE b α F q 1 " HB α pE ˆF q .

Chiralities as polarized models of MLL pol

We now detail what we believe to be the relevant setting to express the internal stability of polarized models of DiLL. Chiralities were introduced by Mellies [START_REF] Melliès | A micrological study of negation[END_REF] after an investigation in game semantics. In this section we recall the definitions of dialogue chiralities and introduce several versions, according to the involutivity of negations functors. § Definition 13 ([28]). A mixed chirality consists in two symmetric monoidal categories pP, b, 1q and pN , `, Kq, between which there are two adjunctions, one of which being strong monoidal:

pP, b, 1q pN op , `, Kq

p´q K P p´q K N % P N % (1)
with a family of natural bijections:

χ p,n,m : N pˆp, n `mq " N pˆpp b n K N q, mq (curryfication) (2)
The natural bijections χ account for the lost monoidal closedness. They must respect the various associativity morphisms by making the following diagrams commute:

N pˆppm `nq

M. Kerjean

XX:7

Multiplicative Linear Logic is the subpart of Linear Logic constructed from the b and `connectives, which is traditionnaly interpreted in a ˚-autonomous category. Interpreting polarized Multiplicative Linear Logic in Chiralities requires an additional family of morphisms, which basically says that there is only one closure operation between the interpretation of negative and positive formulas. Thus one asks for a family of natural isomorphisms in P:

clos p : ´pp K P q » pˆpq K N . closure (4)
The categorical semantics of Linear Logic interprets formulas as objects of a certain category, and proofs as morphisms. Positive formulas of Multiplicative Linear Logic are interpreted in P, negative formulas are interpreted in N . In a negative or dialogue chirality, a proof of $ n 1 , . . . , n n , p is interpreted as an arrow in N pp K P , n 1 `. . . n n q, and a proof of $ n 1 , . . . , n n as an arrow in N pˆ1, n 1 `. . . n n q. Symmetrically, in a positive chirality proofs should be interpreted as arrows in Pppn 1 `... `nn q K N , pq or Pppn 1 `... `nn q K N , ´pKqq.

We refer to [START_REF] Bauer | Chiralités et exponentielles: un peu de différentiation[END_REF] for details on the invariance by cut-elimination of this procedure. § Theorem 15. Dialogue, negative and positive chiralities provide a categorical semantics for polarized MLL.

In Topvec there may not be a shift from positive to negative describing exactly what a double negation would do to an object of P 5 . We thus introduce the following generalisation for chiralities: § Definition 16. A topological chirality takes place between two adjoint categories T and

C . It adds to this first adjunction a strong monoidal contravariant adjunction between a full subcategory of T and a full subcategory of C,

pP, b, 1q pN op , `, Kq

p´q K P p´q K N % T C % ( 5
)
and such that equations 2, 3 and 4 are still validated in T and C respectively.

In topological chiralities, proofs of MLL pol are interpreted exactly following the pattern described previously. The category T might be the category of lcs and C the reflective subcategory of complete lcs (see proposition 18 or theorem 26). However, we also present chiralities in which we have a non-transparent interpretation for ´, or in which C is not even a subcategory of T .

As an example, we briefly revisit existing models of MLL, inherited from models of DiLL, in terms of chiralities. In earlier work [START_REF] Kerjean | Weak topologies for linear logic[END_REF], the author built a model of DiLL in which formulas were interpreted by weak spaces. We argued that the fact that spaces of linear maps endowed with the pointwise convergence topology preserve weak spaces gave this model a polarized flavour. The space E `σ F :" L σ pE 1 w , F q is always endowed with its weak topology ([18, 15.4.7]) and the MLL model described in [START_REF] Kerjean | Weak topologies for linear logic[END_REF] easily refines in a chirality: § Proposition 17. The following adjunctions define a negative chirality: 5 For example when negatives are interpreted by metrisable spaces (Proposition 18), there is no operation on Topvec making a space metrisable.

pTopvec, b w , Rq pWeak op , `σ, Rq Curryfication (Equation 2) is indeed verified, due to the fact that on nuclear complete DF spaces (that is duals of nuclear fréchet spaces), separate continuity implies continuity [23, 40.2.11]. Closure (Equation 4) is exactly the fact that completion preserve the dual.

Compared to the model previously exposed [START_REF] Kerjean | A logical account for linear partial differential equations[END_REF], the interpretation of the shift to negatives ˆas a completion procedure allows to relax the condition on complete Nuclear DF spaces.

Positives formulas are interpreted as Nuclear DF spaces and need not to be completed.

Guided by intuitions of theorem 9, we show that Mackey spaces leave stable the positive constructions, and in particular a certain topological tensor product. The proof is provided in the appendix. § Proposition 19. Consider E, F P Mackey. Then E b µ F is Mackey.

It is however not enough to construct a positive chirality. Consider the following adjunctions, in which ι denotes the inclusion functor and N `µ M :"

pN 1 µ b µ M 1 µ q 1 µ : pMackey, b µ , Rq pTopvec, `µ, Rq p´q 1 µ p´q 1 µ % Mackey Topvec ι p´qµ % 6
Fréchet spaces are metrisable complete lcs, while DF spaces describe their strong duals. Nuclear spaces are the lcs on which several different topological tensor product correspond. Precise definitions can be found in the litterature [18, 12.4, 21.1] M. Kerjean

XX:9

They would define a positive chirality if we had a good characterization of weakly compact sets on E b µ F , allowing us to prove the associativity of `µ. As of today, it is however not the case.

Thus we investigate the interpretation of positive formulas MLL in Mackey spaces. This leads to three models: a first one based on barrelled spaces (section 4) and two others refining it with the notion of bornological spaces (section 5). The goal now is to handle as negatives spaces with some completeness, in order to work with smooth functions and differentiability.

Decomposing reflexivity through polarization

In this section we show that reflexive spaces decompose in a dialogue chirality. Remember that a lcs E is reflexive if and only if is barrelled and semi-reflexive (definition 7). Semi-reflexivity can in fact be characterized in terms of completeness: § Proposition 20. [18, 11.4.1] E is semi-reflexive iff is is weakly quasi-complete: any bounded Cauchy filter in E converges in E σ . Thus E is reflexive iff it is barrelled and weakly quasi-complete.

These requirements enjoy antagonist stability properties: barrelled spaces are stable under inductive limits while weak completeness is preserved by projective limits. In fact, barrelledness and weak quasi-completeness are in duality: § Proposition 21. [18, 11.1.4] A Mackey space is barrelled if and only if its weak dual is quasi-complete.

This proposition is the backbone for the construction of a new chirality between Barr, the full subcategory of barrelled lcs and wqCompl, the full subcategory of weak quasi-complete lcs. We first retrieve a fundamental proposition allowing to prove curryfication (equation 2)

and then state the necessary stability, associativity and monoidality lemmas.

β F q 1 " LpE, F 1 β q.
We denote by pLpE, F qq µ the space LpE, F q endowed with the Mackey topology induced by its predual pE b β F 1 q. § Proposition 25. 1. The Mackey dual of a weak quasi-complete space is barrelled and the weak dual of a barrelled space is quasi-complete.

2.

Consider F a weak and quasi-complete space, and E a barrelled space. Then for any lcs E, L σ pE, F q is quasi-complete and endowed with its weak topology.

3.

Consider E, F barrelled spaces. Then pE b β F q 1 w » L σ pE, F 1 w q.

Consider E, F two weak and quasi complete lcs. Then pL

σ pE 1 µ , F qq µ » E 1 µ b β F 1 β .

5.

The binary operation `w : pE, F q Þ Ñ L σ pE 1 µ , F q is associative and commutative on wqCompl.

6.

Consider F P Weak and E P Mackey. Then LpE w , F q " LpE, F µ q and LpE 1 σ , F q " LpF 1 µ , Eqq 2) is given by proposition 25.7 and closure (Equation 4) by proposition 25.8. § Remark 27. As indicated to the author by Y. Dabrowski, there is in fact of a closure operation making lcs barrelled [30, 4.4.10], which would give a dialogue chirality and not a topological dialogue one. It is however not needed here to interpret proofs of polarized MLL.

Duality with bornological spaces

Bornological spaces were at the heart of the duality in vectorial models of LL [START_REF] Tasson | Sématiques et syntaxes vectorielles de la logique linéaire[END_REF]III.5],

and in the first smooth intuitionistic model of DiLL [START_REF] Blute | A convenient differential category[END_REF]. However, it was shown that in the context of intuitionistic smooth models, bornological topologies were unecessary, and the first model made of bornological and Mackey-complete spaces was refined into a model made only of Mackey-complete space [START_REF] Kerjean | Mackey-complete spaces and power series -a topological model of differential linear logic[END_REF]. We show that bornologicality is in fact the key to make smooth models classical, through polarization.

In this section, we describe two topological chiralities based on bornological spaces.

Section 5.2 offers a polarized extension to Intuitionistic Models of DiLL [START_REF] Blute | A convenient differential category[END_REF], while in section 5.3 we describe a chirality refining [START_REF] Dabrowski | Models of linear logic based on the schwartz ε product[END_REF] which could lead to a more satisfactory interpretation of differentiation.

Bornologies and bounded linear maps

In this section, we recall preliminary material on the more specific subject of vector spaces endowed with bornologies, as exposed in the litterature [START_REF] Hogbe-Nlend | Bornologies and Functional Analysis[END_REF][START_REF] Frölicher | Linear spaces and Differentiation theory[END_REF]. So far, we worked with topological vector spaces, on which the canonical bounded subsets are the one of βpEq.

