Geometric analysis of nonlinear differential-algebraic equations via nonlinear control theory

For nonlinear differential-algebraic equations (DAEs), we define two kinds of equivalences, namely, the external and internal equivalence. Roughly speaking, the word"external"means that we consider a DAE (locally) everywhere and"internal"means that we consider the DAE on its (locally) maximal invariant submanifold (i.e., where its solutions exist) only. First, we revise the geometric reduction method in DAEs solution theory and formulate an implementable algorithm to realize that method. Then a procedure named explicitation with driving variables is proposed to connect nonlinear DAEs with nonlinear control systems and we show that the driving variables of an explicitation system can be reduced under some involutivity conditions. Finally, due to the explicitation, we will use some notions from nonlinear control theory to derive two nonlinear generalizations of the Weierstrass form.


Introduction
Consider a nonlinear differential-algebraic equation (DAE) of the form where x ∈ X is a vector of the generalized states and X is an open subset of R n (or an n-dimensional The maps E : T X → R l and F : X → R l (see the above diagram, where π : T X → X is the canonical projection) are smooth and the word "smooth" will mean throughout this paper C ∞ -smooth . We will denote a DAE of the form (1) by Ξ l,n = (E, F ) or, simply, Ξ. Equation (1) is affine with respect to the velocityẋ, so sometimes it is called a quasi-linear DAE (see e.g., [1,2]) and can be considered as an affine Pfaffian system since the rows E i of E are actually differential 1-forms on X (for linear Pfaffian systems, see e.g. [3]), so E is a R l -valued differential 1-form on X. A semi-explicit DAE is of the form Ξ SE : where x 1 ∈ X 1 is a vector of state variables and x 2 ∈ X 2 is a vector of algebraic or free variables (since there are no differential equations for x 2 ) with X 1 and X 2 being open subsets of R q and R n−q , respectively (or q-and (n − q)-dimensional manifolds, respectively), the maps F 1 : X 1 × X 2 → T X 1 and F 2 : X 1 × X 2 → R l−q are smooth. A linear DAE of the form will be denoted by ∆ l,n = (E, H) or, simply, ∆, where E ∈ R l×n and H ∈ R l×n . Both the semiexplicit DAE Ξ SE and the linear DAE ∆ can be seen as special cases of DAE Ξ. The motivation of studying DAEs is their frequent presence in modelling of practical systems as electrical circuits [2,4], chemical processes [5,6], mechanical systems [7][8][9], etc.
There are three main results of this paper. The first result concerns analyzing a DAE (locally) everywhere (i.e., externally) or considering the restriction of the DAE to a submanifold (i.e., internally), which corresponds to the external equivalence (see Definition 3.1) and the internal equivalence (see Definition 3.9), respectively. The difference between the two equivalences will be illustrated by their relations with the solutions. In order to analyze the existence of solutions, we use a concept called locally maximal invariant submanifold (see Definition 2.2), which is a submanifold where the solutions of a DAE exist and can be constructed via a geometric reduction method shown in Section 2. Note that the geometric reduction method is not new in the theory of nonlinear DAEs, see e.g., [1,2,[10][11][12] and the recent papers [13][14][15]. In the present paper, we will show a practical implementation of this method via an algorithm summarized in Appendix. Note that considering only the restriction of a DAE means that we only care about where and how the solutions of that DAE evolve. However, when a nominal point is not on the maximal invariant submanifold (which is common for practical systems, since an initial point could be anywhere), there are no solutions passing through the point but we still want to steer the solutions to the submanifold and thus we must follow the rules indicated by the "external" form of the DAE, thus considering DAEs everywhere is also important, see our recent publication [16], where we use external equivalence to study jump solutions of nonlinear DAEs.
The second result of this paper is a nonlinear counterpart of the results of [17], in which we have shown that one can associate a class of linear control systems to any linear DAE (by the procedure of explicitation for linear DAEs). In the present paper, to any nonlinear DAE, by introducing extra variables (called driving variables), we can attach a class of nonlinear control systems. Moreover, we show that the driving variables in this explicitation procedure can be fully reduced under some involutivity conditions which explains when a DAE Ξ is ex-equivalent to a semi-explicit DAE Ξ SE .
