Damien Calaque

Derived symplectic geometry

Keywords: Derived geometry, Shifted symplectic structure, Lagrangian morphism, Lagrangian intersection, Virtual class, Critical chart, Darboux theorem, Casson invariant, Donaldson-Thomas invariant, Topological field theory

Derived symplectic geometry studies symplectic structures on derived stacks. Derived stacks are the main players in derived geometry, the purpose of which is to deal with singular spaces, while symplectic structures are an essential ingredient of the geometric formalism of classical mechanics and classical field theory. In addition to providing an overview of a relatively young field of research, we provide a case study on Casson's invariant.

Introduction

Derived geometry One of the purposes of derived geometry is to deal in a satisfying way with spaces that are singular, which is the case for many spaces appearing in algebraic geometry (moduli spaces) and in classical1 physics (spaces of solutions of equations of motion). The reader can consult [START_REF] Toën | Derived algebraic geometry[END_REF] and [START_REF] Anel | The geometry of ambiguity: an introduction to the ideas of derived geometry[END_REF] for extensive overviews of the history, results and main ideas of derived geometry. Foundations of derived algebraic geometry have been laid out by [START_REF] Toën | Homotopical algebraic geometry. II. Geometric stacks and applications[END_REF] and [START_REF] Lurie | Derived algebraic geometry[END_REF]. For a treatment of (as well as references for) derived differential geometry we recommend the work of [START_REF] Steffens | Derived C ∞ -geometry I: Foundations[END_REF].

Very roughly, modern derived geometry combines two ways of introducing more flexibility (using homotopy theory, or higher category theory) into the geometry of spaces:

(a) It generalizes local/test objects (affine schemes, euclidean spaces, Stein manifolds, . . . ) by adding homological/homotopical data to them (for instance, in algebraic geometry, one can replace rings with simplicial rings, or connective commutative differential graded algebras). This line of thoughts can be seen as coming out of an attempt to give an actual geometric content to the intersection formula from [START_REF] Serre | Algèbre locale. Multiplicités[END_REF], as explained for instance by [START_REF] Lurie | Derived algebraic geometry[END_REF] and [START_REF] Toën | Derived algebraic geometry[END_REF].

(b) It allows to glue local objects up to specified identifications. This led to the concept of a stack, introduced by [START_REF] Giraud | Cohomologie non abélienne[END_REF], and ultimately to the higher stacks of [START_REF] Simpson | Algebraic (geometric) n-stacks[END_REF], who was somehow continuing the work of [START_REF] Grothendieck | Pursuing stacks (à la poursuite des champs)[END_REF]. We refer to [START_REF] Mestrano | Stacks. In New spaces in mathematicsformal and conceptual reflections[END_REF] for a gentle introduction to stacks and higher stacks.

These two ways of introducing more flexibility aim to address two different sources of singularities/pathologies: (a) bad intersections (and, more generally, fiber products) and (b) bad quotients. [START_REF] Kontsevich | Enumeration of rational curves via torus actions[END_REF] is one of the first authors who combined both aspects to solve enumerative problems in algebraic geometry, using pre-derived geometric tools. As a complement to the excellent introduction to the thesis of [START_REF] Lurie | Derived algebraic geometry[END_REF], where derived (algebraic) geometry is motivated through the lens of Bézout's theorem (that is an intersection problem) and deformation theory, we also recommend the introduction to the paper of [START_REF] Steffens | Derived C ∞ -geometry I: Foundations[END_REF], where the need for derived geometry is motivated by the study of solution spaces to elliptic PDEs and their local Kuranishi models.

Symplectic geometry Symplectic geometry is a natural geometric setting for the hamiltonian formulation of classical mechanics, as one can learn for instance from [START_REF] Souriau | Structure des systèmes dynamiques[END_REF][START_REF] Arnol'd | Mathematical methods of classical mechanics[END_REF]; most phase spaces appear to be symplectic manifolds (or variations of these, like Poisson manifolds).

Recall that a symplectic manifold is a smooth manifold X equipped with a 2form ω ∈ Ω 2 (X) that is non-degenerate (meaning that the induced bundle map T X → T * X is an isomorphism) and closed (meaning that d dR ω = 0). Observe that this definition makes sense in the algebro-geometric context only if X is a smooth algebraic variety. The cotangent bundle T * M of a manifold M is an example of a symplectic manifold, with ω can = d dR λ, where λ is the tautological 1-form on T * M . Symplectic manifolds do not have local invariants: it follows from a theorem of [START_REF] Darboux | Sur le problème de Pfaff[END_REF] that every symplectic manifold is locally symplectomorphic to T * R n equipped with ω can .

Lagrangian submanifolds play a crucial role in symplectic geometry: recall that a lagrangian submanifold L ⊂ X in a symplectic manifold (X, ω) is lagrangian if ω |L = 0 and the induced map T L → T * L/X is an isomorphism. Generalizing Darboux's theorem, [START_REF] Weinstein | Symplectic manifolds and their Lagrangian submanifolds[END_REF] proved that in the neighborhood of a lagrangian submanifold L every symplectic manifold is symplectomorphic to a neighborhood of the zero section of T * L. Thus lagrangian submanifolds can naturally be interpreted as generalized configurations of a classical mechanical system. Lagrangian submanifolds pop up everywhere: graphs of closed 1-forms, graphs of symplectomorphisms (an example of which is the time one flow of a hamiltonian vector field), conormal bundles, zero loci of moment maps, . . . This led [START_REF] Weinstein | Symplectic geometry[END_REF] to follow the symplectic creed claiming that "everything is a lagrangian submanifold", and envision a symplectic category whose objects are symplectic manifolds and morphisms are lagrangian correspondences (i.e. lagrangian submanifolds of a product (X 1 × X 2 , p * 1 ω 1 -p * 2 ω 2 )). At this point, the need to deal with singular/pathological spaces in symplectic geometry should be obvious:

• The zero locus µ -1 (0) of a moment map µ might be singular.

• The above example is actually a lagrangian correspondence between the original symplectic manifold X and its symplectic reduction X red , where X red is a quotient of µ -1 (0) and thus could be even more singular.

• The composition in the symplectic category involves taking fiber products, that might not be well-behaved. A traditional way to deal with that is by applying a small geometric perturbation, but there are issues with this approach: (a) it cannot always be used in algebraic geometry and (b) geometric perturbations are not functorial.

A leitmotiv of derived geometry is to replace geometric perturbations with homological perturbations for computing fiber products. Homological perturbations can be made functorial (in a higher categorical sense), and make sense in the algebro-geometric context, resolving both issues.

Derived symplectic geometry The ancestors of shifted symplectic structures on derived stacks are the odd symplectic structures on super-manifolds and Q-manifolds appearing in the work of [START_REF] Schwarz | Geometry of Batalin-Vilkovisky quantization[END_REF], [START_REF] Alexandrov | The geometry of the master equation and topological quantum field theory. Internat[END_REF] and in other mathematical physics publications on the geometry of the Batalin-Vilkovisky formalism. These excellent works and also the beautiful treatment by [START_REF] Costello | Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4[END_REF], using L ∞ -spaces and elliptic moduli problems, have two drawbacks:

(1) None of the two defining properties of a symplectic structure (closedness and nondegeneracy) are homotopy invariant.

(2) The geometric objects they consider only capture infinitesimal symmetries.

Both issues are dealt with by the formalism of derived geometry: it indeed encompasses global symmetries (stacks have been invented for that purpose) and is homotopy invariant by definition.

Let us be a bit more specific about the homotopy invariance issue: all ways of doing pre-derived geometry somehow involve spaces having a commutative differential graded algebra of functions, and whose tangent at a point is a cochain complex rather than just a vector space. Therefore, differential forms have an internal cohomological degree in addition to the usual form degree, and there are two differentials at play: the internal differential δ and the de Rham differential d dR . If we have an equivalence f : X -→Y (e.g. a quasi-isomorphism of differential graded manifolds in the sense of [START_REF] Kapranov | Injective resolutions of BG and derived moduli spaces of local systems[END_REF], that is a morphism inducing an isomorphism at the level of the cohomology of the differential graded algebra of functions), then:

• The pull-back of a 2-form ω Y on Y that is strictly non-degenerate (meaning that the induced map T Y → T *
Y is an isomorphism of complexes) might not be strictly non-degenerate.

• A strictly closed 2-form ω X on X (meaning that d dR ω X = δω X = 0) may not be the pull-back of a strictly closed 2-form ω Y on Y . [START_REF] Pantev | Shifted symplectic structures[END_REF] introduce a very flexible notion of symplectic structure on derived stacks, that addresses all the above issues. As we see below, their nondegeneracy condition only requires a quasi-isomorphism, and the closedness is relaxed up to coherent homotopies (where all homotopies are actually part of the structure).

In what follows, whenever we don't give a specific reference for something, the reader shall attribute it to [START_REF] Pantev | Shifted symplectic structures[END_REF]. Note that we also always work over a field of characteristic zero.

Shifted symplectic structures

The tangent space of a derived Artin stack X is no longer a vector bundle, but a perfect complex of O X -modules, denoted T X (whose dual is denoted L X ).

Here O X denotes the sheaf of functions on X. A 2-form of degree n on X is a cochain map ω 0 :

∧ 2 T X → O X [n]. Note that ω 0 is closed under the differential δ of the complex Ω 2 (X) = Γ(X, ∧ 2 L X ) of 2-forms on X; in particular, it induces a morphism of complexes T X → L X [n]. We say that ω 0 is non-degenerate if this induced morphism T X → L X [n]
is a quasi-isomorphism, meaning that it becomes an isomorphism at the level of cohomology.