One can also work with bornologies as the primary structure, and from that construct 0-neighbourhoods as those which absorb any element of the bornology. We denote by LpE, F q the vector space of all bounded linear maps between E, F P Bornvec.

Is is endowed with the bornology of all equibounded sets of functions, that is sets of functions sending uniformly a bounded set in E to a bounded set in F . § Definition 30. We consider the category BornVec of vector spaces endowed with a convex Hausdorff vector bornology, with linear bounded maps as arrows.

While the converse is not true, a linear continuous map is always bounded. ToppB E q and F endowed with ToppB F q.

The interaction between bornologies and topologies is best described through the following adjunction [15, 2.1.10]:

BornVec Topvec

Top Born

%

In the light of section 3, as the domain of a left-adjoint functor, spaces with bornologies should interpret positive connectives while lcs are better suited to interpret negatives. We will refine this intuition through the category of bornological lcs, which is the co-reflective category arising through the previous adjunction. § Proposition 33. [18, 13.1.1] A lcs E is said to be bornological if one of these following equivalent propositions is true:

For any other lcs F , any bounded linear map

f : E / / F is continuous, that is LpE, F q " LpE, F q,
2. E is endowed with the topology Top ˝BornpEq,

E is Mackey, and any bounded linear form

f : E / / K is continuous.
We denote by bTopVec the category of bornological lcs and continuous (equivalently bounded) linear maps between them7 . Equivalently to bTopVec, one can consider topological spaces in BornVec, that is spaces in BornVec which are invariant under Born ˝Top. This are the vector spaces with a convex vector bornology which consists exactly of all the sets absorbed by all the bornivorous subsets.

necessarily a continuous one. The previous adjunction should lead to a polarized model of DiLL with Complete Schwartz spaces, in which smooth maps are defined on ultrabornological spaces, and their differential is thus immediately continuous. Indeed, lifting this model -to higher-order will lead to functions having an ultrabornological space as codomain, and thus to bounded functions to be continuous.

Conclusion

This work presented several chiralities of topological vector spaces, and refines four preexisting smooth models of Differential Linear Logic. We show that chiralities are a good setting for mostly preexisting yet intricate results in the theory of topological spaces, and that this mathematical theory sheds light on previously unseen computational behaviours. Indeed, the following features are observed here:

Two distinct negations (in Theorem 26, Theorem 40 and Theorem 45).

A non-transparent interpretation of the positive shift ´in Theorem 26 and Theorem 40.

Chiralities which are not dialogue chiralities but which feature a negation involutive only on the negatives (Proposition 18) or on the positive (Theorem 40).

The chirality of barrelled spaces and weakly quasi-complete lcs is the most elegant one, as any topological operation corresponds to a logical operation. The situation is less clear in the case of bornological spaces, which are a good interpretation for the positives but on which the interpretation of the negatives undergoes closure operations. In particular, the role of Mackey-completion is not clear. While it allows to interpret positive formulas by ultrabornological (thus barrelled) spaces, it results in a completeness condition on positive formulas, while completeness is usually understood as the characteristic of negative formulas.

The results exposed here could lead to developments in the theory of programming languages involving linear negations [7] [6]. But most of all, we believe that chiralities are a good setting for the non-linear part of Differential Linear, and the models presented here would serve as a basis for models of higher-order differential computations. Indeed, there is no categorical semantics of DiLL reflecting the symmetry of its exponential laws. We conjecture that chiralities should model the interaction between positives and negatives, as well as the interaction between linear proofs and non-linear proofs in DiLL. In the linear-non-linear chirality, the exponential would model the strong monoidal adjunction, while dereliction and codereliction would be modelled as shifts.

  α pE ˆF, Rq and BpE ˆF, Rq. § Definition 11. The projective tensor product E b π F is the finest topology on E b F making the canonical bilinear map E ˆF Ñ E b F continuous. The α-tensor product E b α F is the finest topology on E b F making the canonical bilinear map h : E ˆF Ñ E b F α-hypocontinuous.

%%

  in which ι denotes the inclusion functor.More recently, in order to find the good setting in which to interpret non-linear proofs as the usual smooth function of real analysis, we constructed a polarized model of DiLL[START_REF] Kerjean | A logical account for linear partial differential equations[END_REF] [21] in which positive formulas are interpreted as complete nuclear DF spaces and negative formulas are constructed as nuclear Fréchet spaces6 . § Proposition 18. For its multiplicative part, the distribution model of DiLL organises into the following negative topological chirality: pNdf, bπ , Rq pNf op , b, Rq in which ˜denotes the completion of a lcs, bπ denotes the completion of the projective tensor product, Ndf the category of nuclear DF spaces, Nf the category of Nuclear Fréchet spaces, Compl the category of complete lcs and Ẽ the completion of the lcs E.

  The proofs are given in appendix. § Proposition 22. [5, III.5.3.6 ] When E and F are barrelled, every separately continuous bilinear map on E ˆF is β-hypocontinuous. § Proposition 23. The bounded tensor product b β preserves barrelled spaces and is associative on Barr. § Proposition 24. Consider E and F two barrelled lcs. Then pE b

7 .

 7 Consider E P Barr and F P wqCompl and G P W eak . Then:LpE, L σ pF 1 µ , Gqq " LpE b β F 1 µ , Gq 8.For any E P Mackey and thus any E P Barr, pE w q 1 µ » pE 1 w q µ .

	As the others, the proof of the preceding proposition is detailled in the appendix. Let
	us however insist on the fact that the remarkable stability properties are quite inherent to
	barrelledness : for example, the second point is proven thanks to Banach-Steinhauss theorem,
	which precisely holds for function with barrelled spaces as codomains.
	§ Theorem 26. Barrelled spaces and weak quasi-complete spaces organise in the following
	topological dialogue chirality:
		p´q 1 σ			p´qσ
	pBarr, b β , Rq	%	pwqCompl op , `w, Rq	Mackey	%	Weak
					p´qµ
		p´q 1 µ
	in which curryfication (Equation

  Definition 29. A bounded map is a map for which the image of a bounded set is bounded.

§ Definition 28. Consider E a vector space. A bornology on a vector space E is a vector bornology if it is stable under addition and scalar multiplication. It is convex if it is stable under convex closure, and Hausdorff if the only bounded sub-vector space in B is t0u. M. Kerjean XX:11 §

  Thus we have a functor Born : Topvec / / BornVec mapping any lcs E to the same vector space endowed with its bornology βpEq, and a linear continuous function to itself. § Definition 31. Consider E P BornVec with bornology B E . Then a subset U Ă E is said to be bornivorous if for every B P B E there is a scalar λ P K such that B Ă λU .

We consider the functor Top : BornVec / / Topvec which maps E to the lcs E with the topology of generated by bornivorous subsets, and which is the identity on linear bounded functions. § Proposition 32. A linear bounded map between two vector spaces E and F endowed with respective bornologies B E and B F defines a linear continuous maps between E endowed with

Beware of the difference between spaces of BornVec which are not endowed with a canonical bornology, and bornological lcs of bTopVec.

Note that equicontinuity only depends on the topology of E and not on the choice of a bornology on E

For a Mackey space, being ultrabornological is also equivalent for the strong nuclearification of its Mackey dual to be complete. Thus everything done here in terms of Schwartz spaces could be done in term of nuclear spaces, as for unpolarized smooth models of DiLL

A lcs can be endowed with several Schwartz topologies which preserve the dual, while bornologification for example depends only of the dual pair pE, E 1 q as bounded and weakly bounded set correspond.

XX:12 Chiralities in topological vector spaces § Proposition 34. [START_REF] Frölicher | Linear spaces and Differentiation theory[END_REF] bTopVec is a co-reflective category in Top and tBornVec is reflective in BornVec.

bTopVec Topvec U Top˝Born %

BornVec tBornVec

Born˝Top ι % in which U and ι denotes forgetful functors, leaving objects and maps unchanged. § Proposition 35. [30, 11.3] Consider E and F two bornological lcs. Then E b β F is bornological.

As b B is associative and commutative on tBornVec (see [15, 3.8

%

This however is not enough to have a chirality: we do not have a suitable interpretation for the dual of b β which would be associative on all Mackey spaces, and not just on duals of bornological spaces. More generally, bornological spaces do not verify a duality theorem with some kind of complete spaces, or at least not some kind involving duals which preserves reflexivity [18, 13.2.4]. One solution detailed detailed in section 5.2, is to add a suitable notion of completeness. The other solution, in section 5.3 is to refine our setting, and consider ultrabornological spaces.

Convenient vector spaces classically

To the notion of bornology corresponds a good notion of completeness, enforcing the convergence of Cauchy sequences with respect the norms generated by bounded subsets. § Definition 37. [START_REF] Frölicher | Linear spaces and Differentiation theory[END_REF] Consider V an absolutely convex and bounded subspace of a E P BornVec. We denote by E V the vector space generated by V . It is a normed vector space when endowed with the gauge:

An absolutely convex and bounded subset V is said to be a Banach disk when E V is complete for its norm. E is said to be Mackey-Complete when every absolutely convex and bounded subset is a banach disk.

Equivalently, the definition of Mackey-Completeness extends to Topvec when one considers the bounded sets of βpEq. We choose the notation Mco to denote the full subcategory of Topvec made of Mackey-complete lcs. Mackey-complete spaces are the heart of several smooth models of DiLL [START_REF] Blute | A convenient differential category[END_REF][START_REF] Dabrowski | Models of linear logic based on the schwartz ε product[END_REF][START_REF] Kerjean | Mackey-complete spaces and power series -a topological model of differential linear logic[END_REF].