It is well-known (see e.g., [18], [19]) that any linear DAE ∆ of the form (3) is ex-equivalent (via linear transformations) to the Kronecker canonical form KCF. In particular, if ∆ is regular, i.e., the matrices E and H are square (l = n) and |sE − H| ≡ 0, ∀s ∈ C, then ∆ is ex-equivalent (also via linear transformations) to the Weierstrass form WF [20] (see (18) below). The studies on normal forms and canonical forms of DAEs can be found in [18,[20][21][22][23] for the linear case and in [15,24,25] for the nonlinear case. The last result of this paper is to use such concepts as zero dynamics, relative degree and invariant distributions of the nonlinear control theory [26,27] to derive nonlinear generalizations of the WF. In the linear case, canonical forms as the KCF and the WF are closely related to a geometric concept named the Wong sequences [28] (see Remark 2.6 below). In [22], relations between the WF and the Wong sequences have been built and in [23], the importance of the Wong sequences for the geometric analysis of linear DAEs is reconfirmed. In the present paper, we propose generalizations of the Wong sequences for nonlinear DAEs and show their importance in analyzing structure properties.
This paper is organized as follows. In Section 2, we discuss the existence of solutions of DAEs by revising the geometric reduction method. In Section 3, we compare the notions of external equivalence and internal equivalence and discuss the uniqueness of DAEs solutions via the notion of internal regularity. In Section 4, we propose the explicitation (with driving variables) procedure to connect nonlinear DAEs to nonlinear control systems. In Section 5, we show when a nonlinear DAE is externally equivalent to a semi-explicit one and how this problem is related to the explicitation procedure. Two nonlinear generalizations of the Weierstrass form are given in Section 6. Finally, Section 7 and Section 8 contain proofs and the conclusions, respectively. In Appendix of Section 9, we show a recursive algorithm which implements the geometric reduction method.
The following notations will be used throughout the paper. We use R n×m to denote the set of real valued matrices with n rows and m columns, GL (n, R) to denote the group of nonsingular matrices of R n×n and I n to denote the n × n-identity matrix. For a linear map L, we denote by rank L, ker L and Im L, the rank, the kernel and the image of L, respectively. Denote by T x M the tangent space of a submanifold M of R n at x ∈ M and by C k the class of functions which are k-times differentiable.

dfm
. For two column vectors v 1 ∈ R m and v 2 ∈ R n , we write (v 1 , v 2 ) = [v T 1 , v T 2 ] T ∈ R m+n .

The geometric reduction method revisited
In this section, we revise the geometric reduction method in the DAEs solution theory, other formulations of this method can be consulted in Section 3.4 of [2], Chapter IV of [1] and [13] for DAEs and [14] for DAE control systems. We start from the definition of a solution for a DAE. interval I such that for all t ∈ I, the curve x(·) satisfies E (x(t))ẋ(t) = F (x(t)).
Throughout this paper, we will be interested only in solutions of Ξ that are at least C 1 . A given point x 0 is called consistent (or admissible) if there exists at least one solution x(·) of Ξ satisfying ) for a certain t 0 ∈ I, we will denote by S c the consistency set, i.e., the set of all consistent points. x(t) ∈ M for all t ∈ I. Given a point x p ∈ X, we will say that a submanifold M containing x p is then M is a locally invariant submanifold.
The proof is given in Section 7.1.
Remark 2.4. Note that the assumption that dim E(x)T x M = const. of Proposition 2.3 is not a necessary condition to conclude that M is an invariant submanifold, but it excludes singular points of DAEs and helps to view a DAE as an ordinary differential equation (ODE) defined on the invariant submanifold. Take the following DAE for an example: Nevertheless, for any x 0 ∈ M = R, there exists a unique solution x(t) satisfying x(0) = x 0 , namely, · · · M c 0 of U k−1 for a certain k ≥ 1, has been constructed. Define recursively Then either x p / ∈ M k or x p ∈ M k , and in the latter case, assume that there exists a neighborhood U k of x p such that M c k = M k ∩ U k is a smooth embedded submanifold (which can always be assumed connected by taking U k sufficiently small).
