Reduction methods for quasilinear differential-algebraic equations

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–8) Reduction methods for quasilinear differential-algebraic equations Ricardo Riaza1 1Depto. de Matem´atica Aplicada TTI, ETSI Telecomunicaci´on, Universidad Polit´ecnica de Madrid. E-mail: [email protected]pm.es. Palabras clave: Differential-algebraic equation, index, reduction, singularity Resumen Geometric reduction methods for differential-algebraic equations (DAEs) aim at an iterative reduction of the problem to an explicit ODE on a lower-dimensional submanifold of the so-called semistate space. This approach usually relies on certain algebraic (typically constant-rank) conditions holding at every reduction step. When these conditions are met the DAE is called regular. We discuss in this contribution several recent results concerning the use of reduction techniques in the analysis of quasilinear DAEs, not only for regular systems but also for singular ones, in which the above-mentioned conditions fail. 1. Outline Quasilinear autonomous differential-algebraic equations (DAEs) are implicit ODEs of the form A(x)x0=f(x),(1) where A∈C∞(W0,Rn×n) is a rank-deficient matrix-valued mapping, f∈C∞(W0,Rn), and W0is an open set in Rn. We summarize in Section 2 below the geometric reduction approach of Rabier and Rheinboldt for the analysis of quasilinear systems of the form (1). In Section 3 we recast this framework in a local manner, aimed at the analysis of singular problems carried out in Section 4. In order to provide the reader with a self-contained discussion, we address here the main ideas and refer him/her to [4] and to the forthcoming title [8] for details, specially concerning several results which are stated without proof in this communication. 1 Ricardo Riaza 2. The reduction framework of Rabier and Rheinboldt Stemming from the seminal paper [7] by Rheinboldt, reduction methods are essentially based on the work of Rabier and Rheinboldt [1, 4], and Reich [5, 6]. We summarize below the approach of Rabier and Rheinboldt for the quasilinear DAE (1) following [4]. Any C1solution to (1) must obviously lie on the set W1={x∈W0/ f(x)∈imA(x)}.(2) Define F:TW0≃W0×Rn→Rnas F(x, p) = A(x)p−f(x),and consider the set M0=F−1(0) = {(x, p)∈TW0/ A(x)p−f(x) = 0}.Denoting by π:Rn×Rn→Rnthe projection onto the first factor we have W1=π(M0). If x(t) solves (1), it follows that the pair (x(t), x0(t)) must belong to M0. Via the subimmersion theorem, the following global conditions make W1an r-dimensional submanifold of W0(cf. [4]): (G1) A(x) has constant rank r1≤nfor all x∈W1. (G2) F(x, p) is a submersion on its zero set M0. Now, provided that x(t) is a solution to (1), thereby lying entirely on W1, the pair (x(t), x0(t)) must also belong to the tangent bundle T W1. This means that x0(t) must be tangent not only to W0but also to W1itself. Hence, the pair (x(t), x0(t)) needs to be in the intersection M1=TW1∩M0and, in particular, x(t) must lie on W2=π(M1). Letting F1=F|T W1, we can describe M1=TW1∩M0as F−1 1(0), whereas the set W2 reads W2={x∈W1/ f(x)∈imA(x)|TxW1}.If the analogs of assumptions G1 and G2 hold when applied to F1and A(x)|TxW1,W2will be an r2-dimensional manifold with r2= rkA(x)|TxW1, and the same reasoning can be performed one step further. This way, if the the above-mentioned working assumptions hold at every step, the procedure yields two sequences of smooth manifolds which will eventually stabilize, namely, M0⊃M1⊃...