M. Kerjean

XX:13

In particular, in work by Blute Ehrhard and Tasson [4] formulas were interpreted by convenient spaces, that is bornological lcs which are also Mackey-complete

Ultrabornological and Schwartz spaces

In this section, we refine the previous chirality into a finer one, to get closer to objects used in the first classical non-polarized smooth models of DiLL [START_REF] Dabrowski | Models of linear logic based on the schwartz ε product[END_REF]. Convenient spaces are a particular case of ultrabornological spaces, that is spaces which are bornological with respect to a stricter class of bounded subsets. § Definition 41. [18, 11.1] A lcs E is said to be ultrabornological when its 0-neighbourhoods are exactly the one absorbing all Banach disks.

Let us denote by ubTopVec the full subcategory of ultrabornological spaces. If we denote by uBorn : Topvec / / BornVec the functor mapping a lcs E to the same vector space endowed with the bornology of its Banach disks, we have a coreflective subcategory: V stands for an equicontinuous subset of E 9 (proposition [18, 10.4.1]). We denote by Schw the full subcategory of Schwartz lcs.

We denote by S : Topvec / / Schw the functor mapping a lcs to the same lcs endowed with the topology of uniform convergence on the sequences in E

%

in which refers to Schwartz' ε product [START_REF] Schwartz | Théorie des distributions à valeurs vectorielles[END_REF].

The cartesian closed category of smooth maps, interpreting the non-linear proofs of Linear Logic in [START_REF] Dabrowski | Models of linear logic based on the schwartz ε product[END_REF], was based on the same pattern than the smooth maps in the bornological setting [START_REF] Kriegl | The convenient setting of global analysis[END_REF]. In that context, differentiation leads to a bounded linear function, and not then there is a 0-neighbourhood V in G such that h´1 pV q is not a 0-neighbourhood in E b α F .

A Proofs omitted in the paper

The topology on E b F generated by the α topology still makes h α α-hypocontinuous as h ´1 α p h´1 pV qq " h ´1pV q, and we obtain a contradiction. đ

Proof of Proposition 19. For the purpose of this proof, we will denote by pEbF q µ the tensor product of two lcs, endowed with the Mackey topology induced by its dual HB µ pE ˆF q. By proposition 12, we have that pE b µ F q 1 " HB µ pE ˆF q, thus the topology of E b µ F is coarser that the one of pE b F q µ . Let us show that the canonical bilinear map h : E ˆF / / pE b F q µ is µ-hypocontinuous. Then as pE b µ F q is defined as the finest topology on the tensor product making h µ-hypocontinuous, we will have that that E b µ F is finer that the one of pE b F q µ and our proof will follow.

Consider K a weakly compact absolutely convex subset of E. Let us show that the family of functions hpK, ´q is equicontinuous. As F is endowed with its Mackey topology, continuity from F to pE b F q µ is equivalent to weak continuity from F to E b F endowed with the weak topology induced by HB µ pE ˆF q. Consider P pE b F q 1 µ . By proposition 9 we have that ˝h P HB µ pE ˆF q, and thus the family ˝hpK, _q is equicontinuous. Equicontinuity By the fact that on barrelled spaces, β-hypocontinuity and separate continuity correspond, we have that these two space are linearly isomorphic. đ

XX:18 Chiralities in topological vector spaces

Proof of Proposition 24. By proposition 12 we have pE b β F q 1 " HB β pE ˆF q. Let us show that HB β pE ˆF q " LpE, F 1 β q. Consider h P HB β pE ˆF q. For any x P E, we have by the fact that hypocontinuity implies separate continuity that hpx, _q P F 1 , and by hypocontinuity that x Þ Ñ hpx, _q is continuous from E to F 1 β . Conversely, any P LpE, F 1 β q is hypocontinuous by proposition 22. đ

Proof of Proposition 25. 1. This follows from proposition 21 and from the fact that weak duals are weak spaces, and Mackey duals are Mackey spaces by proposition 9.

2.

That L σ pE, F 1 W q is endowed with its weak topology follows from [18, 15.4.7]. Quasicompleteness follows from the fact that bounded sets of L σ pE, F 1 W q are the simply bounded ones, that bounded Cauchy filters converge pointwise thanks to the quasicompleteness of F , and that this limit function defined pointwise is continuous thanks to the Banach-Steinhauss theorem [18, 11.1.3] applied to E.

3.

By proposition 12 we have that pE b β F q 1 " HB β pEˆq. As on barrelled spaces βhypocontinuous functions and separately continuous functions correspond, we have in turn that pE b β F q 1 " HB β pEˆq " BpE ˆF q " LpE.F 1 w q. The linear homeomorphism pE b β F q 1 w » L σ pE.F 1 w q follows from the fact that the latter space is endowed with its weak topology [18, 15.4.7].

4.

As L σ pE 1 µ , F q is induced by the weak topology induced by E 1 b F 1 , we have that

As E 1 µ and F 1 µ are both barrelled spaces, it follows from propositions 19 and 23 that E 1 µ b β F 1 β is Mackey and linearly homeomorphic to pE 1 b F 1 q µpLpE 1 µ ,F qq .

5.

Associativity and commutativity follow from the fact that L σ pE 1 , F q is endowed with the weak topology induced by E 1 b F 1 .

6.

The first point follows from proposition 9 (and is in fact part of the proof to the adjunction 9). The second point follows from [18, 8.6

8.

Both pE w q 1 µ and pE 1 w q µ correspond algebraically to the vector space E 1 . The former is endowed with the topology of uniform convergence of σpE w q compact subsets of E. The second is endowed with the topology of uniform convergence on σppE 1 w q 1 q compact subsets of E. As E " pE As bornological lcs are in particular Mackey, we have that pBornpTopp´q 1 µ q 1 µ q conv is the identity on Conv.

For E P Conv and F P Mco, one has by the diverse adjunctions at stakes: LpToppEq 1 µ , F q » LpF 1 µ , pT oppEq 1 µ q 1 µ q » LpF 1 µ , ToppEqq » LpBornpF 1 µ q, Eq » Lp { BornpF 1 µ q M , Eq. Thus the contravariant adjunction is proved.

One has easily that pE bM B F q 1 B » L β pToppEq, F 1 B q, and thus p´q 1 B is indeed a strong monoidal functor.


François Montel
email: §francois.montel@univ-pau.fr

Hai Hoang

Guillaume Galliero

Linking up Pressure, Chemical Potential and Thermal Gradients

Petroleum reservoirs are remarkable illustrations of the impact of a thermal gradient on fluid pressure and composition. This topic has been extensively studied during the last decades to build tools that are required by reservoir engineers to populate their models. However, one can get only a very limited number of representative samples from a given reservoir and assessing connectivity between all sampling points is often a key issue. In some extreme cases, the whole reservoir fluid properties must be derived from a single point to define the field development plan. To do so, available models are usually not satisfactory as they need too many parameters and so cannot be considered as predictive tools. We propose in this work a comprehensive approach based on the irreversible thermodynamics principles to derive the relationships between pressure, chemical potentials and thermal gradients in porous media. It appears that there is no need for additional assumptions, it is just a matter of a making the right choices along theoretical developments. One of the most important steps is to express the full pressure gradient. As a final result, we obtain the chemical potential gradients for all components of the mixtures that can be easily translated in term of compositions through Equation of State modelling. The most important features of the final expressions are: (i) the species relative separation in a thermal field is sensitive to the relative diffusion coefficients at stationary state.

In porous media, the separation is sensitive to the permeability when the overall mobility is similar to diffusive mobility; (ii) the magnitude of the separation depends on the residual entropy of the species; (iii) the separation is not simply balanced by the average residual entropy. The balance is modified by the relative diffusion mobility of the components; (iv) in low permeability porous media, the thermal gradient induces a pressure gradient proportional to the fluid residual entropy. As a validation, the proposed approach has been applied on a reservoir fluid subjected to a geothermal gradient and compared with non-equilibrium molecular dynamics simulation results at the stationary state.

I. Introduction

The objective of this paper is to provide a general framework for the evaluation of the impact of a thermal gradient on fluid pressure and composition, i.e. thermodiffusion and related phenomena, in a porous medium at the stationary state. The main driver for this work is the development of a generic macroscopic formalism to describe the fluid distribution in petroleum reservoirs before production but many other applications of the proposed approach could be found in natural and industrial processes, see for example the recent review by Köhler and Morozov [1]. In petroleum industry, this topic is discussed in detailed in many recent textbooks [START_REF] Firoozabadi | Thermodynamics and Applications in Hydrocarbon Energy Production[END_REF][START_REF] Esposito | Compositional Grading in Oil and Gas Reservoir, 1 st Edtion[END_REF], and correlations for the thermodiffusion parameters, despite their intrinsic limitations [1], are commonly used by reservoir engineer to populate their reservoir models.

In some field cases, the geothermal gradient induces compositional gradients of the same order of magnitude than the gravitation [START_REF] Ghorayeb | [END_REF][5]. Since the gravitational segregation is well quantified, the current practice in the petroleum industry consists in tuning the thermodiffusion parameters in order to match the available compositional field data. However, it happens quite often that the fluid column species distribution must be established from a very limited number of samples, sometimes only one as it is the case when data are coming from the discovery well.