Remark 2.6. For a linear DAE ∆ = (E, H) of the form (3), define a sequence of subspaces (one of the Wong sequences [28]) by If we apply the iterative construction of M k by (4) to the DAE ∆, we get M c k = V k , ∀k ≥ 0. Thus the sequence of submanifolds M k can be seen as a nonlinear generalization of the sequence V k .
The following proposition shows that the geometric reduction method above can be used to construct locally maximal invariant submanifold M * and to deduce that the consistency set S c , on which the solutions exist, coincides locally with M * . Proposition 2.7. In the geometric reduction method of Definition 2.5, there always exists k * ≤ n such that either k * is the smallest integer for which x p / ∈ M k * +1 or k * is the smallest integer In the latter case, we assume that dim E(x)T x M c k * +1 = const. in a neighborhood U * ⊆ U k * +1 of x p in X and then (i) x p is consistent and M * = M c k * +1 is a locally maximal invariant submanifold around x p .
(ii) M * coincides locally with the consistency set S c , i.e., M * ∩ U = S c ∩ U * (take a smaller U * if necessary).
The proof is given in Section 7.1. Note that the geometric method can be implemented in practice via an algorithm which we propose in Appendix of the present paper and the results of Proposition 3.3 and Theorem 6.1 below will be based on that algorithm.

External equivalence, internal equivalence and internal regularity
Two linear DAEs Eẋ = Hx andẼẋ =Hx are called externally equivalent [17] or strictly equivalent [19], if there exist constant invertible matrices Q and P such that QEP −1 =Ẽ and QHP −1 =H. Analogously, we define the external equivalence of two nonlinear DAEs as follows.
Definition 3.1 (external equivalence). Two DAEs Ξ l,n = (E, F ) andΞ l,n = (Ẽ,F ) defined on X andX, respectively, are called externally equivalent, shortly ex-equivalent, if there exist a diffeomorphism ψ : X →X and Q : X → GL(l, R) such that where ψ * Ẽ and ψ * F denote the pull-back [3] of the R l -valued differential 1-formẼ onX and R l -valued functionF (0-form) onX, respectively, that is, The ex-equivalence of two DAEs will be denoted by Ξ ex ∼Ξ. If ψ : U →Ũ is a local diffeomorphism between neighborhoods U of x p andŨ ofx p , and Q(x) is defined on U , we will speak about local ex-equivalence.
The following observation relates ex-equivalence with solutions.
Remark 3.2. The ex-equivalence preserves trajectories, i.e., for two DAEs Ξ is a solution of Ξ passing through x 0 = x(t 0 ), thenx = ψ • x is a solution ofΞ passing throughx 0 = ψ(x 0 ); but even if we can smoothly conjugate all trajectories of two DAEs, they are not necessarily ex- . Then for both DAEs Ξ 1 and Ξ 2 , the maximal invariant submanifold is M * = (x 1 , x 2 , x 3 ) ∈ R 3 | x 2 = x 3 = 0 and for any (x 10 , x 20 , x 30 ) = (0, 0, x 30 ) ∈ M * , the unique solution of both systems is x 1 (t) = x 2 (t) = 0, x 3 (t) = x30 1−x30t . Nevertheless, the DAEs are not ex-equivalent since the distribution ker E 1 is involutive but the distribution ker E 2 is not (clearly, the ex-equivalence of two DAEs preserves the involutivity of ker E 1 and ker E 2 since if Ξ 1 ex ∼ Ξ 2 , via Q and ψ, then ker E 2 = ∂ψ ∂x ker E 1 ). Now we use the algorithm presented in the Appendix to implement the geometric reduction method being a practical application of Proposition 2.7 and to show that any DAE Ξ has isomorphic solutions with an "internal" DAE Ξ * defined on its locally maximal invariant submanifold M * . In the statement of Proposition 3.3, we refer to the submanifold M * = M * k+1 , the neighborhood U * = U * k+1 , the coordinates (z * ,z 1 , . . . ,z k * ) on U * , and the DAE Ξ * r * ,n * = (E * , F * ) defined on M * by the algorithm of the Appendix, where E * = E k * +1 : M * → R r * ×n * , F * = F k * +1 : M * → R r * , n * = n k * = n k * +1 , r * = r k * +1 come from Step k * + 1 of the algorithm. Suppose that Assumptions 1 and 2 of the algorithm in Appendix are satisfied. Then M c k , for k = 0, . . . , k * + 1, given by (4) of the geometric reduction method are smooth connected embedded submanifolds and dim E(x)T x M * = const. for all x ∈ M * ∩ U * . Thus by Proposition 2.7, x p ∈ M * is a consistent point and M * is a locally maximal invariant submanifold around x p , given by M * = {x |z 1 (x) = 0, . . . ,z k * (x) = 0}. Then for the DAE Ξ * r * ,n * = (E * , F * ), defined by the algorithm, given on M * by where z * = z k * +1 = z k * are local coordinates on M * , we have rank E * (z * ) = r * , ∀z * ∈ M * , i.e., Moreover, the DAE Ξ * has isomorphic solutions with Ξ l,n , i.e., there exists a local diffeomorphism Ψ : U * → Ψ(U * ), Ψ(x) =ẑ = (z * ,z) = (z * ,z 1 , . . . ,z k * ), transforming the set of all solutions of Ξ l,n on U * into that ofΞl ,n = (Ê,F ) on Ψ(U * ), wherel = r * + (n − n * ),n = n, given bŷ Ξ : The proof is given in Section 7.2. The analysis of Proposition 3.3 shows clearly the reason behind For any DAE Ξ l,n = (E, F ), there may exist some redundant equations (in particular, some trivial algebraic equations 0 = 0 and some dependent equations). In the linear case, we have defined the full rank reduction of a linear DAE (see Definition 6.4 of [17]). We now generalize this notion of reduction to nonlinear DAEs to get rid of their redundant equations.
Definition 3.5 (reduction). For a DAE Ξ l,n = (E, F ), assume rank E(x) = const. = q. Then there is of full row rank q, denote QF = F1 F2 . Assume that rank DF 2 (x) = const. =l − q ≤ l − q. Then the full row rank reduction, shortly reduction, of Ξ, denoted by Ξ red , is the DAE whereF 2 : X → Rl −q with DF 2 being all independent rows of DF 2 .
Remark 3.6. Clearly, since the choice of Q(x) is not unique, the reduction of Ξ is not unique either.
Nevertheless, since Q(x) preserves the solutions, each reduction Ξ red has the same solutions as the original DAE Ξ.
For a locally invariant submanifold M , we consider the local M -restriction Ξ| M of Ξ, and then we construct a reduction of Ξ| M and denote it by Ξ| red M . Notice that the order matters: to construct Ξ| red M , we first restrict and then reduce while reducing first and then restricting will not give Ξ| red M but another DAE Ξ red | M , which may have redundant equations as seen from the following example.
Then the M * -restriction of Ξ, by Definition 3.4, is Ξ| M * : The definition of the internal equivalence of two DAEs is given as follows.
Definition 3.9. (internal equivalence) Consider two DAEs Ξ = (E, F ) andΞ = (Ẽ,F ), and fix two points x p ∈ X andx p ∈X. Let M * andM * be two locally maximal invariant submanifolds of Ξ andΞ, around x p andx p , respectively. Assume that dim E(x)T x M * = const. for x ∈ M * around x p and dimẼ(x)TxM * = const. forx ∈M * aroundx p . Then, Ξ andΞ are called locally internally equivalent, shortly in-equivalent, if Ξ| red M * andΞ| red M * are ex-equivalent, locally around x p andx p , respectively. Denote the in-equivalence of two DAEs by Ξ in ∼Ξ.
Remark 3.10. Under the assumption that dim E(x)T x M * and dimẼ(x)TxM * are constant, by The dimensions l and n, related to Ξ, and l andñ related toΞ are not required to be the same. However, if Ξ andΞ are in-equivalent, then by definition, Ξ| red M * = Ξ * r * ,n * andΞ| red M * =Ξ * r * ,ñ * are locally ex-equivalent and thus the dimensions related to them have to be the same, i.e., r * =r * and n * =ñ * (and l * = r * =r * =l * since all reductions of Ξ andΞ are of full row rank). Now we will study the uniqueness of solutions of DAEs with the help of the notion of internal equivalence (some other results of uniqueness of DAE solutions can be consulted in e.g., [11,12]).