Definition 1 An n-shifted symplectic (or, simply, n-symplectic) structure on X is a sequence ω := (ω 0 , ω 1 , ω 2 , . . . ), where ω k belongs to the complex Ω k (X) of k-forms on X, satisfying two conditions:

• The leading term ω 0 is a 2-form of degree n such that is non-degenerate (as explained above, this means that the induced morphism of complexes

T X → L X [n]
becomes an isomorphism at the level of cohomology).

• For every k ≥ 0, d dR (ω k ) + δ(ω k+1 ) = 0, meaning that ω 0 is closed under the de Rham differential up to coherent homotopies. In other words, ω is an n-cocycle in the truncated de Rham complex

  k≥2 Ω k (X), d dR + δ   .
First examples Because of the non-degeneracy condition, a scheme equipped with a 0-symplectic structure must be smooth. Conversely, all symplectic algebraic varieties provide examples of 0-symplectic structures.

If G is an affine algebraic group, then any non-degenerate symmetric invariant pairing c ∈ S 2 (g * ) G on the Lie algebra g of G defines a 2-symplectic structure on the classifying stack BG.

It was proven in [START_REF] Calaque | Shifted cotangent stacks are shifted symplectic[END_REF] that the n-shifted cotangent stack T * [n]X of an Artin stack X is indeed n-symplectic. In particular, for every affine algebraic group G, T * [1](BG)

g * /G is 1-symplectic.
Quotient stacks of quasi-symplectic groupoids, as defined by [START_REF] Xu | Momentum maps and Morita equivalence[END_REF], are also 1-symplectic, according to [START_REF] Calaque | Derived stacks in symplectic geometry[END_REF]. In particular, for a reductive group G, G /G ad is 1-symplectic.

The moduli of objects, in the sense of [START_REF] Toën | Moduli of objects in dg-categories[END_REF], of a d-Calabi-Yau category, is (2 -d)-symplectic, as witnessed by [START_REF] Toën | Derived algebraic geometry[END_REF] and shown by [START_REF] Brav | Relative Calabi-Yau structures II: shifted Lagrangians in the moduli of objects[END_REF]. This implies in particular that the derived moduli of coherent sheaves on a Calabi-Yau d-fold is (2 -d)-symplectic.

The AKSZ construction Following [START_REF] Alexandrov | The geometry of the master equation and topological quantum field theory. Internat[END_REF], [START_REF] Pantev | Shifted symplectic structures[END_REF] developed a systematic way of producing new shifted symplectic structures on mapping stacks via transgression: if (X, ω) is an n-symplectic derived stack and Σ is a nice enough derived stack equipped with a fundamental d-class [Σ] then there is a (n-d)-symplectic structure on the mapping stack Map(Σ, X), given by [Σ] ev * ω, where ev : Σ × Map(Σ, X) → X is the evaluation morphism.

An important source of examples is when Σ = M B is the Betti stack associated with a closed oriented d-manifold M , and G is a reductive group: then the derived stack Loc(M, G)

:= Map(M B , BG) of G-local systems on M is (2 -d)-symplectic 2 . Whenever M = S 1 , Loc(S 1 , G)
G /G ad and we recover the 1-symplectic structure on G /G ad , as shown by [START_REF] Safronov | Quasi-Hamiltonian reduction via classical Chern-Simons theory[END_REF].

Virtual count on (-1)-symplectic derived schemes Let X be a derived scheme equipped with a (-1)-shifted symplectic structure. The non-degeneracy condition for the symplectic structure imposes that the cotangent complex of X has amplitude [-1, 0], meaning that X is quasi-smooth. [START_REF] Schürg | Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes[END_REF] proved that for a quasi-smooth derived scheme X the cotangent complex L X induces a perfect obstruction theory in the sense of [START_REF] Behrend | The intrinsic normal cone[END_REF] on the underived truncation t 0 (X), allowing to define a well-behaved virtual fundamental class. [START_REF] Pantev | Shifted symplectic structures[END_REF] noticed that the non-degeneracy condition actually guarantees that the obstruction theory is symmetric in the sense of [START_REF] Behrend | Symmetric obstruction theories and Hilbert schemes of points on threefolds[END_REF]: in this case [START_REF] Behrend | Donaldson-Thomas type invariants via microlocal geometry[END_REF] proved that the virtual fundamental class does not depend on the choice of symmetric obstruction theory, and when Y = t 0 (X) is proper the virtual count (the degree of the virtual class) is obtained as a weighted Euler characteristic associated with a constructible function canonically defined on Y .

Finally, [START_REF] Brav | A Darboux theorem for derived schemes with shifted symplectic structure[END_REF] prove a Darboux theorem for (-1)-symplectic derived schemes, saying that they are locally equivalent to derived critical loci (see below).

The above discussion for derived schemes generalizes without problem to derived Deligne-Mumford stacks (i.e. derived orbifolds).

Lagrangian morphisms

In this section we present a far reaching generalization of the notion of a lagrangian submanifold that is well-suited for derived geometry (and encompasses various situations): the one of a lagrangian morphism (or, more accurately, lagrangian structure on a morphism).

Let f : L → X be a morphism of derived Artin stacks, and let ω be an n-symplectic structure on X.

Definition 2 A lagrangian structure on f (relatively to ω) is a homotopy η between ω and 0 (meaning that ω = (d dR +δ)(η)) in the truncated de Rham complex such that the map

T X → L L/X [n -1] induced by η 0 is a quasi-isomorphism.
Note that the leading term η 0 of the homotopy is a homotopy between ω 0 and 0 in the complex of 2-forms: ω 0 = δη 0 . This means that the sequence

T L → f * L X [n] → L L [n] is null-homotopic, leading to a map T X → L L/X [n-1].
The derived symplectic creed "Everything is a lagrangian morphism". The inclusion of a lagrangian subvariety into a symplectic variety is an example of a lagrangian morphism. [START_REF] Calaque | Lagrangian structures on mapping stacks and semi-classical TFTs[END_REF] provides more surprising examples of lagrangian morphisms:

• An n-symplectic structure on X is the same as a lagrangian structure on X → * , where * is equipped with the trivial (n + 1)-symplectic structure.

• If G is an affine algebraic group, X is a symplectic G-variety, and µ : X → g * is a moment map, then the induced morphism X /G → g * /G between quotient stacks carries a lagrangian structure.

• If G is reductive and X is a quasi-hamiltonian G-variety with group-valued moment map µ : X → G, then X /G → G /G ad also carries a lagrangian structure.

Calaque ( 2015) proves that lagrangian correspondences (i.e. lagrangian morphisms with codomain a product (X 1 × X 2 , p * 1 ω 1 -p * 2 ω 2 )) compose well. This allows to define a (shifted and derived) version of the symplectic category envisioned by Weinstein, whose objects are n-symplectic derived stacks and morphisms are lagrangian correspondences.

Virtual count for lagrangian intersections In particular, composing two lagrangian morphisms L 1 → X ← L 2 , with X being n-shifted symplectic, gives a lagrangian structure on L 1 × X L 2 → * , and thus a (n -1)-symplectic structure on L 1 × X L 2 . For n = 0 we get a (-1)-symplectic structure on the derived intersection of two lagrangian subvarieties, and the virtual count gives the intersection number of L 1 and L 2 in X. An even more specific situation is when X = T * M , L 1 is the zero section, and L 2 is the graph of d dR f for a function f ∈ O(M ): in this case the derived lagrangian intersection is the so-called derived critical locus of f , and the virtual count is given as the Euler characteristic of the hypercohomology of (a shift by dim(M ) of) the twisted de Rham complex (Ω M , d dR -d dR f ∧), that is a perverse sheaf on the reduced ordinary critical locus. [START_REF] Sabbah | Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems[END_REF] prove that, up to a shift by dim(M ), the twisted de Rham complex is equivalent to the sheaf of vanishing cycles associated with f , which plays a crucial role for motivic extensions and categorifications of the virtual count.

The AKSZ construction as a TFT A relative version of the AKSZ contruction is proven by [START_REF] Calaque | Lagrangian structures on mapping stacks and semi-classical TFTs[END_REF] and leads, for every d ≥ 0 and every n-symplectic derived stack X, to an oriented d-dimensional Topological Field Theory (TFT), given by Map (-) B , X , and taking values in the symplectic category (with shift n -d + 1). [START_REF] Calaque | The AKSZ construction in derived algebraic geometry as an extended topological field theory[END_REF] prove that this oriented TFT is fully extended in the sense of [START_REF] Lurie | On the classification of topological field theories[END_REF].

Case study: Casson invariant

We refer to [START_REF] Akbulut | Casson's invariant for oriented homology 3-spheres[END_REF] for a detailed exposition, following Casson's original proposal from his talk at MSRI in 1985. Let M be a com-pact oriented 3-manifold. We consider the moduli space of irreducible SU (2)representations of the fundamental group of M : R irr (M ) := Hom irr π 1 (M ), SU (2) SU (2) .

Irreducible representation are the ones with centralizer Z SU (2) = Z/2Z. For simplicty, we assume that M is an integral homology sphere (this implies in particular that the only reducible representation is the trivial one).

Definition of the Casson invariant Let now M = H 1 Σ H 2 be a Heegaard splitting of M : H 1 and H 2 are handlebodies with boundary Σ, a closed oriented surface. It turns out that R irr (H 1 ) and R irr (H 2 ) are lagrangian submanifolds of R irr (Σ), that is a symplectic manifold of dimension 6g -6, where g is the genus of Σ. Even though R irr (M ) is not a manifold, it is the intersection of R irr (H 1 ) and R irr (H 2 ), leading to define the Casson invariant λ(M ) as ((-1) g times) half the intersection number of R irr (H 1 ) and R irr (H 2 ) in R irr (Σ).