And it is of primary importance to get indisputable estimation of the thermodiffusion parameters to assess the fluid column species distribution and consequently oil and gas accumulation. A typical critical situation is the case of a discovery well crossing an oil pool, down-dip a large accumulation, for which a crucial question arises: is there a gas cap in the reservoir or not and if any where is it located? The symmetrical critical situation is encountered when the discovery well crosses the gas phase only with a possible oil rim down-dip.

Thus, building a predictive model to describe fluid distribution in oil and gas reservoirs at the stationary state is a challenge that may be a game changer for the petroleum industry. Following a previous work on the subject [6], in which the conditions for application of the stationary state model was discussed, the proposed article is a step towards that direction. The article is organized as follows. In section II, the theoretical framework is provided. Details on the new predictive approach is described in section III. Then, the methodology is applied and validated on a practical case in section IV. Finally, the main outcomes of this work are summarized in section V.

II. Theoretical framework

To derive the conservation equations and the related flux-forces relations as proposed in this work, our main hypothesis are: A comprehensive discussion on these assumptions when dealing with oil and gas reservoir fluids can be found in Ref. [6].

Material balance equation

To write the material balance equation per species, we consider an elementary representative volume of the porous medium filled with a reservoir fluid, in which ni is the number of moles of component "i" in this elementary volume.

𝜕𝑛 𝑖 𝜕𝑡 + ∇𝐉 𝐢 = 𝟎 (1)
where, Ji is the total molar flux of each component, which is a linear function of the thermodynamic forces, composed of three contributions (advection, thermodiffusion and molar diffusion). It can be expressed in terms of molar quantities in the absence of an inertial term (hypothesis "a") [6,[START_REF] De Groot | Nonequilibrium Thermodynamics[END_REF]:

𝐉 𝐢 = -𝑛 𝑖 𝐿 𝑃 ( V∇𝑃-𝑀 𝐠 𝑇 ) + 𝑛 𝑖 𝐿 𝑖𝑞 ∇ ( 1 𝑇 ) + 𝑛 𝑖 ∑ 𝐿 𝑖𝑗 ( 𝑀 𝑗 𝐠 𝑇 + ∇ (- 𝜇 𝑗 𝑇 )) 𝑗 ( 2
)
where P is the pressure, T the temperature, 𝜇 𝑖 is the molar contribution of component "i" to the free energy of the fluid in the porous medium, it is the chemical potential in absence of specific interactions between the component "i" and the solid phase [6], g is the gravitational acceleration M is the molecular weight, V is the molar volume of the fluid, and Lp, Liq, Lij are phenomenological coefficients related to advection (i.e. barodiffusion), thermodiffusion and molar diffusion respectively.

From the formulation of the molar flux per species, it is possible to deduce the advective fluid velocity which is simply the sum of the advective contributions of all components:

𝐔 = - 1 𝜌 ∑ 𝑛 𝑖 𝑀 𝑖 𝐿 𝑃 ( V∇𝑃-𝑀 𝐠 𝑇 ) 𝑖 = - 𝐿 𝑝 𝑇 (∇𝑃 -𝜌 𝐠) (3)
where, ρ is the density. The sum of the thermodiffusive and diffusive contributions is nil in a closed system. However, it should be kept in mind that, in many petroleum reservoirs, there exist some stationary light gas component fluxes coming from deep sources through water saturated surrounding rocks, i.e. open systems. In such situations, the sum of these contributions are not nil at the stationary state [6].

Energy equation

If we neglect the viscous dissipation, the energy conservation equation can be written as:

𝜕𝐻 𝑚 𝜕𝑡 = ∇ • 𝐉 𝐓 (4)
where Hm is the enthalpy of the saturated porous medium and the total heat flux JT is expressed by: 𝐉 𝐓 = 𝐿 𝑞𝑞 ∇ ( (5) in which ones recognizes a pure conduction contribution, an advective term and a thermodiffusion (Dufour) term with the corresponding phenomenological coefficients Lqq, Lp ,Lqj, respectively. H is the enthalpy defined as, 𝐻 = ∑ 𝑛 𝑖 ℎ 𝑖 𝑖 , where hi is the partial molar enthalpy of component i.

The contribution of the molar flux of component "i" to the heat flux can be written as:

𝐉 𝐪𝐢 = -
in which ℎ 𝑖 * is the molar enthalpy of component i carried out during molar diffusion. As for the difference between overall permeability and individual mobility [6], we distinguish between the enthalpy of the component in the bulk fluid flow ℎ 𝑖 and the enthalpy transported by the same component, in and out the elementary volume, through a diffusion process, ℎ 𝑖 * . This difference is crucial. A single molecule carries only its own enthalpy. The energy balance of a single molecule in and out the elementary volume is not equal to its partial enthalpy in the fluid phase multiplied by the temperature difference if it moves alone independently from other molecules. ℎ 𝑖 * differs from the partial enthalpy of the component "i" .

From the previous equations, i.e. Eqs. (5)(6), it is straightforward to deduce that the total heat flux can be written as :

𝐉 𝐓 = 𝐿 𝑞𝑞 ∇ (

Molar flux

The reciprocal Onsager relations imply that the contribution of molar diffusive fluxes to the total heat flux must be consistent with the contribution of the thermal gradient to the molar diffusive fluxes. The coefficients are equal when the right thermodynamic forces and conjugate variables are selected [START_REF] De Groot | Nonequilibrium Thermodynamics[END_REF]. Therefore, in the proposed formulation, the phenomenological thermodiffusion coefficients are:

𝐿 𝑖𝑞 = 𝐿 𝑞𝑖 = 1 𝑛 𝑖 ∑ 𝑛 𝑗 𝐿 𝑖𝑗 ℎ 𝑖 * 𝑗 (8)
and the component fluxes, Eq. ( 2), can be expressed as :

𝐉 𝐢 = -
It should be noted that a similar approach was initially proposed by Haase [START_REF] Haase | Thermodynamics of irreversible processes[END_REF] and applied to petroleum reservoir by Pedersen and Lindeloff in 2003 [START_REF] Pedersen | Simulations of Compositional Gradients in Hydrocarbon Reservoirs Under the Influence of a Temperature Gradient[END_REF].

It is possible to go even further by using the fact that all the thermodynamic properties of a component i can be derived from its chemical potential and introducing the chemical potential gradient at constant temperature and the molar entropy obtains:

∇𝜇 𝑖 𝑇 = -𝑠 𝑖 ∇T+∇ 𝑇 𝜇 𝑖 𝑇 (10)
Finally, the driving force for molecular diffusion can be expressed as:

∇ (-

𝜇 𝑖 𝑇 ) = - ∇ 𝑇 𝜇 𝑖 𝑇 -ℎ 𝑖 ∇ ( 1 𝑇 ) (11)
By combining Eqs. ( 9) and ( 11), the component fluxes can be written as:

𝐉 𝐢 = -𝑛 𝑖 𝐿 𝑃 ( V∇𝑃-𝑀 𝐠 𝑇 ) + 𝑛 𝑖 ∑ 𝐿 𝑖𝑗 (( 𝑛 𝑗 𝑛 𝑖 ℎ 𝑖 * -ℎ 𝑗 ) ∇ ( 1 𝑇 ) - ∇ 𝑇 𝜇 𝑗 𝑇 + 𝑀 𝑗 𝐠 𝑇 ) 𝑗 (12)
The cross phenomelogical diffusion terms Lij are very difficult to get and can be considered to be small compared to the diagonal terms. This is of course not true for Fickian cross diffusion terms, even if they are usually small compared to the diagonal terms. The impact of chemical potential derivative matrix can even become singular at the critical point. Therefore, we consider: 𝐿 𝑖𝑗 = 0 if 𝑖 ≠ 𝑗 and 𝐿 𝑖𝑖 = 𝐿 𝑖 if 𝑖 = 𝑗. In addition, if we use the following quantities:

𝑞 𝑖 = ℎ 𝑖 -ℎ 𝑖 * 𝑇 and 𝑄 = ∑ 𝑥 𝑖 ℎ 𝑖 -ℎ 𝑖 *
𝑇 𝑖 (13) The final expressions of the molar fluxes become:

𝐉 𝐢 = -𝑛 𝑖 𝐿 𝑃 ( V∇𝑃-𝑀 𝐠 𝑇 ) + 𝑛 𝑖 𝐿 𝑖 𝑇 (𝑞 𝑖 ∇𝑇 -∇ 𝑇 𝜇 𝑖 + 𝑀 𝑖 𝐠) (14)

III. Practical formulations

Pressure and chemical potential gradients

General case in an open system

The purpose of this section is to derive pressure and chemical potential gradients within the framework provided in the previous section. To do so, we first sum the component fluxes divided by Li which gives

𝑀 𝜌 ∑ 𝐉 𝐢 𝐿 𝑖 𝒊 = - 𝐿 𝑃 𝐿 𝐷 ( V∇𝑃-𝑀 𝐠 𝑇 ) + 𝑄 𝑇 ∇T - 1 𝑇 ∑ 𝑥 𝑖 (∇ 𝑇 𝜇 𝑖 -𝑀 𝑖 𝐠) 𝒊 ( 15
)
where xi is the mole fraction of component i and LD, the average diffusion mobility, is given by:

1 𝐿 𝐷 = ∑ 𝑥 𝑖 𝐿 𝑖 𝑖 (16)
Applying the Gibbs-Duhem relationship [START_REF] De Groot | Nonequilibrium Thermodynamics[END_REF] in the thermal field and the gravity field yields:

∑ 𝑥 𝑖 (∇ 𝑇 𝜇 𝑖 -𝑀 𝑖 𝐠) 𝒊 = ∑ (∇ 𝑇 𝜇 𝑖 -𝑀 𝑖 𝐠) 𝒊 + 𝑆∇𝑇 = 𝑉∇𝑃 -𝑀𝐠 (17)
Then, it is possible to deduce the pressure gradient:

∇𝑃 -𝜌g = 𝜌 𝑀 𝑄∇𝑇-𝑇 ∑ J i 𝐿 𝑖 𝑖 𝐿 𝑃 𝐿 𝐷 +1 (18)
and using Eq.( 14) and Eq.( 18) we obtain the chemical potential gradient at constant temperature:

∇ 𝑇 𝜇 𝑖 -𝑀 𝑖 𝐠 = (𝑞 𝑖 - 𝐿 𝑝 𝐿 𝑖 ( 𝑄 𝐿 𝑃 𝐿 𝐷 +1 )) ∇𝑇 - 𝑀 𝜌 𝑇 ( J i 𝑥 𝑖 𝐿 𝑖 - ∑ 𝐉 𝐣 𝐿 𝑖 𝒋 𝐿 𝑖 ( 𝐿 𝑃 𝐿 𝐷 +1) ) (19)
It should be emphasized that, in our approach, the link between thermodynamic properties and transport properties is build-in. The stationary state is a balance between the various forces weighted by the mobility of each molecule.