We will say that a solution x : I → M * of a DAE Ξ satisfying x(t 0 ) = x 0 , where t 0 ∈ I and x 0 ∈ M * , is maximal if for any solutionx :Ĩ → M * such that t 0 ∈Ĩ,x(t 0 ) = x 0 and x(t) =x(t), ∀t ∈ I ∩Ĩ, we haveĨ ⊆ I.
x ∈ M * around x p . Then the following conditions are equivalent: for z * ∈ M * ∩ U , where U is a neighborhood of x p and f * is a smooth vector field on M * ∩ U .
The proof is given in Section 7.2.
Remark 3.13. Theorem 3.12 is a nonlinear generalization of the results on the internal regularity of linear DAEs in [17] (see also [29], where the internal regularity is called autonomy). As stated in Theorem 6.11 of [17], a linear DAE ∆ = (E, H), given by (3), is internally regular if and only if the maximal invariant subspace M * of ∆ (i.e., the largest subspace such that HM * ⊆ EM * ) satisfies dim M * = dim EM * . A nonlinear counterpart of the last condition is (ii) of Theorem 3.12 and thus M * is a natural nonlinear generalization of M * . Observe that M * is the limit of M k as V * is the limit of V k , defined in Remark 2.6. Moreover, we have shown in [17] that the maximal invariant subspace M * = V * , where V * coincides with the limit of the Wong sequence V k defined in Remark 2.6.
Step 1: We have rank E(x) = r 1 = 4 on U 1 = X. Since E is already in the desired form, set It is clear that x p ∈ M 1 and M c 1 = M 1 ∩ U 1 = M 1 is a locally smooth connected embedded submanifold and n 1 = dim M c 1 = 4. Then choose new coordinatesz 1 = (x 1 ,x 3 ) = ( x1 x6 , x 3 + x 5 ) and keep the remaining coordinates z 1 = (x 2 , x 4 , x 5 , x 6 ) unchanged. The system in new coordinates, denotedΞ 1 , take the form By settingz 1 = (x 1 ,x 3 ) = 0, we get the reduction of M c 1 -restriction ofΞ 1 (see Definition 3.4 and 3.5) as Step 2: Consider the DAE It is clear that 6 −x 6 x 2 +x 4 ) and keep the remaining coordinates z 2 = (x 5 , x 6 ) unchanged.
For the system in new coordinates, denotedΞ 2 , by a similar procedure as in Step 1, we can define the reduction of M c 1 -restriction ofΞ 2 as Step 3: It can be observed that dim M c 3 = n 3 = 1 and by a similar construction as at former steps, we have Step 4: , thus k * = 3 and the algorithm stops at Step k * + 1 = 4. Therefore, by Proposition 3.3, . Hence the DAE Ξ is internally regular around x p by definition, which illustrates the results of Theorem 3.12 since dim M * = n 4 = dim E(x)T x M * = r 4 = 1, and Ξ is in-equivalent to the ODE:ẋ 6 = −x 6 .

Explicitation with driving variables of nonlinear DAEs
The explicitation (with driving variables) of a DAE Ξ is the following procedure.
• For a DAE Ξ l,n = (E, F ), assume that rank where (11) is given by the differential inclusion:ẋ • Since ker E(x) is a distribution of constant rank n − q, choose locally m = n − q independent vector fields g 1 , . . . , g m on X such that ker E(x) = span {g 1 , . . . , g m } (x). Then by introducing driving variables v i , i = 1, . . . , m, we parametrize the affine distribution f (x) + ker E 1 (x) and thus all solutions of (12) are given by all solutions (corresponding to all controls v i (t) ∈ R) oḟ • Form a matrix g(x) = [g 1 (x), . . . , g m (x)]. Then, we rewrite equation (13) . We claim, see Proposition 4.5 below, that all solutions of DAE (11) (and thus of the original DAE Ξ) are in one-to-one correspondence with • To (14), we attach the control system Σ = Σ n,m,p = (f, g, h), given by Σ : where n = dim x, m = dim v, p = dim y. Clearly, m = n − q and p = l − q (we will use these dimensional relations in the following discussion). In the above way, we attach a control system Σ to a DAE Ξ (actually, a class of control systems, see Proposition 4.2 below). x p ∈ X and assume that rank E(x) = const. locally around x p . Then, by a (Q, v)-explicitation we will call any control system Σ = Σ n,m,p = (f, g, h) given by (15) with F2(x) . The class of all (Q, v)-explicitations will be called shortly the explicitation class. If a particular control system Σ belongs to the explicitation class of Ξ, we will write Σ ∈ Expl(Ξ).