⊃Mν=Mν+1 and W0⊃W1⊃...⊃...⊃Wν⊇Wν+1 =Wν+2.(3) The dimensions of these manifolds are given by the ranks n > r1> . . . > rν=rν+1 =rν+2. The smallest integer νsuch that either Mν=∅or Mν6=∅and rν=rν+1 is the geometric index of (1). In index-νproblems with Mν6=∅, the manifold Wν+1 turns out to be open in Wν. Always under the above-stated working assumptions, this manifold comprises all the smooth solutions of the DAE, which can be described in terms of a vector field uniquely defined on Wν+1. Details can be found in Theorems 23.2 and 24.1 of [4]. Via local parametrizations, solutions of index-νDAEs can be also locally described in terms of reduced equations, in a way similar to the one detailed in Section 3 below. 3. A local approach The above-summarized framework provides a nice approach for the analysis of quasilinear DAEs when the global assumptions G1 and G2 above hold at every reduction step. But its obvious drawback is the exclusion of DAEs for which these assumptions are 2 Reduction methods for quasilinear DAEs not met. In order to accommodate these singular problems, let us recast the construction above in a local manner. A point x∗∈W0is called regular with geometric index zero for the DAE (1) if A(x∗) is a non-singular matrix. The behavior on the (open) set of index zero points trivially amounts to that of the explicit ODE x0=A−1(x)f(x). Definition 1 A point x∗∈W0is said to be 0-regular for the DAE (1) if x∗∈W1and the following two conditions hold: (R1) A(x)has constant rank r1≤nin some neighborhood of x∗. (R2) Fis a submersion at (x∗, p∗), for some p∗satisfying A(x∗)p∗=f(x∗). The set of 0-regular points will be denoted by Wreg 1. By construction Wreg 1is open in W1and therefore the sets Wreg 1and W1coincide locally around any 0-regular point. More precisely, for all x∗∈Wreg 1there exist a neighborhood Uof x∗such that Wreg 1∩U=W1∩U. Henceforth we will abbreviate this kind of relation as Wreg 1 loc =W1. The constant rank condition in R1 above is a local version of G1 in page 2, with the slightly stronger requirement that the rank is constant within a whole neighborhood (say ˜ U0) of x∗in W0. If this locally constant rank r1verifies r1< n, then there will exist another open neighborhood ˆ U0⊆˜ U0∩Uof x∗and a smooth matrix-valued map H∈C∞(ˆ U0,R(n−r1)×n) such that keH(x) = imA(x)∀x∈ˆ U0,see e.g. Lemma 22.1 in [4]. Note that H(x)A(x) = 0 and rkH(x) = n−r1on ˆ U0. Now v∈imA(x)⇔H(x)v= 0 for x∈ˆ U0, allowing for the local implicit description of Wreg 1 loc =W1as Wreg 1∩ˆ U0=W1∩ˆ U0= {x∈ˆ U0/ H(x)f(x) = 0}. The submersion condition R2 requires rkF0(x∗, p∗) = n. This is a key hypothesis because it characterizes the situations in which the product H(x)f(x) is a submersion, as stated in Lemma 1 below. Lemma 1 Let x∗∈W1. Assume that A(x)has constant rank r1, with 0< r1< n, in some open neighborhood ˜ U0of x∗, and let the matrix-valued map H∈C∞(ˆ U0,R(n−r1)×n) verify keH(x) = imA(x)∀x∈ˆ U0⊆˜ U0. Then H(x)f(x)is a submersion at x∗if and only if F(x, p)is a submersion at (x∗, p∗)for some (hence any) p∗satisfying A(x∗)p∗=f(x∗). In this setting, the local description of Wreg 1 loc =W1as the zero set of H(x)f(x) locally yields a smooth structure on this set and paves the way for the following local reduction. Theorem 1 Let x∗∈W0be a 0-regular point for (1), and denote by r1the locally constant rank of A(x)around x∗. If r1>0, then there exists an open neighborhood U0⊆W0⊆Rn of x∗such that (i) Wreg 1∩U0=W1∩U0admits a smooth r1-dimensional parametrization x=ϕ1(ξ) with surjective ϕ1: Ω1→Wreg 1∩U0; (ii) there exists a C∞matrix-valued mapping P1:U0→Rr1×nverifying that P1(x) im A(x) is an isomorphism imA(x)↔Rr1for all x∈U0. 