There are some interesting features in Eq. ( 19):

• The separation in a thermal field is sensitive to the relative diffusion coefficients at stationary state. In porous media, the separation is sensitive to the permeability (through LP coefficient) when the overall mobility LP is similar to the average diffusive mobility LD.

• The magnitude of the separation depends on qi quantities: difference between the enthalpy of the component in the bulk flow and the enthalpy of the same component moving independently through a diffusion process.

• External fluxes impact the compositional distribution, as expected from fluxes, cf. eq. [START_REF] Rosenfeld | [END_REF]. External fluxes values that just cancel the thermodiffusion are given by 𝐉 𝐢 = 𝜌 𝑀 𝑥 𝑖 𝐿 𝑖 𝑞 𝑖 𝑇 ∇𝑇

Closed system

In a closed system, the previous relations reduce to:

∇ 𝑇 𝜇 𝑖 -𝑀 𝑖 𝐠 = (𝑞 𝑖 - 𝐿 𝑝 𝐿 𝑖 ( 𝑄 𝐿 𝑝 𝐿 𝐷 +1 )) ∇𝑇 (20) ∇𝑃 -𝜌g = 𝜌 𝑀 ( 𝑄 𝐿 𝑝 𝐿 𝐷 +1 ) ∇𝑇 (21)
As expected, summation of Eq. ( 20) verifies Gibbs-Duhem [START_REF] De Groot | Nonequilibrium Thermodynamics[END_REF] relation:

∑ 𝑥 𝑖 (∇ 𝑇 𝜇 𝑖 -𝑀 𝑖 𝐠) 𝑖 = Q 𝐿 𝑝 𝐿 𝐷 +1 ∇𝑇 = 𝑀 𝜌 (∇𝑃 -𝜌g) = 𝑉∇𝑃 -𝑀g (22)

Permeable porous media case

In case of a high overall mobility (LP >>LD), Eq.(20) reduce to:

∇ 𝑇 𝜇 𝑖 -𝑀 𝑖 𝐠 = (𝑞 𝑖 - 𝐿 𝐷 𝐿 𝑖 𝑄) ∇𝑇 (23)
and according to Eq.( 21) the pressure gradient becomes hydrostatic: ∇𝑃 = 𝜌g. This is true only if LP >>LD , we will see in §3.3 that this is true for most of the petroleum reservoirs except for the unconventional play (Oil and Gas shale). Practical expressions for LP and LD are given in $3.3. Thus, in permeable porous media, the separation results from a balance between two forces. The thermal force pushes the molecules but the diffusion process holds it back.

Low permeability porous media case

In case of low overall mobility (tight porous media) Eq.(20) reduce to:

∇ 𝑇 𝜇 𝑖 -𝑀 𝑖 𝐠 = 𝑞 𝑖 ∇𝑇 (24) ∇𝑃 -𝜌g = 𝜌 𝑀 𝑄 ∇𝑇 (25)
Thus, in low permeability porous media, the thermal gradient induces the highest possible pressure gradient. Interestingly, without gravity, the pressure gradient becomes:

∇𝑃 = 𝑄 𝑉 ∇𝑇 (26)
Applied to a pure compound this just means that ∇𝜇 = 0 and it leads to the Clapeyron equation [START_REF] Poling | The properties of Gases and Liquids, 5 th Edition[END_REF] along an interface if Q=Sres. (molar residual entropy). It is a strong indication that 𝑞 𝑖 should be related to the molar residual entropy of component "i" in a mixture. We will come back on that important point latter.

Aerial thermal gradient

In the case of a liquid vapour interface subjected to an aerial thermal gradient and a gravity, see

At this point, we assume that the entropy of the component "i' alone diffusing in and out the elementary volume is its ideal entropy at reference T and V conditions. Such a clear link between thermodiffusion and residual entropy is strongly supported by recent results obtained from molecular simulations of isotopic mixtures [START_REF] Hoang | [END_REF] and is consistent with what known for other transport properties [START_REF] Rosenfeld | [END_REF][13]. It is also compulsory to be consistent with the Clapeyron equation along a vapour/liquid interface.

But it remains an assumption until we could make a clear evaluation of the heat transfer through molecular diffusion.

It is then possible to deduce that:

𝑞 𝑖 =
And, starting from a reference point at which the fluid properties are known, the new compositions are obtained by solving a set of linear equations.

Link with usual transport properties

The phenomenological coefficients defined so far, Li, Lp ,Lqj are not practical quantities. Thus, in the following we link these coefficients with more usual quantities. The phenomenological advection coefficient is simply deduced from the fact that the advective fluid velocity without chemical and thermal forces is described by a simple Darcy's law [6]:

𝐔 = - 𝑘 𝜂 (∇𝑃 -𝜌 𝐠) (38)
where, k is the permeability of the porous medium and  the viscosity of the fluid. From Eq.

(3), therefore, it is possible to deduce that :

𝐿 𝑃 = 𝜌 𝑀 𝑇 𝑘 𝜂 = 𝑃 𝑍𝑅 𝑘 𝜂 (39)
The phenomenological diffusion coefficient Li can be linked to the usual effective diffusion coefficient Di, only when dealing with ideal mixtures: 𝐿 𝑖 = 𝐷 𝑖 /𝑅.

The average diffusion mobility LD is linked to usual effective diffusion coefficient Di of the components [6], therefore:

𝐿 𝑖 𝐿 𝐷 = ∑ 𝑥 𝑘 𝐿 𝑘 𝑘 𝐿 𝑖 = 𝐷 𝑖 ∑ 𝑥 𝑘 𝐷 𝑘 𝑖 (40)
In the general case, using eq. (37), we have :

𝐿 𝑖 [(1 + 𝑥 𝑖 𝜕 log 𝜑 𝑖 𝜕𝑥 𝑖 ) ∇𝑥 𝑖 + 𝑥 𝑖 ∑ 𝜕 log 𝜑 𝑖 𝜕𝑥 𝑘 ∇𝑥 𝑘 𝑘≠𝑖 ] ≡ 𝐷 𝑖 𝑅 ∇𝑥 𝑖 (41)
And the evaluation of the phenomenological diffusion coefficient Li cannot be made straightforwardly from usual diffusion coefficients. However, for a dilute component in a solvent mixture, far from critical conditions, we can approximate our mobility coefficients by :

𝐿 𝑖 = 𝐷 𝑖 𝑅(1+𝑥 𝑖 𝜕 log 𝜑 𝑖 𝜕𝑥 𝑖 ) (42)
In dense fluids, the effective infinite dilution diffusion coefficient of a species is roughly inversely proportional to the section of the molecule, i.e. a volume at an exponent  of about 2/3. For practical application with an equation of state, as will be seen in the following, we assume that 𝐷 𝑖 is proportional to 𝑏 𝑖 -𝜏 , where 𝑏 𝑖 is the covolume of the studied component [START_REF] Poling | The properties of Gases and Liquids, 5 th Edition[END_REF] defined as 𝑏 𝑖 ∝ 𝑇𝑐 𝑖 /𝑃𝑐 𝑖 where 𝑇𝑐 𝑖 and 𝑃𝑐 𝑖 are the critical temperature and the critical pressure of component i respectively. With this formulation and assuming that all components are diluted, we can deduce that:

𝐿 𝑖 𝐿 𝐷 = 𝐿 𝑖 ∑ 𝑥 𝑘 𝐿 𝑘 𝑘 = ( 𝑇𝑐 𝑖 𝑃𝑐 𝑖 ) -𝜏 (1+𝑥 𝑖 𝜕 log 𝜑 𝑖 𝜕𝑥 𝑖 ) ∑ 𝑥 𝑘 𝑘 (1+𝑥 𝑘 𝜕 log 𝜑 𝑘 𝜕𝑥 𝑘 ) ( 𝑇𝑐 𝑘 𝑃𝑐 𝑘 ) -𝜏 (43)
We use (𝑇𝑐 𝑖 /𝑃𝑐 𝑖 ) -𝜏 for practical reasons, the right quantities are the square of molecular size.

In addition, since LP and LD are both inversely proportional to the fluid viscosity, it is interesting to notice that the LP / LD ratio in equations ( 35) and ( 36) is independent of the fluid viscosity.

In dense fluid, this ration depends mainly on the ratio between the size of the molecules and the size of the pore throat.