Derived symplectic nature of the Casson invariant Computing the intersection number instead of looking at the naive intersection R irr (M ) suggests one should consider the derived intersection. Also note that the Casson invariant is defined as half the intersection number, which is due to the hidden stacky nature of the moduli space: every irreducible representation carries a trivial Z/2Z symmetry. The Casson invariant is in fact a virtual count on the connected component of the derived stack Loc M, SU (2) that does not contain the trivial local system. Indeed, the underived truncation of Loc M, SU (2) is R irr (M ) /(Z/2Z) * /SU(2) . The (-1)-symplectic structure on Loc M, SU (2) therefore induces a symmetric obstruction theory on R irr (M ) /(Z/2Z) , allowing for a virtual count.

An important result is that λ(M ) does not depend on the choice of Heegaard splitting M = H 1 Σ H 2 . This is a direct consequence of the fact that Loc -, SU (2) defines an oriented 3-dimensional TFT with values in the derived version of Weinstein's symplectic category, as recalled in the previous section. The original definition of Casson for λ(M ) becomes a computation, that uses the fact that the (-1)-symplectic moduli can be obtained as a lagrangian intersection (thanks to the excision property of the TFT). [START_REF] Taubes | Casson's invariant and gauge theory[END_REF] gives a gauge theoretic definition of the Casson invariant, for which we provide a derived symplectic interpretation. A first observation is that there is a morphism M dR → M B of 3-oriented derived stacks that leads to an equivalence of (-1)-symplectic derived stacks between Loc(M, G) := Map(M B , BG) and Flat(M, G) := Map(M dR , BG), for G = SU (2).

Gauge theoretic approach

Then one identifies Flat(M, G) with the derived critical locus of the Chern-Simons functional S that is defined on the (infinite dimensional) moduli stack

Conn(M, G) of G-connections: S(A) := M tr d dR A ∧ A + 2 3 A ∧ A ∧ A .
In order to be fully accurate, one shall say that S is not well-defined on this moduli space, as it is not gauge invariant: there exists a constant c such that for a gauge g ∈ C ∞ (M, G), S(A g ) = S(g -1 d dR g + g -1 Ag) = S(A) + cZ. Hence, even though S is not defined on the moduli space, its exterior derivative d dR S defines a closed 1-form on Conn(G), so that we can still talk about the derived critical locus of S (that is the derived intersection of the zero section with the graph of d dR S).

Because of infinite dimensional issues, it is not easy at all to define correctly a (possibly virtual) count for the derived critical locus of S. This is where analytic techniques come into the game: [START_REF] Taubes | Casson's invariant and gauge theory[END_REF] uses elliptic regularity and Fredholm theory to reduce locally to finite dimension. In fact, he constructs local potentials and finds critical charts (sometimes called Chern-Simons charts) in the sense of [START_REF] Brav | A Darboux theorem for derived schemes with shifted symplectic structure[END_REF], whose Darboux theorem in fact already provides such critical charts.

It is [START_REF] Tu | Casson invariants via virtual counting[END_REF] who first gave a virtual count formulation of the gauge theoretic approach to the Casson invariant, using L ∞ -spaces and Kuranishi charts.

Conclusion

An important topic of current research is the quantization of shifted symplectic structures. The study of their geometric quantization has been initiated by [START_REF] Safronov | Shifted geometric quantization[END_REF], while the one of their deformation quantization has been initiated by [START_REF] Calaque | Shifted Poisson structures and deformation quantization[END_REF]. The latter required to develop a whole new theory of shifted Poisson structure, which was independently done by [START_REF] Pridham | Shifted Poisson and symplectic structures on derived N -stacks[END_REF].

The virtual count on (-1)-symplectic derived schemes can already be seen as a quantization process. General quantization principles suggest that a (-1)symplectic structures shall be quantized by a cochain complex, whose Euler characteristic would give back the virtual count. [START_REF] Brav | Symmetries and stabilization for sheaves of vanishing cycles[END_REF] prove that if a scheme X 0 admits local critical charts with a global choice of appropriate signs for their glueing (such an X 0 is called an oriented d-critical locus), then one can construct a perverse sheaf on X 0 that is locally equivalent to the sheaf of vanishing cycles of the local potential; the Euler characteristic of the hypercohomology of this sheaf gives back the virtual count on X 0 .

Examples of such oriented d-critical loci are given as underived truncations of (-1)-symplectic derived schemes equipped with a square root of the canonical sheaf (such a square root exists for instance whenever there is a lagrangian foliation structure on the (-1)-symplectic derived scheme). [START_REF] Pridham | Deformation quantisation for (-1)-shifted symplectic structures and vanishing cycles[END_REF] provides a detailed explanation of the relation between the above and deformation quantization of (-1)-symplectic structures. This circle of ideas can be applied to recover a holomorphic version of the Casson invariant, originally introduced by [START_REF] Thomas | A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations[END_REF], that "counts" sheaves on a Calabi-Yau 3-fold, and that is the starting point of the theory of Donaldson-Thomas invariants.

Here, classical is understood as opposed to quantum.

A similar result holds in derived differential geometry if one replaces the reductive group with a compact Lie group.

Acknowledgments This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 768679).


In this perceptive study, Noëlle Cuny sets out to examine the function of the body in D. H. Lawrence's writings, bringing to the fore a web of metaphors based on the themes of pathology and deformity, and thereby dismissing views that the Lawrencian body is only sexual or erotic.

Focusing on four novels, The Rainbow, Women in Love, Kangaroo and The Plumed Serpent, she shows how the living body and fiction echo one another, and therefore, how, from this perspective, the metaphor of "the body in becoming" (le corps en devenir), coined from Deleuze's terminology, aptly depicts the development of Lawrence's fiction. Thus, she says, "the body in becoming" cannot fit one definition; rather, it oscillates between contrary forces which are unstable and ephemeral within the tensions of the narrative, the resulting dynamics combining cohesion and discrepancy, corruption and paroxysm. To sustain her point, Noëlle Cuny devotes interesting pages to how Lawrence appropriates the ancient rhetorical figure of the ailing body inherited from the Hebrew Prophets and points to how he responds to his readings of Frazer, Darwin and others, by resorting to the weaknesses of the body to express social malaise.

The novels considered in the book were written in two distinct periods of the author's life: the first a happy one in England, the second more bitter in exile. But going beyond mere biographical considerations, Cuny contends that taking into account scientific breakthroughs as well as emerging ideologies contemporary to Lawrence is fundamental to understanding the function of the body in the novels. Thus, The Rainbow and Women in Love stage the construction of the self in the heart of modern industry, while Kangaroo and The Plumed Serpent depict the regeneration of the collective body in lands suffering from their history. To support her interesting findings, the study also contains incursions into Lawrence's short stories, essays and poems, and looks at some of his paintings. Excerpts from early versions of novels are also included.

Before launching into the study itself, Cuny devotes a chapter to the body in literature, identifying the liberating influences which enabled Lawrence to build his own vision of corporeality. She traces the evolution of critical views on the body, more specifically in feminist and Marxist criticism. Set against the profuse critical studies devoted to Lawrence, Cuny's contribution casts new light on the body as a metaphor for the evolution of the literary works of an author whose own body was ailing.

In the first part, starting with The Rainbow, Cuny analyses the narrative components of the mystery (in the mystical sense) which frees Ursula Brangwen's body from her biological and social background. She studies how the alchemical metaphor of resolution disseminated throughout the novel expresses the character's endeavour to achieve singularity. Irresolute matter, she contends, is, as the blood of generations of Brangwens, that which has not been purified, and atavism, whose various bodily forms are studied, is identified as being the obstacle to the resolution, or fulfilment, of the potentiality of each character. As the characters deprived of a nucleus are prisoners of history, achieving singularity implies getting rid of scoriae, a term which evokes coal and also introduces the web of analogies linked to the production of steel. Indeed, steel is ridden of the coal and phosphor which make it brittle, two elements to be found repeatedly in The Rainbow which refer to the industrial expansion of the period when the novel was written. After having mentioned the eugenic connotations of such images, Cuny highlights the dual polarity of coal and steel between which "the body in becoming" oscillates.

The second chapter in this first part illustrates how, in Women in Love, the quest for the atom is voiced through an aesthetics of fragmentation. Focusing on the increasing mechanization of society, the chapter shows how such fragmentation leads to the reinterpretation of the human. A case in point, Gerald Crich, the director of the mine, incarnates this process when he reduces his employees to a mechanical and Taylorist function, and also when he subjects the body to a form of "exo-Darwinism", replacing it or extending it with tools. Cuny stresses that the merging of the human with the machine, characteristic of modernist aesthetics, also governs the manifestations of affect. Indeed, she shows that the semantic theme of electricity is disseminated in the narration of amorous intercourse, arguing that this form of energy, which affects the nervous system, is aggressive and destructive.

Cuny shows that language, like "the body in becoming", also undergoes fragmentation. The polysemic play on terms such as "concentration", "reduction" and "resolution", which all designate processes of purification, unfolds in a pervasive semantic web. As the web evolves from one novel to the next, the reader is enticed to find the new meaning(s) of language. Cuny concludes that the new meanings confer on the human metabolism a salvaging function. That's why, by means of a typically Lawrencian syncretism of different mythologies, the center and the origin of the world are shifted to the digestive system, the bowels being the organ of reduction par excellence. The energy which the characters send out is assimilated to phosphor, to radioactivity, to the quanta of particles, so many images to express communication between two beings, all of which with hindsight seem to be borrowed from the scientific breakthroughs of a period in full mutation.