3 Ricardo Riaza For any such ϕ1, P1,x(t)is a solution of (1) within U0if and only if x(t)∈Wreg 1 loc =W1 for all tand ξ(t) = ϕ−1 1(x(t)) is a solution of A1(ξ)ξ0=f1(ξ), ξ ∈Ω1⊆Rr1(4) with A1(ξ) = P1(ϕ1(ξ))A(ϕ1(ξ))ϕ0 1(ξ),f1(ξ) = P1(ϕ1(ξ))f(ϕ1(ξ)). In the index one setting described by item (a) of Definition 2 below, the reduction (4) can be rewritten as a explicit ODE in some neighborhood of ξ∗, possibly smaller than Ω1. Definition 2 A point x∗∈W0is called regular with geometric index one for (1) (a) either if it is 0-regular with n > r1>0and A1(ξ∗)is non-singular for some (hence any) reduction pair (P1, ϕ1)satisfying x∗=ϕ1(ξ∗); (b) or if it is 0-regular with r1= 0. The set of index one points will be denoted by Wind1. For points which are not index one, the same procedure can be applied to (A1(ξ), f1(ξ)) in (4). Introduce r2= rkA1,W2={x∈Wreg 1/ f(x)∈imA(x)|TxWreg 1} ⊆ Wreg 1⊆W1or, in local coordinates, V2={ξ∈Ω1/ f1(ξ)∈imA1(ξ)} ⊆ Ω1⊆Rr1which yields a local description of W2as ϕ1(V2). A 0-regular point ξ∗of (A1, f1) will define x∗=ϕ1(ξ∗) as a 1-regular point of (A, f). Recursively, we are naturally led to the following notion. Definition 3 A point x∗∈W0is called regular with geometric index ν,ν≥1, for (1) (a) either if it is (ν−1)-regular with n > r1> r2> . . . > rν>0, and the matrix Aν(u∗) is non-singular, for some (hence any) reduction sequence (P1, ϕ1),...,(Pν, ϕν)satisfying x∗=ϕ1◦ · · · ◦ ϕν(u∗); (b) or if it is (ν−1)-regular with n > r1> . . . > rν= 0. The set of index-νpoints will be denoted by Windν. Solutions of the original DAE (1) near a given (ν−1)-regular point with rν>0 are mapped bijectively into those of the reduced equation Aν(u)u0=fν(u), as stated in Theorem 2 below which naturally extends Theorem 1. We denote as Uthe neighborhood of x∗given by ϕ1◦...◦ϕν−1(Uν−1), where Uν−1is such that the last-step parametrization ϕν is onto Vν∩Uν−1. In particular, under an index-νassumption an explicit ODE reduction is possible and thereby local unique solvability properties follow from the corresponding theory for explicit ODEs. Theorem 2 Assume that x∗∈W0is a (ν−1)-regular point for (1) with rν>0,ν≥1, and let Aν(u)u0=fν(u), u ∈Ων⊆Rrν(5) be a ν-th step reduction of (1), given by a sequence of reduction pairs (P1, ϕ1), . . . , (Pν, ϕν), on a neighborhood Ωνof u∗= (ϕ1◦ · · · ◦ ϕν)−1(x∗). Then x(t)is a solution of (1) within Uif and only if x(t)∈Wreg ν loc =Wνfor all tand u(t) = (ϕ1◦...◦ϕν)−1(x(t)) solves (5). Moreover, if x∗is index ν, then Aν(u)is non-singular on some neighborhood of u∗ within Ων, and in that neighborhood the reduction (5) can be rewritten in the explicit form u0=A−1 ν(u)fν(u).(6) 4 Reduction methods for quasilinear DAEs Equation (6) is a local state space description of the DAE behavior. Of course, different reduction pairs will yield different state-space descriptions, although all of them can be proved to be C∞-conjugate. 4. Singularities In the light of the result in Section 3, the manifold sequence (3) can be replaced by W0⊇Wreg 1⊇Wreg 2⊇...⊇Wreg n,(7) which, around an index-νpoint, will stabilize as W0⊃Wreg 1⊃...