For typical reservoir fluid composition and typical reservoir conditions, the ratio LP / LD becomes close to unity when the permeability is in the range 10 to 100 nanoDarcy which corresponds to the permeability of shale. In the case of usual reservoir permeability this ratio is very large and equation ( 36) can be applied to get the compositional gradient.

Practical resolution algorithm

For practical applications, one can use an equation of state (EoS) model, in that work a classical Peng-Robinson EoS with Peneloux volume translation [START_REF] Poling | The properties of Gases and Liquids, 5 th Edition[END_REF], to compute all quantities [5,14].

The simplest formulation is to calculate the composition at a given depth, (Z), and temperature,

as
where fugacity coefficients have to be evaluated at the same temperature. If the gravity forces are not balanced, the new pressure P is equal to the reference pressure plus the weight of the fluid column. Thus, the contribution of the pressure to the fugacity ratio can be included in the gravity term:

𝑥 𝑖 𝑥 𝑖 𝑟𝑒𝑓 = 𝜑 𝑖 𝑟𝑒𝑓 𝜑 𝑖 𝑒 ((𝑀 𝑖 -ρ𝒗 𝒊 )𝐠+ (𝑞 𝑖 - 𝐿 𝐷 𝐿 𝑖 𝑄)∇𝑇) (𝑍-𝑍 𝑟𝑒𝑓 ) 𝑅𝑇 (45)
Where

𝑣 𝑖 = 𝑅𝑇 𝑃 + 𝑅𝑇 𝜕 log 𝜑 𝑖 𝜕𝑃 (46)
In this case, the fugacity at the point of interest must be calculated at the same pressure than the reference point.

This set of equations clearly shows the competition between gravity and thermodiffusion forces.

A perfect balance is obtained if:

(𝑀 𝑖 -ρ𝒗 𝒊 ) 𝐠 + (𝑞 𝑖 - 𝐿 𝐷 𝐿 𝑖 𝑄) ∇𝑇 = 0 (47)
And external fluxes values that just cancel the thermodiffusion are given by:

𝐉 𝐢 = 𝜌 𝑀 𝑥 𝑖 𝐿 𝑖 𝑞 𝑖 𝑇 ∇𝑇 (48)

IV. Application and validation

In this section, the proposed approach has been applied on a realistic reservoir fluid in reservoir conditions to determine the influence of thermodiffusion on species vertical distribution. More precisely, a fluid column of 450m height (located between -5325m and -5775m) subjected to a geothermal gradient of 0.029K/m and gravity has been modeled. The temperature and the pressure at -5525m are 87.5°C and 641 bar, respectively.

Parameterisation of the studied system

It is worth mentioning that validation on a real field case is almost impossible because of the uncertainty on fluid samples. There are also various factors that may impact the compositional gradient [6] and it is rather difficult to be sure that the stationary state is reached. Therefore, to apply and verify the proposed approach we used a realistic fluid composition but, instead of using real field data, the influence of gravity and thermal gradient were evaluated through Non-Equilibrium Molecular Dynamics (NEMD) simulations based on the methodology described in refs. [14,15].

Furthermore, modern analytical technics give us a huge number of components and it is not possible to use a full fluid description of reservoir fluid in NEMD simulation. Thus, we used a thermodynamic lumping technique, to reduce the number of components of the studied oil to a manageable value (8 pseudo-components in this case as shown in Table 1). More precisely, lumping was achieved a standard approach based on volatility under engineering constraints (groups correspond to boiling points cuts). The small amount of nitrogen was lumped with methane (N2C1) and the heavy ends C20+ was divided in two cuts (CN1 and CN2).

The properties of the pseudo-components (critical properties, Tc and Pc, and acentric factors, ) were deduced from the constraint of iso-chemical potential at initial reservoir conditions because EOS parameters of the pseudo component must be the same than calculated values from EOS mixing rules based on initial reservoir fluid compositions. The transcription of the properties of the pseudo-component into molecular models used in molecular dynamics has been achieved using a top-down strategy based on the corresponding states, similarly to what done in ref. [16]. More precisely, we have used the Lennard-Jones chains model to represent the pseudo-component in which the molecular parameters (: sphere diameter, : potential depth and N: number of segments) were adjusted so as to yield exactly the critical temperature and the critical pressure, except for CO2. All molecular parameters so defined are provided in Table 2. For CO2 we have used the force field developed in ref. [17].

Classical Lorentz-Berthelot combining rules were used to define the cross molecular parameters. This coarse grained molecular model, even if perfect, is known to describe reasonably well thermodiffusion and PVT properties of hydrocarbons mixtures [5,14]. )

Pseudo

As an initial value we have chosen to take  = 0.6 as done in the correlation of Wilke and Chang which is known to apply well on the hydrocarbon homologue series [START_REF] Poling | The properties of Gases and Liquids, 5 th Edition[END_REF]. It should be recalled that the applicability of this relation is based on the fact that most of the component can be considered as "diluted" in the reservoir fluid. It is obviously the case of the components of the liquid fraction but it is not the case of some light components (CH4 and CO2 in our example).

We assume that such an approximation remains valid for these small molecules in the reservoir fluid.

Results

As mentioned previously, a fluid column of 450m height was simulated by NEMD subject to gravity with, and without, a realistic thermal gradient of 0.029K/m to build the reference solution used to validate the thermodynamic model developed in this work. NEMD compositional profiles of some of the pseudo component are shown in Figs. 1 and2. Then, computations have been achieved using the proposed thermodynamic model. The composition at depth -5525m was used as a reference for EOS modelling, EOS calculations make use of a reference depth composition, NEMD simulation use an average composition. For sake of comparison we use the composition at 5525m obtained from anisothermal NEMD simulation as the reference composition for EOS calculation. Since the average composition depends on the profile, it is not possible to get the same reference composition for the different cases. First, it is interesting to notice, see Figs. 2 and3, that this realistic thermal gradient leads to an almost perfect compensation of the gravity force by the thermodiffusion force. Even, if this behavior has been noticed for other systems, such as acid gas [5] or synthetic oil [14], such a behavior is particularly striking for the heavy cuts (CN1 and CN2) which are highly segregated when gravity alone is applied.

Second, Figures 2 and3 show a qualitative agreement between results from NEMD simulations and those obtained from the proposed thermodynamic models with or without thermal gradient.

Thus, as the results obtained by the two approaches without gravity are similar, this indicates that the thermodynamic model proposed in this work, despites its intrinsic assumptions, captures well the influence of thermodiffusion on the compositional profiles without any fitting parameters. However, it should be pointed out that the sensitivity to exponent  is large as shown in Fig. 4 and that a better fit could be obtained for  = 0.59. From this application we can deduced that our approach is good enough for practical application in the petroleum industry. But it may be due to the large number of components in the reservoir fluids and it could be necessary for other mixtures to account for cross phenomenological coefficients in Eq. ( 2). One of the key figures is that the molecular diffusive mobility appears only as divided by the average diffusive mobility (Eq.( 43)). Therefore the evaluation of this ratio do not require information on the P,T,  dependency of these coefficients. We just need to know how these coefficients vary with the properties of the molecule itself. The other part of the thermal force is a simple thermodynamic property and accurate enough when the EOS model have been tuned on a large set of experimental data.

Conclusions

Thanks to an irreversible thermodynamic approach, we developed expressions providing the chemical potential gradients and the pressure gradient induced by the geo-thermal field in a petroleum reservoir. In the final expressions, the chemical potential gradients are proportional to the temperature gradient, and the proportionality coefficients are molar residual entropy of each component balanced by the average residual entropy multiplied by the relative diffusion mobility of the components.

The obtained relations imply that the species relative separation in a thermal field is sensitive to the relative diffusion coefficients at stationary state. In porous media, the separation is sensitive to the permeability when the overall mobility is similar to diffusive mobility. In addition, the magnitude of the separation depends on the residual entropy of the species and the separation is not simply balanced by the average residual entropy. The balance is modified by the relative diffusion mobility of the components. Another interesting feature of the proposed relations is that in low permeability porous media, the thermal gradient induces a pressure gradient proportional to the fluid residual entropy.

Combined with an equation of state and a classical correlation for transport properties, the proposed relations do not need fitting parameters to model the impact of the thermal gradient on pressure and compositional gradients. As a test, the proposed approach has been applied on an eight pseudo-components reservoir fluid subjected to a geothermal gradient. First results in terms of compositional profiles versus depth have shown to be in good agreement with nonequilibrium molecular dynamics simulation results at the stationary state.

  a. the average fluid velocity is slow enough to allow linearization of the fluid transport equation and to neglect inertial terms, b. the local equilibrium hypothesis is valid in the fluid phase, c. the fluid phase is homogeneous, d. the transport and conservation equations can be homogenized in a simple manner regarding thermodynamic variables and properties.

Fig 1 ,

 1 Fig 1, which is sometimes found in large fields, it is possible to derive the tilt angle from Eq.18 by applying this equation in both phase along the interface.

Fig. 1 : 3 . 2 .