The second part of the book deals with the collective body. In Kangaroo, two political projects develop against the backdrop of the apathy of Australians at odds with their history. The homoerotic friendship which links the members of the military group named the Diggers stages the body through masculine embraces, which become so many manifestations of a will for universal fusion. The unflattering description of the body of the Diggers' charismatic leader, Ben Cooley, alias Kangaroo, conveys the ambivalent reaction of Somers, the main character who must decide whether or not to join the movement. His attraction to this body, whose enormous marsupial-like belly symbolizes the magnetic centre of the cosmos, leads to disgust. When Kangaroo dies, a bullet having perforated his belly during a riot, disgust reaches its climax. Cuny further explores the metaphors of roundness and adiposity which convey the tensions at play in the collective social body. She associates such corporeal excesses, which characterize the bourgeois masses in the novel, with a cerebral void. The key to the rebirth of man is therefore what Somers calls "the fearful god who enters us from below", terms reminiscent, as Cuny points out, of Lawrence's essays, in which he explicitly adopted the coccygeal centre as the starting point for the nervous flux.

In The Plumed Serpent, Lawrence daringly merges the political and the mystic by staging the accession to power of priesthood. Cuny argues that this new era is brought about to revive the ancient physiological humours. The body is therefore subjected to primitive rituals which glorify the interior life. The chapter analyses more specifically the ritual of unction which delineates the bodies of the chosen ones and enables them to achieve a form of secret communication, or communion, akin to jouissance. She also turns our attention to the female body which undergoes the cult's sexual rituals. As it is denied access to sexual satisfaction, she asks whether the sexual renewal staged in the novel does not aim at promoting the very cerebral sexuality which Lawrence spoke out against in some of his essays.

In her conclusion, in order to synthesize her findings, Cuny extends her analysis of "the body in becoming" to Lady Chatterley's Lover. She demonstrates how the body here undergoes a perpetual refining process in order to achieve the essential and the truth. Thus the body is the means to convey an aesthetics of mutation, of conversion whose vocabulary is permeated both by alchemy and science.

The reader of this book will appreciate the impact that contemporary industrial and scientific breakthroughs had on Lawrence's writing. These breakthroughs provide webs of metaphors which converge to express a modernist view of corporeality. Annexed to this thorough study, an index of themes for each novel highlights the vast scope of the subject. Summaries of the novels discussed emphasize the role of the body in each of the narratives. To conclude, this book skilfully restores the body to its complex metaphorical function in Lawrence's art, a restoration that is fully symbolized on the book's cover through the choice of Lawrence's painting entitled "Leda", itself an allegory of the mysticism of the body.


KKL (Kazantzis-Kravaris/Luenberger) observers are based on the idea of immersing a given nonlinear system into a target system that is a linear stable filter of the measured output. In the present paper, we extend this theory by allowing this target system to be a nonlinear contracting filter of the output. We prove, under a differential observability condition, the existence of these new KKL observers. We motivate their introduction by showing numerically the possibility of combining convergence speed and robustness to noise, unlike what is known for linear filtering.

On the existence of KKL observers with nonlinear contracting dynamics (long version)

Victor Pachy * Vincent Andrieu * * ,1 Pauline Bernard * * * Lucas Brivadis * * * * Laurent Praly * * *

INTRODUCTION.

We consider nonlinear systems of the form ẋ = f (x), (1a) y = h(x), (1b) where x lying in R n is the state of the system, y lying in R is the measured output, and f and h are smooth maps. The synthesis of state observers for such systems is a major topic in control theory, and numerous methods have been developed over the years. Interested readers can refer to [START_REF] Bernard | Observer design for continuous-time dynamical systems[END_REF] for an overview of existing methods. Among these methods, the so-called KKL approach has recently garnered significant attention due to its generality and the weak assumptions required ensuring its existence. This methodology, originating from the seminal work of [START_REF] Luenberger | Observing the state of a linear system[END_REF] for linear systems, has been extended for nonlinear systems first locally in [START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF]; [START_REF] Shoshitaishvili | Singularities for projections of integral manifolds with applications to control and observation problems[END_REF] and then globally in [START_REF] Kreisselmeier | Nonlinear observers for autonomous Lipschitz continuous systems[END_REF]; [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF]; [START_REF] Brivadis | Further remarks on KKL observers[END_REF]. The strategy consists in finding a positive integer m and a nonlinear mapping T : R n → R m such that if (x, y) is a solution of (1), then z = T (x) satisfies ż = Az + By, (2) where A ∈ R n×m is a Hurwitz matrix and B ∈ R m is a vector. In that case, d dt (z -T (x)) = A(z -T (x)), 1 The research of this author was funded in whole or in part by the National Research Agency (ANR) under the project titled "ANR-23-CE48-0006-02."

hence an asymptotic exponential approximation of T (x) can be obtained by running the linear filter of the output (2). Then, if T admits a uniformly continuous left-inverse T inv : R m → R n , an observer of x can be defined as x = T inv (z) and on has lim t→+∞ (x(t) -x(t)) = 0.

Different kind of sufficient conditions for the existence of such mappings T and T inv are given in [START_REF] Andrieu | On the existence of a kazantzis-kravaris/luenberger observer[END_REF].

In this paper, we propose to extend the class of admissible filters of the output for the design of the observer. More precisely, we replace (2) by ż = kσ(z, y)

(3) where k is a positive real number and σ ensures that (3) has exponentially contracting dynamics (in a sense to be defined), ensuring in particular that the distance between any pair of solutions sharing the same y exponentially decreases towards 0. This is a natural extension, since, similar to the linear case, if one finds a mapping T such that z = T (x) is solution to (3), an asymptotic exponential estimation of T (x) can be obtained, by running (3) from any initial condition. Then, as with the linear filter (2), if T admits a left-inverse T inv : R m → R n , an observer can be defined as x = T inv (z) (4) and one has lim t→+∞ (x(t) -x(t)) = 0 provided that T inv is uniformly continuous. This study is motivated by three main reasons: nonlinear filters may (i) give access to better observer performance, for instance allowing to combine convergence speed and robustness to noise or increase robustness to model uncertainties; (ii) bridge the gap between the fields of observer design and recursive neural networks in machine learning; (iii) give more flexibility in the research of an analytical expression of the map T .

Regarding item (i), while it is well-known that in the context of linear dynamics with Gaussian noise, the use of linear correction gains, as provided by a Kalman filter, is optimal [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF]), the same cannot be said when departing from this context. For instance, it is typically interesting to introduce nonlinear phenomena such as saturations in the case of sporadic noise, or, dead zones in the case of high-frequency and lowamplitude noise [START_REF] Tarbouriech | Lmi-based stubborn and dead-zone redesign in linear dynamic output feedback[END_REF]. Furthermore, the interest in non-linear observers for achieving robustness to model errors is typically highlighted in the context of homogeneous observers or sliding mode observers (see for instance [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF]). Moreover, convergence speed and robustness to noise are usually antinomic, unless varying/switching gains are introduced or even switches among different observers, depending on the size of the output error, but with thresholds that may be hard to tune [START_REF] Petri | Hybrid multi-observer for improving estimation performance[END_REF]; [START_REF] Chong | Parameter and state estimation of nonlinear systems using a multi-observer under the supervisory framework[END_REF]; [START_REF] Esfandiari | Bank of highgain observers in output feedback control: Robustness analysis against measurement noise[END_REF]. Our goal in considering (3) is to allow such performance, without requesting any canonical form of the system (1), or having any online threshold to handle.

This also brings us to item (ii) since it is well-known in the machine learning community that the addition of nonlinear terms can enhance the expressiveness of neural networks and improve performance. Actually the resemblance between KKL observers -where the system output is fed to a series of filters and the estimate is recovered from those internal states via a nonlinear map to be found -and recursive neural networks (RNNs) is uncanny (see [START_REF] Janny | Deep KKL: Data-driven output prediction for non-linear systems[END_REF] for a comparison). In fact, when no explicit expression of the map T is available, it has been proposed to learn an approximate model of it via neural networks in [START_REF] Ramos | Numerical design of Luenberger observers for nonlinear systems[END_REF]; [START_REF] Niazi | Learning-based design of Luenberger observers for autonomous nonlinear systems[END_REF]; [START_REF] Buisson-Fenet | Towards gain tuning for numerical KKL observers[END_REF]; [START_REF] Peralez | Deep learning-based Luenberger observer design for discrete-time nonlinear systems[END_REF]; [START_REF] Janny | Deep KKL: Data-driven output prediction for non-linear systems[END_REF]. Allowing nonlinear contractions in KKL could pave the road towards understanding this similarity and providing theoretical foundations to the convergence of RNNs as well as guidelines in terms of dimensions.

However, all those methods require a significant (offline) computational load and suffer from a curse of dimensionality, sometimes at the expense of asymptotic precision on the online estimate. Our third goal (iii) is thus to enlarge the class of systems for which an explicit expression of the transformation T can be obtained by providing more flexibility in the target dynamics (3).

From a theoretical point of view, the existence of an immersion T of any system (1) into dynamics (3) is typically guaranteed, for instance exploiting the theory of uniformly convergent systems in [START_REF] Pavlov | The uniform global output regulation problem[END_REF]. The challenge rather lies in proving its injectivity under observability conditions. This paper achieves a first step by showing this injectivity for a particular structure of (3), consisting of m sufficiently fast parallel filters and under an assumption of strong differential observability of order m.

The article is structured as follows. Firstly, we present a general theorem that establishes the existence of a Lipschitz injective application T which, when applied on the state trajectory is a specific solution to the nonlinear filter equation for sufficintly large parameter k. The next section provides a demonstration, presented as a series of propositions, with proofs provided in the Appendix. In the following section, we offer a preliminary illustration of these results within a robustness context. Lastly, the conclusion is provided.