⊃Wreg ν loc =Wreg ν+1 loc =Windν. This point of view allows us to accommodate singular points within this framework, under the working assumptions S1 and S2 below. Points in Wk+1 −Wreg k+1 are called inner k-singular points, whereas those in Wk+1 −Wk+1 are called boundary k-singular points. Assumption S1 below is aimed to cover cases in which the constant rank assumption R1 in Definition 1 fails after the k-th reduction step, that is, on Ak(ζ), k= 0 standing for rank deficiencies in A(x). This can be the case for both inner and boundary k-singularities. Assumption S1 describes situation very often found in practice in which, despite the rank deficiency, imAk(ζ) admits a smooth extension or continuation Lk(ζ) on a neighborhood of the singularity, with Lk(ζ) = imAk(ζ) on some dense subset of that neighborhood. By an r-dimensional C∞-space L(x) on Uwe mean an x-dependent linear space which is spanned by rbasis mappings depending smoothly on x∈Uor, equivalently, such that Sx∈U{x} × L(x) has an r-dimensional vector bundle structure. Assumption S1. Let x∗be a k-singularity for (1), with k≥0, and consider the k-th local reduction Ak(ζ)ζ0=fk(ζ). Write x∗=ϕ1◦...◦ϕk(ζ∗). There exists an open neighborhood ˜ Uk⊆Ωk⊆Rrkof ζ∗and, for some ˜rk+1 ≤rk, an ˜rk+1-dimensional C∞-space Lk(ζ) defined on ˜ Uksuch that imAk(ζ) = Lk(ζ)on some dense subset of ˜ Uk. It may happen in particular that ˜rk+1 =rk: in this case Assumption S1 expresses that Akis non-singular on a dense subset of ˜ Uk, since Lk(ζ) = Rrkmeets the requirements. We may speak in this situation of a “last-step” singularity. This is essentially the context considered by Rabier and Rheinboldt in [2, 3, 4]. There is no need for further reduction of the DAE, and Theorem 4 will apply. This will be a particular instance of a singular index kproblem. If ˜rk+1 < rk, from the structure of Lk(ζ) there must exist an open neighborhood ˆ Uk⊆˜ Ukof ζ∗∈Ωk⊆Rrkand a smooth, maximal rank matrix-valued map Hk(ζ)∈ R(rk−˜rk+1)×rkwith keHk(ζ) = Lk(ζ) on ˆ Uk, so that v∈Lk(ζ) if and only if Hk(ζ)v= 0 for ζ∈ˆ Uk. Note that, if x∗is an inner k-singular point, the set Vk+1 ={ζ∈Ωk/ fk(ζ)∈ imAk(ζ)}cannot be guaranteed to admit a local parametrization near ζ∗, in the terms holding at k-regular points. Near (inner or boundary) k-singularities we will work instead with the set ˜ Vk+1 ={ζ∈ˆ Uk/ fk(ζ)∈Lk(ζ)}={ζ∈ˆ Uk/ Hk(ζ)fk(ζ) = 0} ⊆ Ωk(8) which not even locally can be identified with Vk+1. But in the setting defined by Assumption S1, we have Vk+1 ∩ˆ Uk⊆Vk+1 ∩ˆ Uk⊆˜ Vk+1.(9) 5 Ricardo Riaza Indeed, by the density hypothesis in Assumption S1 we have imAk(ζ)⊆Lk(ζ) = keHk(ζ) for all ζ∈ˆ Uk, showing that Vk+1 ∩ˆ Uk⊆˜ Vk+1. The relations (9) then follow from the fact that ˜ Vk+1 is closed in ˆ Uk. Set ˜ Wk+1 =ϕ1◦...◦ϕk(˜ Vk+1), the obvious analog of (9) holding for Wk+1, Wk+1 and ˜ Wk+1. The relation depicted in (9) suggests that ˜ Vk+1 may also accommodate a reduction around boundary k-singularities. Indeed, under Assumption S2 below, ˜ Vk+1 will admit a local ˜rk+1-dimensional parametrization. Assumption S2. Let x∗=ϕ1◦...◦ϕk(ζ∗)be a k-singularity for (1), with k≥0. If Assumption S1 holds with ˜rk+1 < rk, let ˆ Uk⊆˜ Uk⊆Ωkbe an open neighborhood of ζ∗ such that Hk∈C∞(ˆ Uk,R(rk−˜rk+1)×rk)verifies keHk(ζ) = Lk(ζ)∀ζ∈ˆ Uk.Then Hk(ζ)fk(ζ) is a submersion at ζ∗. As stated above, Assumption S2 applies to both inner and boundary k-singular points. Inner ones verify ζ∗∈Vk+1, and hence they admit solutions p∗to Ak(ζ∗)p∗−fk(ζ∗) = 0: Assumption S2 then holds if the submersion condition in item R2 of Definition 1 is met in the current context. More precisely, if x∗is an inner k-singularity for (1), Assumption S1 is met with ˜rk+1 < rk, and Fk(ζ, p) = Ak(ζ)p−fk(ζ) is a submersion at (ζ∗, p∗) for some p∗satisfying Ak(ζ∗)p∗=fk(ζ∗),then Assumption S2 can be checked to hold. Assumptions S1 and S2 make it possible to perform a reduction of singular quasilinear DAEs, as detailed below. Theorem 3 Let x∗=ϕ1◦...