 132 Fig.1: Interface subjected to gravity and thermal gradient

Figure 2 :

 2 Figure 2: Compositional profile of the two lightest compounds of the oil.

Figure 3 :

 3 Figure 3: Compositional profile of the two heaviest compounds of the oil.

Figure 4 :

 4 Figure 4: Compositional profile of CO2, sensitivity tests on 

  𝑛 𝑖 ℎ 𝑖 𝐿 𝑃 (

	V∇𝑃-𝑀 𝐠 𝑇	) + 𝑛 𝑖 ℎ 𝑖 * ∑ 𝐿 𝑖𝑗 ( 𝑗	𝑀 𝑗 𝐠 𝑇	+ ∇ (-	𝜇 𝑗 𝑇 ))

  a function of a reference point "ref" where all properties are known. Starting from Eq. (36)

	we get:
	𝑥 𝑖 𝑥 𝑖 𝑟𝑒𝑓 =	𝑃 𝑟𝑒𝑓 𝑃	𝜑 𝑖 𝑟𝑒𝑓 𝜑 𝑖	𝑒	(𝑀 𝑖 𝐠+(𝑞 𝑖 -𝐿 𝐷 𝐿 𝑖	(𝑍-𝑍 𝑟𝑒𝑓 ) 𝑄)∇𝑇) 𝑅𝑇

Table 1 :

 1 Thermodynamic properties of pseudo-components used to describe the oil.

	The usual properties of

Table 2 :

 2 Molecular parameters of pseudo-components used to describe the oilThe last parameter to define is the exponent  appearing in the mobility ratio of Eq. (43):

	-
	components	N	 (j.mol -1 )	 (Å)
	N2C1	1	1209	3.697
	C2	2	1438	3.289
	C3C4	2	1831	3.760
	C5C10	3	2277	4.095
	C11C19	5	2501	4.149
	CN1	8	2798	4.222
	CN2	10	3459	4.468

Acknowledgements

The authors would like to thank all the colleagues who have contributed to this study and especially Alexander Shapiro from DTU for fruitful discussions. We are also very grateful to Total S.A. for its support during many years.


Boutheina Maalej
email: maalej.boutheina@gmail.com

Ahmed Chemori
email: ahmed.chemori@lirmm.fr

Nabil Derbel
email: n.derbel@enis.rnu.tn

Adaptive Control of a Lower Limb Exoskeleton Dedicated to Kids' Rehabilitation

Keywords: Rehabilitation, Cerebral palsy, Exoskeletons, Classical adaptive control, L 1 adaptive controller

In this chapter, four adaptive controllers have been proposed to control a 2-DOF exoskeleton dedicated to kids' rehabilitation. These control laws are implemented at the hip and the knee joints. In fact, tracking the gait scheme with an intense and a precise work may allow children to increase their brain plasticity. Through the proposed study, it is shown that the augmented L 1 adaptive controller is robust regards to parametric variations. Besides, to validate this controller, different scenarios and simulations were carried out to prove its effectiveness.

Introduction

Robotic systems have recently impacted the human life. Indeed, human-robot interaction in several domains is considered as one of the most remarkable achievement all over the world. Nowadays, exoskeletons can be considered as a significant example of human-robot interaction. Initially, exoskeletons were developed for military applications. Then, for industrial applications and medical uses. Since walking is a key feature of independency, restoring safe walking is one of the main goals of robotic rehabilitation. There exist several disabilities that may cause the movement disorders and affect the brain which is the responsible of the body functions. Any damage in the brain tissue before, during or after childbirth may affect certain areas of the brain. Besides, depending on the degree of injury, it can cause permanent disorders, characteristic of a non-progressive injury. Among the potentially determining factors of irreversible brain damage, the most frequently observed include infections of the nervous system, hypoxia (lack of oxygen), head injuries, etc. Children with cerebral motor disorders represent 3 per 1000 newborns. In this context, several medical applications appear in order to help paralyzed kids to restore their locomotion.

In this work, we consider the case of lower limb rehabilitation using exoskeletons in order to assist children movements. In the sequel, we will be interested in kids who have between 2 and 13 years old. Hence, we propose to implement robust controllers that can be adapted to the difference of children morphologies [START_REF] Refai | Adaptive control of a human-driven knee joint orthosis[END_REF]. An adaptive control is a controller with adjustable parameters. In fact, it can adapt to changes in the process dynamics and the disturbance characteristics. Moreover, adaptive techniques can also be used to provide automatic tuning of controllers. In this chapter, four control laws are proposed to solve the problem of parametric variations. The chosen controllers include two nonlinear state feedback adaptive controllers, the L 1 adaptive control and augmented L 1 adaptive control. The first approach, based on Slotine works [START_REF] Slotine | On the adaptive control of robot manipulators[END_REF], consists in a PD feedback part and a full dynamic feedforward tacking into consideration parametric variations. The second approach is also based on Slotine works [START_REF] Slotine | On the adaptive control of robot manipulators[END_REF]. It consists in adding an integral control action to the previous approach in order to eliminate undesirable steady-state position errors. The third approach consists in applying L 1 adaptive control proposed by Naira Hovakimyan [START_REF] Hovakimyan | L1 adaptive control theory guaranted robustness with fast adaptation[END_REF]. The fourth approach is the augmented L 1 adaptive control [START_REF] Refai | Augmented L1 adaptive control of an actuated knee joint exoskeleton: from design to real-time experiments[END_REF] which combines the L 1 adaptive control with a Proportional-Integral control to eliminate the time lag and to improve the tracking performances. Simulation results will be presented with a comparative study which aims to show the robustness of the augmented L 1 adaptive control.

Description and Modeling of Exoskeletons

Exoskeletons are made in order to help kids suffering from several diseases, such as the cerebral palsy, to restore their walk again by a cyclical and alternative rhythmic activities. The basic idea is to control the lower limb exoskeleton (Fig. 1) at the hip and knee joints.

The dynamic model includes the human limb and the exoskeleton, it is written as follows [START_REF] Ghezal | Design and robust control of a 2 DOFs lower limb exoskeleton[END_REF]:

M(q) q + C(q, q) q + G(q) = τ (1)
with, [START_REF] Bennehar | L1 adaptive control of parallel kinematic manipulators: design and real-time experiments[END_REF] where: q = [q 1 q 2 ] T ∈ R 2 represents the position vector of the hip and knee joints, respectively, q = [ q1 q2 ] T ∈ R 2 is the speed vector, q = [ q1 q2 ] T ∈ R 2 is the acceleration vector, M(q) ∈ R 2 is the inertia matrix, which is symmetric, uniformly bounded and positive definite, C(q, q) q ∈ R 2 represents the Coriolis, and centrifugal forces and torques, G(q) ∈ R 2 denotes the gravity torques, τ ∈ R 2 is the vector of actuator torques,

M 11 = 1 8 l 2 1 m 1 + 1 4 m 2 1 4 l 2 2 +l 2 1 + l 1 l 2 cos q 2 + 1 2 m s (l 2 1 +l 2 2 k 2 2 + 2l 1 l 2 k 2 cos q 2 ) + 1 2 k 2 1 l 2 1 m t + 1 2 (I 1 + I 2 + I s + I t ) M 12 = M 21 = 1 2 m 2 1 2 l 2 2 +l 1 l 2 cos q 2 + 1 2 m s (l 2 2 k 2 2 +2l 1 l 2 k 2 cos q 2 )+(I 2 + I s ) M 22 = 1 8 m 2 l 2 2 + 1 2 m s l 2 1 + l 2 2 k 2 2 + 1 2 (I 2 + I s ) C 11 = 1 2 m 2 + m s k 2 l 1 l 2 sin q 2 C 12 = - 1 2 m 2 + m s k 2 l 1 l 2 sin q 2 C 21 = 1 4 m 2 + 1/2m s k 2 l 1 l 2 sin q 2 C 22 = 0 G 1 = - 1 2 m 1 + m 2 + k 1 m t + m s gl 1 sin q 1 -(1/2m 1 + m s k 2 )gl 2 sin(q 1 + q 2 ) G 2 = - 1 2 m 1 + m s k 2 gl 2 sin(q 1 + q 2 )

Adaptive Control [12]

Let consider that q d is the desired joint position and qd is the desired velocity. The main idea is to develop a globally stable adaptive controller which is obtained from a Lyapunov stability analysis. The Lyapunov function associated to the system is: K p , Γ : symmetric positive definite matrices q = qq d : represents the position tracking error q = q -qd : represents the velocity tracking error

V (t) = 1 2 ˙ q T M(q) ˙ q + a T Γ a + q T K p q (2)
The derivative of V with respect to time is:

V (t) = ˙ q T M ¨ q + 1 2 ˙ q T M ˙ q + a T Γ ˙ a + q T K p ˙ q = ˙ q T τ -C(q, q) q -G(q) -M qd + ˙ q T 1 2 ( Ṁ -2C) + C ˙ q + a T Γ ˙ a + ˙ q T K p q = ˙ q T τ -M(q) qd -C(q, q) qd -G(q) + K p q + a T Γ ˙ a (3)
Using the property of skew symmetric matrix, we have:

d dt ( qT M q) = qT τ -G(q) (4) qT 1 2 Ṁ -C q = 0 ( 5 )
Considering the following control law:

τ = M qd + C(q, q) + G(q) -K p q -K D ˙ q (6)
gives

V = ˙ q T M(q) qd + C(q, q) qd + G(q) -K D ˙ q + a T Γ ˙ a ( 7
)
where:

486709_1_En_6_Chapter
M(q) = M(q) -M(q) = i M i a i (8) C(q, q) = C(q, q) -C(q, q) = i C i a i (9) G(q) = G(q) -G(q) = i G i a i ( 10
)
Then, let's write:

M(q) qd + C(q, q) qd + G(q) = Y a ( 11
)
where Y = Y (q, q, qd , qd ) is an n × m matrix. Therefore:

V = -˙ q T K D ˙ q + a T [Γ ˙ a + Y T ˙ q] ( 12
)
Assuming that variations of the unknown vector a can be neglected, we obtain:

V = -˙ q T K D ˙ q ≤ 0 ( 13
)
using the adaptive law of the parameter vector a:

˙ a = -Γ -1 Y T ˙ q ( 14

Adaptive Control with an Integral Action [12]

The idea is to eliminate the steady-state position errors by restrict them to lie on a sliding surface.