Notation. For a subset O of R n , we denote by O + δ the set O + δ = {x ∈ R n | ∃x 0 ∈ O, |x -x 0 | ≤ δ}.
When needed, to exhibit the dependency on initial conditions x ∈ R n , we denote, when defined, X(x, t) the (unique) solution to (1a) at time t, initialized at x. By L f h we denote the Lie derivative of h alongside the vector field f , i.e., L f h(x) = ∂h ∂x (x)f (x), for all x ∈ R n . Given a map g : R n → R p , we define ∆g : λ obtained by keeping only the monomials of power 1 λ j with j ∈ {0, . . . , i} (resp. j = i) in p(λ) m .

R n × R n → R p by ∆g(x a , x b ) = g(x a ) -g(x b ) for (x a , x b ) ∈ R n × R n

MAIN RESULT

We assume (i) the system solutions of interest, initialized in some set X 0 ⊂ R n , are defined on R ≥0 and remain in a compact set X ⊂ R n , and (ii) the system (1) is strongly differentially observable of order m on X , as detailed next. Assumption 1. There exists a compact set X ⊂ R n such that for all solution of (1) such that x(0) ∈ X 0 then x(t) ∈ X for all t ≥ 0. Assumption 2. There exists an integer m ≥ 1 such that the map H m : R n → R m defined by:

H m : x →      h(x) L f h(x) . . . L m-1 f h(x)      (5)
is k H -Lipschitz injective on X for some constant k H > 0.

The latter assumption is guaranteed as soon as H m is injective and its jacobian is full-rank on X , i.e., when H m is an injective immersion on X .

In this paper, we show that the system (1) can then be transformed through a left-invertible change of coordinates into a contracting filter (3) with σ consisting of m parallel contracting filters of the output in the form

σ(z, y) =    λ 0 σ(z 0 , y) . . . λ m-1 σ(z m-1 , y)    (6)
where λ 0 , . . . , λ m-1 are positive distinct scalars, and σ : R × R → R satisfying:

Assumption 3. The map σ : R × R → R is of class C m+1
and verifies for all (z, y) ∈ R × R,

0 < γ ≤ ∂σ ∂y (z, y) (7a) -β ≤ ∂σ ∂z (z, y) ≤ -α < 0 (7b)
for some constants α, β, γ > 0.

It can be noticed that the map σ defined in (6) then verifies ∂σ ∂z (z, y)

+ ∂σ ∂z (z, y) ⊤ < -µ I m , ∀(z, y) ∈ R m × R, (8)
where µ = -α min 0≤i≤m-1 λ i > 0 which according to (Pavlov et al., 2004, Theorem 1), guarantees that the filter (3) is uniformly convergent in the sense of [START_REF] Pavlov | The uniform global output regulation problem[END_REF]. From this, we show the following result.

Theorem 1. Under Assumptions 1, 2 and 3, for all m-uplet (λ 0 , . . . , λ m-1 ) of distinct positive scalars, there exists k * > 0 such that for all k > k * , there exists a map T : R n → R m that is Lipschitz injective on X and such that, for all solution t → x(t) of (1a) with initial condition in X 0 , t → T (x(t)) is solution to (3) with t → y(t) given by ( 1b) and σ defined in ( 6).

This result is proved in Section 3. Under the conditions of the above theorem, we deduce from the injectivity of T that there exists a continuous map

T inv : R m → R n that is a left-inverse of T on X , namely T inv (T (x)) = x ∀x ∈ X .
For any such left-inverse, any solution (x, z) of ( 1)-( 3) initialized in X 0 × R m , with σ defined in (6), verifies lim t→+∞ (x(t) -x(t)) = 0, with x given by (4) (see for instance (Brivadis et al., 2023, Theorem 1.1)). Note that from the Lipschitz -injectity of T , T inv can even be picked globally Lipschitz, providing exponential convergence of the estimation error as in [START_REF] Andrieu | Convergence speed of nonlinear Luenberger observers[END_REF].

PROOF

Some preliminaries

For each y in R, the function z → σ(z, y) is decreasing and is a bijection from R to R from Assumption 3. Hence, there exists a unique map ψ : R → R such that for all y ∈ R, σ(ψ(y), y) = 0. (9) Moreover, according to the implicit function theorem,

ψ ∈ C m+1 (R, R) since σ ∈ C m+1 (R, R).
Also, it can be observed that, from Assumption 1, the solutions to (1a) initialized in X 0 coincide in positive time with that of the modified dynamics

ẋ = f (x) = χ(x)f (x), (10) with χ a smooth function such that χ(x) = 1 if x ∈ X , χ(x) = 0 if x / ∈ X + δ u for some δ u > 0.
Besides, Assumptions 1 and 2 still hold for (10) with output map (1b). In the rest of the proof, we thus assume without loss of generality that f = f and we consider that Assumption 1, whenever used, ensures (forward and backward) invariance of the compact set X + δ u as well as completeness and boundedness of all solutions.

Construction of T

Proposition 1. Under Assumptions 1 and 3, for all m-uplet (λ 0 , . . . , λ m-1 ) of distinct positive scalars, and for all k > 0 there exists T : R n → R m such that, for all solution t → x(t) of (1a) with initial condition in R n , t → T (x(t)) is solution to (3) with t → y(t) given by (1b) and σ defined in (6).

Proof. With Assumption 1, for all x ∈ R n , the function s → y(s) := h(X(x, s)) is defined, smooth, and bounded for all positive time. According to [START_REF] Pavlov | The uniform global output regulation problem[END_REF], Theorem 1), Assumption 3 guarantees that for all x in R n , the system ż = λσ(z, y) (11) with state z ∈ R and λ > 0 admits a unique bounded solution t → Z(t, y, λ) defined on R, which is uniformly globally asymptotically stable. For all x ∈ R n and all λ > 0, we then define

T (x, λ) = Z(0, s → h(X(x, s)), λ), (12)
and for all λ 0 , . . . , λ m-1 > 0, and all k > 0,

T (x) =     T (x, kλ 0 ) T (x, kλ 1 ) . . . T (x, kλ m-1 )     . (13)
We then show that for any x ∈ R n , t → T (X(x, t)) is solution to (3). To do so, it is sufficient to show that for any λ, k > 0, T (X(x, t), kλ) = Z(t, s → h(X(x, s)), kλ) for all t ∈ R. For all t in R, we have T (X(x, t), kλ) = Z(0, s → h(X(X(x, t), s)), kλ) = Z(0, s → h(X(x, t + s)), kλ). It is thus enough to show that for each t ∈ R, Z(0, s → h(X(x, t + s)), kλ) = Z(t, s → h(X(x, s)), kλ).

(14) Let t ∈ R. For any bounded τ → y(τ ), the system dz dτ

(τ ) = kλσ(z(τ ), y(τ )), ∀τ ∈ R, (15)
is uniformly convergent in the sense of [START_REF] Pavlov | The uniform global output regulation problem[END_REF]. Therefore, it admits a unique bounded solution defined on R that is τ → Z(τ, s → y(s), kλ). Consider the map

z t : τ → Z(τ + t, s → h(X(x, s)), kλ). It is bounded on R and verifies, by definition of Z, dz t dτ (τ ) = kλσ( Z(τ +t, s → h(X(x, s)), kλ), h(X(x, t+τ ))).
So by uniqueness of the bounded solution of (15), z t (τ ) = Z(τ, s → h(X(x, t + s)), kλ). Taking τ = 0 yields (14) and concludes the proof. □

Construction of an approximation of T which is Lipschitz injective

To prove Theorem 1, we need to show that the mapping T obtained from Proposition 1 is Lipschitz injective on X for sufficiently large k. The idea of the proof is to use an approximation of the components of T (i.e. T (•, kλ i )), and more precisely a development in powers of 1 k . The approximation we consider is obtained from a C 1 mapping ϕ : R n → R m , with ϕ = (ϕ 0 , . . . , ϕ m-1 ) where ϕ ℓ are of class C m-i+1 and are defined recursively for x in R n as ϕ 0 (x) = ψ(h(x)), (16) and for ℓ ∈ {1, . . . , m -1}:

ϕ ℓ (x) = κ(h(x)) L f ϕ ℓ-1 (x) - ℓ j=1 1 j! ∂ j σ ∂z j (ψ(h(x)), h(x)) × ℓ1+•••+ℓj =ℓ 1≤ℓ1,...,ℓj ≤ℓ-1 ϕ ℓ1 (x) • • • ϕ ℓj (x) , (17)
where

κ(y) = ∂σ ∂z (ψ(y), y) -1 . ( 18
)
Notice that the definition of

ϕ ℓ (x) for ℓ ≥ 1 involves ϕ i (x) with i ∈ {1, • • • , ℓ -1} only and is independent from λ.
The approximation of T (x, λ) is then defined as

T a (x, λ) = m-1 ℓ=0 ϕ ℓ (x) λ ℓ . ( 19
)
From there, given λ = (λ 0 , . . . , λ m-1 ) and given k > 1, we thus get from ( 13) the following approximation of T

T a (x) = VK -1 ϕ(x), (20) where
K = diag 1, . . . , k m-1 and V is the Vandermonde matrix V =    1 λ -1 0 . . . λ -(m-1) 0 . . . . . . . . . . . . 1 λ -1 m-1 . . . λ -(m-1) m-1    . (21)
Clearly, injectivity of T a may be deduced from the injectivity of ϕ as soon as the λ i 's are all distinct. Actually, in the following proposition, it is shown that with Assumption 2, injectivity of ϕ is ensured on X . Proposition 2. Under Assumption 2, ϕ is Lipschitz injective in X . In other word, there exists a positive real number

k ϕ such that for all (x a , x b ) in X 2 |ϕ(x a ) -ϕ(x b )| ≥ k ϕ |x a -x b | . ( 22
)
The proof of Proposition 2 can be found in Section A.5. A direct consequence is that for all m-uplet (λ 0 , . . . , λ m-1 ) of positive distinct scalars, all k > 1, all (x a , x b ) in X 2 the following inequality is satisfied

|T a (x a ) -T a (x b )| ≥ k ϕ k m-1 ∥V -1 ∥ |x a -x b |, (23) since ∥K∥ ≤ k m-1 , which establishes that T a is Lipschitz injective in X .
3.4 T a is an approximation of order λ -m of T