◦ϕk(ζ∗)be a k-singularity for (1) satisfying Assumptions S1 and S2 with 0<˜rk+1 < rk. Then there exists an open neighborhood Uk⊆ˆ Uk⊆Ωk⊆Rrk of ζ∗such that (i) ˜ Vk+1 ∩Ukadmits an ˜rk+1-dimensional parametrization ζ= ˜ϕk+1(η)with surjective ˜ϕk+1 : Ωk+1 →˜ Vk+1 ∩Uk; (ii) there exists a C∞matrix-valued map ˜ Pk+1 :Uk→R˜rk+1×rkverifying that ˜ Pk+1(ζ) Lk(ζ) yields an isomorphism Lk(ζ)→R˜rk+1 for all ζ∈Uk. For any such ˜ϕk+1,˜ Pk+1,ζ(t)is a solution of the k-th reduction Ak(ζ)ζ0=fk(ζ), ζ ∈Ωk⊆Rrk(10) within Ukif and only if ζ(t)∈˜ Vk+1 for all tand η(t) = ˜ϕ−1 k+1(ζ(t)) is a solution of ˜ Ak+1(η)η0=˜ fk+1(η), η ∈Ωk+1 ⊆R˜rk+1 (11) with ˜ Ak+1(η) = ˜ Pk+1( ˜ϕk+1(η))Ak( ˜ϕk+1(η)) ˜ϕ0 k+1(η),˜ fk+1(η) = ˜ Pk+1( ˜ϕk+1(η))fk( ˜ϕk+1(η)). Proof: The existence of the smooth parametrization ˜ϕk+1 follows from (8) together with Assumption S2, whereas that of ˜ Pk+1 is due to the smooth structure of Lk(ζ) in Assumption S1. Assume that ζ(t) solves (10). Then fk(ζ(t)) ∈imAk(ζ(t)), that is, ζ(t)∈Vk+1 and thus ζ(t)∈˜ Vk+1 for all tby (9). This means that η(t) is well-defined by ζ(t) = ˜ϕk+1(η(t)): premultiplying (10) by ˜ Pk+1( ˜ϕk+1(η(t))) and inserting ζ(t) = ˜ϕk+1(η(t)), ζ0(t) = ˜ϕ0 k+1(η(t))η0(t) in the resulting equation, we obtain (11). 6 Reduction methods for quasilinear DAEs Conversely, the assumption that (11) holds can be written as ˜ Pk+1( ˜ϕk+1(η))Ak( ˜ϕk+1(η)) ˜ϕ0 k+1(η)η0=˜ Pk+1( ˜ϕk+1(η))fk( ˜ϕk+1(η)) or, in terms of ζ= ˜ϕk+1(η), ˜ Pk+1(ζ)Ak(ζ)ζ0=˜ Pk+1(ζ)fk(ζ).(12) If we show that Ak(ζ)ζ0∈Lk(ζ), fk(ζ)∈Lk(ζ), the identity (12) would yield (10) due to the isomorphism ˜ Pk+1(ζ) Lk(ζ):Lk(ζ)→R˜rk+1 . Indeed, the relation Ak(ζ)ζ0∈Lk(ζ) holds trivially due to imAk(ζ)⊆Lk(ζ), whereas ζ= ˜ϕk+1(η)∈˜ Vk+1 means fk(ζ)∈Lk(ζ) by (8).  This result generalizes the one-step local reduction of Theorem 1 to singular points as long as they meet Assumptions S1 and S2. In the setting defined by Theorem 3, a onestep singular reduction is again suitable for assessment for the reduction (11). Defining Vs k+2 ={η∈Ωk+1 /˜ fk+1(η)∈im ˜ Ak+1(η)},Ws k+2 =ϕ1◦...◦ϕk◦˜ϕk+1(Vs k+2)⊆˜ Wk+1, we may naturally extend the singular reduction process beyond the (k+ 1)-th step. This way, instead of the sequence of manifolds W1⊃W2⊃W3... constructed in the regular setting, we build up a sequence of the form W0⊃Wreg 1⊃...⊃Wreg k⊃˜ Wk+1 ⊃...⊃˜ Wν loc =˜ Wν+1,(13) the local stabilization after the ν-th step holding in the setting of Theorem 4 below. The importance of this construction stems from the fact Wk+1 and later on Ws k+2 and subsequent sets may fail to have a C∞structure near an inner k-singularity, whereas the extensions ˜ Wk+1,˜ Wk+2, etc., display a local C∞structure, allowing for a local reduction of the DAE. These manifolds comprise in addition the closures Wk+1,Ws k+2, etc., and therefore may also accommodate boundary singularities. The repeated application of the one-step singular reduction in Theorem 3 yields the following analog of Theorem 2; the meaning of Uparallelizes exactly the one explained there. In the particular case k=ν, the symbols ˜rν,˜ Aν,˜ fνand ˜ Wνbelow must be replaced by rν,Aν, fνand Wν. Since no singular reduction is required for these last-step singular points, in this situation Theorem 4 virtually amounts to Theorem 2, consistently with the fact that the setting of Rabier and Rheinboldt discussed in [2, 3, 4] accommodates last-step singularities. Theorem 4 Let x∗∈W0be a k-singularity for (1), k≥0. Suppose that Assumptions S1 and S2 hold in steps k+ 1,k+ 2, . . . , ν of the singular reduction process described above with n=r0> r1> . . . > rk>˜rk+1 >˜rk+2 > . . . > ˜rν>0,(14) and that Assumption S1 is met in step ν+ 1 with ˜rν= ˜rν+1. Let ˜ Aν(u)u0=˜ fν(u), u ∈Ων⊆R˜rν(15) be a ν-th step reduction of (1) given by a sequence of reduction pairs (P1, ϕ1), . . . , (Pk, ϕk), (˜ Pk+1,˜ϕk+1), . . . , (˜ Pν,˜ϕν)on a neighborhood Ωνof u∗= (ϕ1◦· · ·◦ϕk◦˜ϕk+1◦...◦˜ϕν)−1(x∗). Then x(t)is a solution of (1) within Uif and only if x(t)∈˜ Wνfor all tand u(t) = (ϕ1◦...◦ϕk◦˜ϕk+1 ◦...◦˜ϕν)−1(x(t)) solves (15). 7 Ricardo Riaza The requirement that Assumption S1 holds in the last step with ˜rν= ˜rν+1 >0 amounts to saying that ˜ Aν(or Akif ν=k) is non-singular on some dense subset of ˜ Uν⊆Ων. This means that points in this dense subset are regular with index ν. We speak of a k-singularity x∗as a singular index νpoint when the hypotheses of this Theorem hold. In these situations the DAE (1) can be locally thought of as a singular index νproblem. The difference between Theorem 4 and the regular index νstatement within Theorem 2 is that now ˜ Aν(u∗) will typically be a singular matrix. This may be due to a rankdeficiency arising at any reduction step, not necessarily at the last one. Theorem 4 hence drives the local analysis of a broad family of singular quasilinear DAEs not to the context of explicit ODEs but to the quasilinear ODE setting. This way, not only impasse points but the whole analysis of singular phenomena in [9] can by systematically tackled in the DAE context. Finally, it is worth emphasizing that all the notions introduced above are invariant with respect to local equivalence. Two quasilinear DAEs A(x)x0=f(x), B(y)y0=g(y) defined on Wa 0,Wb 0open in Rn, are said to be C∞-equivalent locally around x∗,y∗if there exist open neighborhoods Ub⊆Wb 0of y∗,Ua⊆Wa 0of x∗, a C∞-diffeomorphism φ:Ub→Ua with φ(y∗) = x∗, and a C∞non-singular matrix-valued mapping E:Ub→Rn×n, such that B(y) = E(y)A(φ(y))φ0(y), g(y) = E(y)f(φ(y)) for all y∈Ub. The relation g(y) = E(y)f(φ(y)) is a contact equivalence between fand g. Note that, for the equivalence of the quasilinear systems, the pair (E, φ) is required to link additionally the matrix mappings A and B. This equivalence relation amounts to a C∞-conjugacy for index-0 cases, that is, for explicit ODEs, thereby explaining the fact that any two local reductions of a quasilinear DAE are C∞-conjugate. Agradecimientos Research supported by Projects MTM2004-5316 and MTM2005-3894 of Ministerio de Educaci´on y Ciencia, Spain. Referencias [1] P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, J. Differential Equations 109 (1994) 110-146. [2] P. J. Rabier and W. C. Rheinboldt, On impasse points of quasi-linear differential-algebraic equations, J. Math. Anal. Appl. 181 (1994) 429-454. [3] P. J. Rabier and W. C. Rheinboldt, On the computation of impasse points of quasi-linear differentialalgebraic equations, Math. Comp. 62 (1994) 133-154. [4] P. J. Rabier and W. C. 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