˙ q + Λ q = 0 ( 15
)
where Λ is a positive definite diagonal matrix.

The virtual reference trajectory is expressed by

q r = q d + Λ qdt ( 16
s = ˙ q r = q -qr = ˙ q + Λ q (19)
The control and the adaptation laws become:

τ = M qr + C(q, q) qr + G(q) -K D s (20) ˙ a = -Γ -1 Y T (q, q, qr , qr )s (21)
We use a Lyapunov function to demonstrate the global convergence of the tracking:

V = 1 2 s T Ms + 1 2 a T Γ a ( 22
)
Its first time derivative leads to: The control law does not contain K p , since the position error q is already included in ˙ q r . Moreover where s becomes equal to 0, q converges to 0.

V = -s T K D s ≤ 0 (
It is well known that the adaptive controller has several limitations such as (i)

the initial value of the parameters, (ii) the persistent excitation of the estimated parameters, and (iii) the stability. Hence, L 1 Adaptive controller is presented in the next subsection which overcome these limitations.

L 1 Adaptive Control [2]

L 1 adaptive control is inspired from the Model Reference Adaptive Control (MRAC), it is structured into four principal parts as illustrated in Fig. 2 The expression of the torque is:

τ (t) = A m r (t) + τ ad (t) ( 24
)
where:

A m ∈ R 2×2 is a Hurwitz matrix τ ad ∈ R 2×2 is the adaptive component
The tracking error is expressed as: Fig. [START_REF] Tucker | Control strategies for active lower extremity prosthetics and orthotics: a review[END_REF] Position, speed and torque control of the knee joint, obtained with a Augmented L 1 adaptive controller in the nominal case. Solid line: actual trajectory, dashed line: desired trajectory with Λ ∈ R 2×2 is a symetric positive definite matrix. It is difficult to know exactly all nonlinearities of the system, hence the adaptive control can be defined in such a way to cancel the effect of these nonlinearities. For that, we consider the state predictor which is based on estimated parameters obtained from the adaptation mechanism:

r = ( q -qd ) + Λ(q -q d ) ( 25
˙ r = A m r (t) + τ ad (t) -[ θ(t) r t ∞ + σ(t)] -K r (t) ( 26
)
where:

r (t) = r (t) -r (t) is the prediction error K ∈ R 2×2 used to reject high frequency noises.
Using the projection method, we obtain the estimate of θ(t) and σ(t):

θ(t) = Γ Proj ( θ(t), P r (t) r t ∞ ) (27) σ(t) = Γ Proj ( σ(t), P r (t)) ( 28
Fig. 14 Influence of -30% of masses and -15% of length uncertainties on the hip joint tracking with augmented L 1 adaptive controller where Γ is the adaptive gain.

Let's P the solution of the algebric Lyapunov equation:

A T m P + P A m = -Q (29)
The adaptive control is as follows:

τ ad (s) = C(s) η(s) (30
0 0.5 1 1.5 2 2.5 3 Time (s) 0 500 1000
Torque control of Joint 1 (Nm)

1 1.2 1.4 1.6 1.8 2 -2 0 2 4
Fig. 15 Influence of -50% of masses and -25% of length uncertainties on the hip joint tracking with augmented L 1 adaptive controller

Augmented L 1 Adaptive Control [2]

The implementation of the classical L 1 adaptive control indicates the presence of a time lag. Hence, the idea is to develop an augmented version of the L 1 adaptive control. Figure 3 shows the block diagram of the augmented L 1 adaptive control. In fact, a linear PI controller is used as an additional part to the filtered control input.

The expression of the torque becomes:

τ (t) = A m r (t) + τ ad (t) + K p (q -q d ) + K i (q -q d )dt ( 31
)
where K p and K i ∈ R 2×2 are diagonal positive definite matrices.

Simulation Results

Lower limb exoskeletons are actuated at the knee and hip joints. To validate the different controllers proposed in the previous section, some simulations are presented.

First, Figs. 4 and5 show respectively the position evolution of the hip and the knee joints, the tracking error, their speed and their applied torques using the first approach of Slotine in the nominal case. Besides, Fig. 6 show the convergence of the estimated unknown parameter to the system parameters.

The second part consists in the adaptive control using an integral action. Results

I AE R = |e|dt |q d |dt (34)
e denotes the tracking error.

The obtained results are summarized in Table 1.

Conclusion and Future Work

In this work, four adaptive controllers have been proposed and implemented to control an exoskeleton for kids rehabilitation. These control laws have been tested for the tracking of walking cycle desired trajectory. This study concerns kids who have between 2 and 13 years old (i.e. with different morphologies). Through numerical simulations, it has been shown that the augmented L 1 adaptive controller is the best one in terms of robustness against these parametric variations. Future works aim to validate the proposed approaches on a real exoskeleton, as well as the application to the targeted kids therapy.

L 1 3 Fig. 1

 131 Fig. 1 Concept of the 2-joint lower limb exoskeleton

Fig. 2 Fig. 3

 23 Fig.2Block diagram of L 1 adaptive control[START_REF] Hovakimyan | L1 adaptive control theory guaranted robustness with fast adaptation[END_REF]

m 1 ,

 1 m 2 , l 1 , l 2 represent the mass and length of thigh and shank segments of the exoskeleton, respectively, m t , m s denote thigh and shank masses of human limb, k 1 , k 2 are the position of the center of mass of thigh and shank segments, respectively, I 1 , I 2 , I s , I t represent the moments of inertia of thigh and shank of the exoskeleton and human limb, respectively, g is the gravity acceleration.

L 1 Fig. 4

 14 Fig. 4 Classical adaptive control of the hip joint. For the position and the speed, dashed line: desired trajectory, solid line: actual trajectory

Fig. 5

 5 Fig. 5 Classical adaptive control of the knee joint. For the position and the speed, dashed line: desired trajectory, solid line: actual trajectory

L 1 Fig. 6

 16 Fig. 6 Evolution of the estimated Parameters for the first approach of Slotine

Fig. 7

 7 Fig. 7 Adaptive control with integral action of the hip joint. For the position and the speed, dashed line: desired trajectory, solid line: actual trajectory

Fig. 8

 8 Fig.[START_REF] Jamshidi | Modelling of human walking to optimise the function of ankle-foot orthosis in Guillan-Barré patients with drop foot[END_REF] Adaptive control with integral action of the knee joint. For the position and the speed, dashed line: desired trajectory, solid line: actual trajectory

Fig. 9

 9 Fig. 9 Evolution of the estimated Parameters for the second adaptive approach of Slotine

)Fig. 10

 10 Fig. 10 Classical L 1 adaptive control of the knee joint. For the position and the speed, dashed line: desired trajectory, solid line: actual trajectory

Fig. 11

 11 Fig. 11 Classical L 1 adaptive control of the hip joint. For the position and the speed, dashed line: desired trajectory, solid line: actual trajectory

L 1 Fig. 12

 112 Fig.[START_REF] Slotine | On the adaptive control of robot manipulators[END_REF] Position, speed and torque control of the hip joint, obtained with a Augmented L 1 adaptive controller in the nominal case. Solid line: actual trajectory, dashed line: desired trajectory

L 1

 1 Adaptive Control of a Lower Limb Exoskeleton Dedicated to Kids'

  ) where η(s) is the Laplace transform of η(t) = θ(t) r t ∞ + σ(t), and C(s) is a bounded-input bounded-output (BIBO) stable strictly proper transfer function.

L 1 Fig. 16

 116 Fig. 16Influence of 30% of masses and 15% of length uncertainties on the hip joint tracking with augmented L 1 adaptive controller

Fig. 17

 17 Fig. 17Influence of 50% of masses and 25% of length uncertainties on the hip joint tracking with augmented L 1 adaptive controller

Fig. 18

 18 Fig. 18Influence of -30% of masses and -15% of length uncertainties on the knee joint tracking with augmented L 1 adaptive controller

Fig. 19 Fig. 20

 1920 Fig. 19Influence of -50% of masses and -25% of length uncertainties on the knee joint tracking with augmented L 1 adaptive controller

Fig. 21

 21 Fig. 21 Influence of 50% of masses and 25% of length uncertainties on the knee joint tracking with augmented L 1 adaptive controller

Table 1

 1 Tracking performance comparison

		Classical adaptive controller	L 1 adaptive controller	Augmented L 1 adaptive controller
		Joint 1	Joint 2	Joint 1	Joint 2	Joint 1	Joint 2
	I AE	0.0037	0.0169	0.0972	0.1578	0.0054	0.0609
	I SE	0.0062	0.0181	0.0057	0.0279	1.2702×10 -4 0.0183
	I AE R	28.2872	3.1664	0.4075	0.2513	0.0228	0.0970

Acknowledgements

The present work is supported by (i) the "Association de Sauvegarde des Handicapés Moteurs -Sfax" (ASHMS), the (ii) Clinical Inverstigation Center (CIC) of the Hospitalo-University Center of Sfax (CHU) Tunisia, (iii) the Laboratory "Control & Energy Managements" (CEMLab) of the "National School of Engineering of Sfax", University of Sfax, Tunisia, and (iv) the Digital Research Center of Sfax, Tunisia (CRNS).