The reason for stating that T a is an approximation of T will be shown in this section. Indeed, we demonstrate that the difference between these two functions is of the order λ -m . In Proposition 1, we have shown that for any λ 0 , . . . , λ m-1 > 0, any k > 0 and any x ∈ R n , t → T (X(x, t)) is solution to (3). From the definitions of σ and T in ( 6) and ( 13), we deduce that T is supposed to verify, for all x ∈ R n and all λ > 0,

L f T (x, λ) = λσ(T (x, λ), h(x)). (24)
In order to establish that T a approximates T , we first study the error in the partial differential equation ( 24) induced by this approximation. Proposition 3. Let ω : R n × R >0 → R be defined as (19). Under Assumptions 1 and 3 there exist two positive real numbers ω and k ω such that

ω(x, λ) = L f T a (x, λ) -λσ(T a (x, λ), h(x)) λ with T a defined in
∀x ∈ X + δ u , ∀λ > 1, |ω(x, λ)| ≤ ω λ m , (25)
and for all (

x a , x b ) ∈ (X + δ u ) 2 , ∀λ > 1, |ω(x a , λ) -ω(x b , λ)| ≤ k ω λ m |x a -x b |. ( 26
)
The proof of Proposition 3 can be found in Section A.1. We introduce R the difference between T and its approximation T a : R(x, λ) = T (x, λ) -T a (x, λ).

(27) Employing the bounds obtained in Proposition 3, the following two propositions establish that R is bounded and is Lipschitz with order 1 λ m : Proposition 4. Under Assumption 1 and 3,

∀x ∈ X + δ u , ∀λ > 1, |R(x, λ)| ≤ ω αλ m . ( 28
)
Proposition 5. Under Assumption 1 and 3, there exist λ * > 1 and a positive real number k R such that for all

(x a , x b ) in (X + δ u ) 2 for all λ > λ * , |R(x a , λ) -R(x b , λ)| ≤ k R λ m |x a -x b |. (29)
The proof of Proposition 4 and 5 can be found in Section A.2 and Section A.4.

Proof of Theorem 1

For k > 1 and for a given m-uplet λ of distinct positive real numbers, let T be given by Proposition 1. For all solution t → x(t) of (1a) with initial condition in X 0 , t → T (x(t)) is solution to (3) with t → y(t) given by (1b) and σ defined in ( 6). With T a given previously, we can introduce the mapping R as

T (x) = T a (x) + R(x), (30) with R(x) = (R(x, kλ 0 ), . . . , R(x, kλ m-1 )
). With Proposition 2, equation ( 23) and Proposition (29), we conclude that for all k such that kλ i > λ * for all i = 0, . . . , m -1,

|∆T (x a , x b )| ≥ 1 k m-1 k ϕ ∥V -1 ∥ - 1 k m-1 i=0 k R λ m i |x a -x b |.
This proves that there exists k * > 0 such that, for any k ≥ k * , x → T (x) given in ( 13) is Lipschitz injective on X which concludes the proof of Theorem 1.

ILLUSTRATION

In this part, we show how a KKL observer with nonlinear filter dynamics may allow to obtain simultaneously fast convergence and robustness to measurement noise. For that, we compare its performance to two KKL observers with linear filter dynamics, one fast and the other slow ż = λa fast (z -y), ż = λa slow (z -y), (31) where λ > 0, and a fast < a slow < 0. When choosing a family (λ i ) 1≤i≤m , the corresponding observers are given by (2) with A fast = a fast diag(λ i ) 1≤i≤m and B fast = -a fast (1, . . . , 1) ⊤ (resp. A slow = a slow diag(λ i ) 1≤i≤m and B slow = -a slow (1, . . . , 1) ⊤ ). It is well-known that when tuning the parameters of linear filters, a compromise has to be found between speed of convergence and robustness to noise. More precisely, modulo the nonlinear left-inversion of T , KKL observers built with fast (resp. slow) linear filters usually exhibit fast (resp. slow) convergence properties but poor (resp. good) robustness to measurement noise.

On the other hand, building a KKL observer based on nonlinear filters allows to consider dynamics of the form ż = λ(a fast (z -y) + (a slow -a fast )tanh(z -y)) (32) which can be checked to verify Assumption 3. When choosing a family (λ i ) 1≤i≤m , the corresponding observer is given by (3) for k = 1 and the corresponding contraction σ as in ( 6). The motivation is the following: On the one hand, for large values of |z -y|, the nonlinear dynamics behave as the fast linear dynamics. Therefore, a fast convergence is expected. On the other hand, for small values of |z -y|, the nonlinear dynamics behave as the slow linear dynamics. Therefore, the same robustness with respect to small perturbations of the measurement y is expected. In other words, the nonlinear KKL observers is expected to take the best of both worlds.

For simulations, we consider a nonlinear Duffing oscillator ẋ1 = x 2 ẋ2 = -0.2x

1 -x 3 1 , y = x 1 , (33)
which verifies Assumptions 1 and 2 with X 0 = [-2, 2] 2 and m = 2. We pick λ 1 = -2, λ 2 = -4, λ 3 = -6, a fast = 5 and a slow = 0.5. For each KKL observer, we create a dataset of points (x, z) approximating (x, T (x)), with the map T such that the image of solutions to (33) by T is solution to the corresponding observer dynamics (2) or (3). This is done by simulating the interconnection of ( 33) with (2) or (3) from a grid of 200 × 200 initial conditions in X 0 × {0}, and storing the obtained pairs (x, z) after a time t l := 20/ min{|λ i |}, needed for the filters to "forget" their initial condition. Values of T (x) for a given x and of T inv (z) for a given z can then be obtained by fetching the closest point in the dataset.

We then propose two simulation scenarios: 1) we initialize all observers with the same initial error randomly picked so that |z(0) -T (x(0))| = 100 and we do not add measurement noise, and 2) we initialize all observers to their correct initial condition T (x(0)) and we add a sinusoidal measurement noise (ν(t) = 0.1 sin(10t)). The first (resp. second) scenario aims at comparing the convergence times (resp. the impact of measurement noise). The numerical results for a particular choice of x(0) are provided in Figure 1. As expected, the KKL observer with nonlinear dynamics converges as fast as the one with fast linear dynamics and seems almost as robust to noise than the one with slow linear dynamics. Minimum, maximum and mean convergence time and gain with respect to noise for (3) defined from (32). Scenario 1: initial error but no noise; Scenario 2 : no initial error but noise.

CONCLUSION

In this article, we have presented a KKL-type observer with non-linear dynamics. In the scenario where the system is differentially observable of order m, we have demonstrated the existence of such an estimation algorithm when the observer's dynamics are structured as m non-linear filters operating in parallel, provided that their dynamics are sufficiently fast. Through a simplified illustration, we have highlighted a potential application of this technique to obtain a better qualitative behavior of the estimate. In a more general context, demonstrating the existence of such an observer for a broader class of contractions would be highly intriguing. However, the techniques employed in this article may not readily adapt to this more general framework.

Appendix A. PROOFS

This part of the paper is not in the version which has been published in [START_REF] Pachy | On the existence of KKL observers with nonlinear contracting dynamics[END_REF].

In this section, we give the technical proofs employed to get the main result. Note that to simplify the presentation, we use the following notation:

σ (j) (x) = ∂ j σ ∂z j (ψ(h(x)), h(x)).

A.1 Proof of Proposition 3

First of all, we have the following technical lemma whose proof can be found in Appendix A.3. Lemma 1. For each x ∈ R n and λ > 0,

L f T a (x, λ) = λ m-1 j=1 σ (j) (x) j! m-1 ℓ=1 ϕ ℓ (x) λ ℓ j ≤m-1 + 1 λ m-1 L f ϕ m-1 (x). (A.1)
Hence, with ( 18), ( 19), ( 16) and (A.1), it yields

ω(x, λ) = m-1 i=1 σ (i) (x) i! m-1 ℓ=1 ϕ ℓ (x) λ ℓ i ≤m-1 -σ ψ(h(x)) + m-1 ℓ=1 ϕ ℓ (x) λ ℓ , h(x) + 1 λ m L f ϕ m-1 (x)
As σ is C m+1 , using ( 9) and a Taylor expansion of σ

around (ψ(h(x)), h(x)), ω(x, λ) = ω 0 (x, λ) λ m -G(x, λ) + 1 λ m L f ϕ m-1 (x) where ω 0 (x, λ) = 1 (m -1)! 1 0 (1 -s) m-1 ∂ m σ ∂z m ψ(h(x)) + s m-1 ℓ=1 ϕ ℓ (x) λ ℓ , h(x) ds m-1 ℓ=1 ϕ ℓ (x) λ ℓ-1 m G(x, λ) = m-1 i=1 σ (i) (x) i! m-1 ℓ=1 ϕ ℓ (x) λ ℓ i - m-1 ℓ=1 ϕ ℓ (x) λ ℓ i ≤m-1
Note that G equals 0 if i = 1 and equals

1 λ m i(m-1) j=0 1 λ j φij (x)
if i > 1, with φij being a certain polynomial of the ϕ ℓ for ℓ = 1, . . . , m -1, therefore this gives

ω(x, λ) = ω 0 (x, λ) + L f ϕ m-1 (x) λ m + 1 λ m m-1 i=2 σ (i) (x) i! i(m-1) j=0 φij (x) λ j . (A.2)
Therefore, using triangular inequality, for x ∈ X + δ u and λ > 0,

|λ m ||ω(x, λ)| ≤ |ω 0 (x, λ)| + |L f ϕ m-1 (x)| + m-1 i=2 1 i! σ (i) (x) i(m-1) j=0 1 λ j | φij (x)|. (A.3)
As σ and ψ are C m+1 , σ (i) is continuous, and therefore bounded on the compact set X + δ u for all i in {1, . . . , m}. Moreover, φkj is a polynomial of the continuous functions ϕ ℓ , and therefore is bounded on the compact set X +δ u . As

ϕ m-1 is C 2 , L f ϕ m-1 is also bounded on X + δ u . Moreover, there exists δ ψ > 0 such that m-1 ℓ=1 ϕ ℓ (x) λ ℓ ≤ δ ψ for all x ∈ X + δ u and all λ > 1. By continuity of ∂ m σ ∂z m , it is bounded on (ψ(h(X + δ u )) + δ ψ ) × h(X + δ u ), uniformly in λ > 1.
We can conclude that there exists ω > 0, independent from λ, such that (25) holds. Note also that ω is C 1 and so are all the previously listed maps. So similarly, it yields,

λ m ∂ω ∂x (x, λ) ≤ ∂ω 0 ∂x (x, λ) + ∂L f ϕ m-1 ∂x (x) + m-1 i=2 1 i! ∂σ (i) ∂x (x) i(m-1) j=0 1 λ j ∂ φij ∂x (x) . (A.4)
Hence, there exists a positive real number k ′ ω , such that for all λ > 1 and x ∈ X + δ u , it yields

λ m ∂ω ∂x (x, λ) ≤ k ′ ω .
Inequality ( 26) is obtained for some k ω > k ′ ω .

A.2 Proof of Proposition 4

The evaluation R(X(x, t), λ) of R along any solution initiated at x ∈ X + δ u satisfies

d dt R(X(x, t), λ) = λ[σ(T a (X(x, t), λ) + R(X(x, t), λ), h(X(x, t))) -σ(T a (X(x, t), λ), h(X(x, t)))] -λω(X(x, t), λ)
With the condition (7b) it yields for all x ∈ X + δ

u R(x, λ) σ(T (x, λ) + R(x, λ), h(x)) -σ(T (x, λ), h(x)) ≤ -αR(x, λ) 2 .
Since moreover, X + δ u is negatively invariant along the flow and with Young's inequality, for all ε > 0 it gives

d dt R(X(x, t), λ) 2 ≤ -2λαR(X(x, t), λ) 2 + λ ε ω(X(x, t), λ) 2 + λεR(X(x, t), λ) 2 = -λαR(X(x, t), λ) 2 + λ α ω(X(x, t), λ) 2
where we chose ε = α for the last equality to hold. By multiplying each side by e λαt and rearranging the terms, we obtain d dt (R(X(x, t), λ) 2 e λαt ) ≤ λ α ω(X(x, t), λ) 2 e λαt . (A.5)

Let δ > 0. Integrating between -δ and 0 and using (25), we get for any x ∈ X + δ u and any λ > 1,

R(x, λ) 2 ≤ R(X(x, -δ), λ) 2 e -λαδ + λ α 0 -δ ω(X(x, s), λ) 2 e λαs ds ≤ R(X(x, -δ), λ) 2 e -λαδ + λ α ω2 λ 2m 0 -δ e λαs ds = R(X(x, -δ), λ) 2 e -λαδ + λω 2 (1 -e -λαδ ) α 2 λ 2m+1 δ → X(x, -δ) is bounded, so with the definition of T in (12), δ → T (X(x, -δ), λ) is bounded, and, by continuity of ϕ ℓ for each ℓ ∈ {0, . . . , m -1}, δ → R(X(x, -δ), λ) is also bounded. Letting δ go to +∞, it yields ∀λ > 1, ∀x ∈ X + δ u , R(x, λ) 2 ≤ ω2 α 2 λ 2m .
We finally get (28).

A.3 Proof of Lemma 1

Define T a,i (x, λ) = i-1 j=0 ϕj (x) λ j , so that T a,m = T a . The proof of Lemma 1 is obtained by recursion. We are going to prove the following property for i ≥ 1:

P(i): For all x ∈ R n and λ > 0, T a,i verifies L f T a,i (x, λ) = λ i-1 j=1 σ (j) (x) j! i-1 ℓ=1 ϕ ℓ (x) λ ℓ j ≤i-1 + L f ϕ i-1 (x) λ i-1 . (A.6) Clearly, L f T a,1 = L f ϕ 0
so the property is true for i = 1. Assume P(i) is true for a certain i ≥ 1. Let us show that P(i + 1) holds. We have

L f T a,i+1 (x, λ) = i ℓ=0 L f ϕ ℓ (x) λ ℓ = L f T a,i (x, λ) + L f ϕ i (x) λ i = λ i-1 j=1 σ (j) (x) j! i-1 ℓ=1 ϕ ℓ (x) λ ℓ j ≤i-1 + L f ϕ i-1 (x) λ i-1 + L f ϕ i (x) λ i . On another hand, we have λ i j=1 σ (j) (x) j! i-1 ℓ=1 ϕ ℓ (x) λ ℓ + ϕ i (x) λ i j ≤i + L f ϕ i (x) λ i = λ i-1 j=1 σ (j) (x) j! i-1 ℓ=1 ϕ ℓ (x) λ ℓ j ≤i + σ (1) (x) λ i ϕ i (x) + σ (i) (x) i! i-1 ℓ=1 1 λ ℓ ϕ ℓ (x) i ≤i + L f ϕ i (x) λ i = λ i-1 j=1 σ (j) (x) j! i-1 ℓ=1 ϕ ℓ (x) λ ℓ j ≤i-1 + i-1 j=1 σ (j) (x) j! i-1 ℓ=1 ϕ ℓ (x) λ ℓ j =i + σ (1) (x) ϕ i (x) λ i + 1 λ i σ (i) (x) i! (ϕ 1 (x)) i + L f ϕ i (x) λ i = L f T a,i+1 (x, λ) + λ i-1 j=1 σ (j) (x) j! i-1 ℓ=1 ϕ ℓ (x) λ ℓ j =i + σ (1) (x) ϕ i (x) λ i + 1 λ i σ (i) (x) i! (ϕ 1 (x)) i - L f ϕ i-1 (x) λ i-1 . Therefore, with ϕ i defined in (17) and noting that i-1 ℓ=1 ϕ ℓ (x) λ ℓ j =i = 1 λ i ℓ1+•••+ℓj =i 1≤ℓ1,...,ℓj ≤i-1 ϕ ℓ1 (x) • • • ϕ ℓj (x)
it yields the result.

A.4 Proof of Proposition 5

To simplify the readability of this proof, in the following the dependencies on λ and h(x) has been removed. Moreover, for all (x a , x b ) in (X + δ u ) 2 , d dt (R(X(x a , t)) -R(X(x b , t)))

= λ σ(T a (X(x a , t)) + R(X(x a , t))) -σ(T a (X(x a , t)))

-(σ(T a (X(x b , t)) + R(X(x b , t))) -σ(T a (X(x b , t))))

-λ ω(X(x a , t)) -ω(X(x b , t)) . On another hand, with Proposition 4, there exists a positive real number B σ,2 such that for all (x a , x b ) in (X + δ u ) 2 and for all λ > 1, and (θ, ρ) in [0, 1] 2 ,

∂ 2 σ ∂z 2 (T a (x b ) + θR(x b ) + ρ[T a (x a ) -T a (x b ) + θ(R(x a ) -R(x b ))) ≤ B σ,2
Hence, since X + δ u is invariant along the flow and with (7a), the last three equations lead to

d dt ∆R(X(x a , t), X(x b , t)) 2 ≤ -2λ(α -B σ,2 |R(X(x b , t))|)∆R(X(x a , t), X(x b , t)) 2 + 2λB σ,2 |R(X(x b , t))|
|∆T a (X(x a , t), X(x b , t))||∆R(X(x a , t), X(x b , t))| + 2λ|∆ω(X(x a , t), X(x b , t))||∆R(X(x a , t), X(x b , t))|, Using Young's inequality twice, we obtain for some ε and ε ′ strictly positive:

d dt ∆R(X(x a , t), X(x b , t)) 2 ≤ -2λ α - ε ′ 2 -B σ,2 |R(X(x b , t))| 1 + ε 2 × ∆R(X(x a , t), X(x b , t)) 2 + λ ε B σ,2 |R(X(x b , t))|∆T a (X(x a , t), X(x b , t)) 2 + λ ε ′ ∆ω(X(x a , t), X(x b , t)) 2 Then, using Proposition 4, d dt ∆R(X(x a , t), X(x b , t)) 2 ≤ -2λ α - ε ′ 2 -B σ,2 ω αλ m 1 + ε 2 × ∆R(X(x a , t), X(x b , t)) 2 + ωλB σ,2 αελ m ∆T a (X(x a , t), X(x b , t)) 2 + λ ε ′ ∆ω(X(x a , t), X(x b , t)) 2 . Choosing ε = α 2 2Bσ,2 ω λ m and ε ′ = α 2 gives d dt ∆R(X(x a , t), X(x b , t)) 2 ≤ - λ 2 α -4B σ,2 ω αλ m ∆R(X(x a , t), X(x b , t)) 2 + ω2 B σ,2 λ α 3 λ 2m ∆T a (X(x a , t), X(x b , t)) 2 + 2λ α ∆ω(X(x a , t), X(x b , t)) 2 .
If we multiply by exp λ 2 α -4B σ,2 ω αλ m t ≡ exp(Θt) on both sides, we obtain (by ommitting the dependency on λ of Θ for now)

d dt ∆R(X(x a , t), X(x b , t)) 2 exp(Θt) ≤ ω2 B σ,2 λ α 3 λ 2m ∆T a (X(x a , t), X(x b , t)) 2 + 2λ α ∆ω(X(x a , t), X(x b , t)) 2 exp(Θt)
We then integrate between -δ < 0 and 0, for (

x a , x b ) ∈ (X + δ u ) 2 and λ > 1, ∆R(x a , x b ) 2 ≤ ∆R(X(x a , -δ), X(x b , -δ)) 2 exp(-Θδ) + 0 -δ ω2 B σ,2 λ α 3 λ 2m ∆T a (X(x a , s), X(x b , s)) 2 exp(Θs)ds + 0 -δ 2λ α ∆ω(X(x a , s), X(x b , s)) 2 exp(Θs)ds
Since by Proposition 3, s → ∆ω(X(x a , s), X(x b , s)) 2 is bounded and s → ∆T (X(x a , s), X(x b , s)) 2 is also bounded, it yields that for all λ > 4B σ,2 ω α 2

1/m the former integrals are well defined when δ goes to +∞. Moreover, since by Proposition 4 and the invariance of (

X + δ u ), δ → ∆R(X(x a , -δ), X(x b , -δ)) 2 is bounded it yields ∆R(x a , x b ) 2 ≤ 0 -∞ ω2 B σ,2 λ α 3 λ 2m ∆T a (X(x a , s), X(x b , s)) 2 exp(Θs)ds + 0 -∞ 2λ α ∆ω(X(x a , s), X(x b , s)) 2 exp(Θs)ds.
Also, the functions ϕ ℓ being C 1 , there exists k Ta such that for all λ > 1 and all (x a , x b ) in (

X + δ u ) 2 , |∆T a (x a , x b )| ≤ k Ta |x a -x b |. Using Proposition 3 with λ > max{ 4B σ,2 ω α 2 1/m , 1} we can write ∆R(x a , x b ) 2 ≤ ω2 B σ,2 λ α 3 λ 2m k 2 T 0 -∞ |X(x a , s) -X(x b , s)| 2 exp(Θs)ds + 2λ α k 2 ω λ 2m 0 -∞ |X(x a , s) -X(x b , s)| 2 exp(Θs)ds. (A.7) Since f is Lipschitz on (X + δ u ), there exists k f > 0 such that for all s < 0 and (x a , x b ) in (X + δ u ) 2 , |X(x a , s) -X(x b , s)| ≤ |x a -x b | + 0 s k f |X(x a , s) -X(x b , s)|ds. Grönwall's lemma gives for t < 0, |X(x a , t) -X(x b , t)| ≤ |x a -x b | exp(-k f t) and reinjecting this in (A.7), ∆R(x a , x b ) 2 ≤ ω2 B σ,2 λk 2 T α 3 λ 2m |x a -x b | 2 0 -∞ exp ([Θ -2k f ] s) ds + 2λ α k 2 ω λ 2m |x a -x b | 2 0 -∞ exp ([Θ -2k f ] s) ds. Let λ * > 1 be large enough such that λ 2 α -4B σ,2 ω αλ m -2k f > 0.
Then, for all λ > λ * , the integrals converge and we get

0 -∞ exp ([Θ -2k f ] s) ds = 1 Θ -2k f .
By integrating the last identity into the previous one and taking the square root, we have for any (x a , x b ) ∈ X 2 and any λ > λ * ,

|∆R(x a , x b )| ≤ 1 λ m ω2 B σ,2 λk 2 T α 2 + 2λk 2 ω α 3 (Θ -2k f ) 1/2 |x a -x b |.
Which can be rewritten as ( 29) for some positive real number k R .

A.5 Proof of Proposition 2

We now establish Lipschitz injectivity of ϕ. For ϕ 0 , under Assumption 3, we have Then, an immediate recursion from ( 16)-( 17) shows that there exist C 1 functions P 1 , . . . , P m-1 such that for all x ∈ X + δ u , λ > 0, and ℓ ∈ {1, . . . , m -1}, ϕ ℓ is written as ϕ ℓ (x) = κ(h(x)) ℓ dψ dy (h(x))L ℓ f h(x) + P ℓ (H ℓ (x)). where H ℓ = (h, . . . , L ℓ-1 f h).

To simplify the expressions, when there is no ambiguity, we write g(h) = g • h for the composition of functions.

For ℓ ≥ 1 and (x a , x b ) ∈ X 2 , the triangular inequality gives We want to lower bound the first term, and upper bound the other two. The output h(x) is bounded for x in the compact X , and as h and f are regular enough and x lies in a compact, L ℓ f h(x) is bounded too. Moreover, κ is C m and dψ dy is C m , therefore they are bounded and Lipschitz on the compact set h(X ). It follows that κ l is Lipschitz on the compact set h(X ). Moreover, using (A.8) and the definition of κ with Assumption 3, Therefore we get for all (x a , x b ) ∈ X 2

  , and we say that g is (k-)Lipschitz injective on S ⊂ R n , if there exists k > 0 such that for all (x a , x b ) ∈ S × S, |∆g(x a , x b )| ≥ k|x a -x b |. Given a polynomial in powers of 1 λ in the form p(λ) = ℓ j=0 pj λ j and given an integer m, the notation [p(λ)]| m ≤i (resp. [p(λ)]| m =i ) represents the polynomial in powers of 1

  Fig. 1. Estimation error in x and z coordinates from two KKL observers with linear dynamics (2) with slow and fast pairs (A slow , B slow ) and (A fast , B fast ) defined from (31), and by a KKL observer with nonlinear dynamics(3) defined from (32). Scenario 1: initial error but no noise; Scenario 2 : no initial error but noise.

  Note that given (z a , r a , z b , r b ) in R 4 , the function σ being C 2 , we haveσ(z a + r a ) -σ(z a ) = 1 0 ∂σ ∂z (z a + θr a )dθ r a and (σ(z a + r a ) -σ(z a )) -(σ(z b + r b ) -σ(z b )) + θr a )dθ [r a -r b ] + θr a )dθ [r a -r b ] ∂z 2 (z b + θr b + ρ[z a -z b + θ(r a -r b )])× (z a -z b + θ(r a -r b ))dρ dθ r b .

  that ∀(x a , x b ) ∈ X 2 , |∆ϕ 0 (x a , x b )| ≥ µ 0 |∆h(x a , x b )|.

  |∆ϕ ℓ (x a , x b )| ≥ ∆(κ ℓ (h) dψ dy (h)L ℓ f h)(x a , x b )) -|∆P ℓ (H ℓ )(x a , x b )|.The first term can be decomposed this way∆(κ ℓ (h) dψ dy (h)L ℓ f h)(x a , x b ) ≥ κ ℓ dψ dy (h(x a ))∆L ℓ f h(x a , x b ) -L ℓ f (h(x b ))κ(h(x a )) ℓ ∆ dψ dy (h(x a ), h(x b )) -L ℓ f (h(x b ))dψ dy (h(x b ))∆κ ℓ (h(x a ), h(x b )) .

  x a , x b ))| -ρ ℓ |∆h(x a , x b )| for some ρℓ ≥ 0.Moreover, P ℓ is C 1 on the compact set H ℓ (X ), so we have∀(x a , x b ) ∈ X 2 , |∆P ℓ (H ℓ )(x a , x b ))| ≤ k P ℓ |∆H ℓ (x a , x b )| for some k P ℓ > 0. So we finally get ∀(x a , x b ) ∈ X 2 , |∆ϕ ℓ (x a , x b )| ≥ µ ℓ |∆L ℓ f (h(x a ), h(x b ))| x a ), h(x b ))|for some µ ℓ > 0 and ρ ℓ ≥ 0.To conclude the proof, we show the Lipschitz injectivity of Dϕ with D = diag(d 0 , d 1 , . . . , d m-1 ) for some d i > 0 to be picked. We compute|∆(Dϕ)(x a , x b )| = m-1 ℓ=0 d ℓ |∆ϕ ℓ (x a , x b )| ≥ m-1 ℓ=0 d ℓ µ ℓ |∆L ℓ f h(x a , x b )|i )|∆L ℓ f h(x a , x b ))|.We want to fixd ℓ µ ℓ -m-1 i=ℓ+1 d i ρ i = 1 for all ℓ = 0, . . . , m-Since all µ i are non-zero, the matrix is invertible. Hence there exists a suitable choice of d i . Assumption (2) of differentiable observability gives for any (x a , x b ) ∈ X 2 |∆(Dϕ)(x a , x b )| ≥ |∆H m (x a , x b )| ≥ k H |x a -x b |.Hence, we readily get that for all (x a , x b ) ∈ X 2 ,|∆ϕ(x a , x b )| ≥ k H max d i |x a -x b |.

  are given in Table1. The convergence time is computed in Scenario 1 as the first time after which the error remains below tolerance thresholds, determined by the precision of the approximation of T and T inv . The robustness to measurement noise is quantified in Scenario 2 by dividing the 2-norm of the steady state error by the amplitude of the noise. We refer the reader to[START_REF] Brivadis | KKL observers with nonlinear dynamics[END_REF] to experiment the simulations.

			Fast linear Slow linear Nonlinear
	Convergence time	Min Max Mean	0.78 0.87 0.83	6.29 7.72 6.79	1.52 3.15 2.27
	Gain w.r.t. noise	Min Max Mean	7.23 8.07 7.57	0.94 1.53 1.15	1.56 2.37 1.95
	Table 1. Comparison of KKL observers with
	slow/fast linear dynamics and KKL observer
	with nonlinear dynamics in terms of conver-
	gence time and gain with respect to noise of
			|x -x|.
	100 random x(0)