Notice that a given Ξ has many (Q, v)-explicitations since the construction of Σ ∈ Expl(Ξ) is not unique: there is a freedom in choosing Q(x), E † 1 (x), and g(x). As a consequence of this nonuniqueness of construction, the explicitation Σ of Ξ is a system defined up to a feedback transformation, an output multiplication and a generalized output injection (or, equivalently, a class of systems).
and an output multiplicationỹ = η(x)y, which map where α, β and η are smooth matrix-valued functions of appropriate sizes, γ = (γ 1 , . . . , γ p ) is a p-tuple of smooth vector fields on X, and β and η are invertible.
The proof is given in Section 7.3. Since the explicitation of a DAE is a class of control systems, we will propose now an equivalence relation for control systems. An equivalence of two nonlinear control systems is usually defined by state coordinates transformations and feedback transformations (e.g. see [26,27]), and sometimes output coordinates transformations [30]. In the present paper, we define a more general system equivalence of two control systems as follows.
If ψ : U →Ũ is a local diffeomorphism between neighborhoods U of x p andŨ ofx p , and α, β, γ, η are defined locally on U , we will speak about local sys-equivalence.
Remark 4.4. The above defined sys-equivalence of two nonlinear control systems generalizes the Morse equivalence of two linear control systems (see [17,31]).
The following proposition shows that solutions of any DAE are in a one-to-one correspondence with solutions of its (Q, v)-explicitation.
Proposition 4.5. Consider a DAE Ξ l,n = (E, F ) and let a control system Σ n,m,p = (f, g, h) be a ) is a solution of Σ respecting the output constraints y = 0, i.e., a solution of (14).
The proof is given in Section 7.3. The following theorem is a fundamental result of the present paper, which shows that sys-equivalence for explicitation systems (control systems) is a true counterpart of the ex-equivalence for DAEs.
Theorem 4.6. Consider two DAEs Ξ l,n = (E, F ) andΞ l,n = (Ẽ,F ). Assume that rank E(x) and rankẼ(x) are constant around two points x p andx p , respectively. Then for any two control systems The proof is given in Section 7.3. In order to show how the explicitation can be useful in the DAEs theory, we discuss below how the analysis of DAEs of Sections 2 and 3 is related to the notion of zero dynamics of nonlinear control theory. For a nonlinear control system Σ n,m,p = (f, g, h) and a nominal point x p , assume h(x p ) = 0. Recall its zero dynamics algorithm [26,27].
Step 1: set For a control system Σ = (f, g, h), a smooth embedded connected submanifold N containing a point An output zeroing submanifold N * is locally maximal if for some neighborhood U of x p , any other Remark 4.7. (i) It is shown in [26] that N k is invariant under feedback transformations. Then consider a control systemΣ = (f ,g,h), given by applying a generalized output injection and an output multiplication to Σ, i.e.,f = f + γh,g = g,h = ηh, where γ : X → R n×p and η : X → GL(p, R).
we haveÑ k = N k for k ≥ 0, which means that N k of the zero dynamics algorithm is invariant under generalized output injections and output multiplications.
(ii) The sequence of submanifolds N c k of the zero dynamics algorithm is well-defined for the class Expl(Ξ), i.e., does not depend on the choice of Σ ∈ Expl(Ξ). Since by Proposition 4.2 any two systems Σ, Σ ′ ∈ Expl(Ξ) are equivalent via a v-feedback, a generalized output injection, and an output multiplication, then by the argument in item (i) above, we haveÑ k = N k .
(A2) For Σ, the submanifold N c k of the zero dynamics algorithm above is smooth, embedded, connected and dim G( are equivalent for each k ≥ 1. Assume that either (A1) or (A2) holds, then the maximal invariant of [26]).
The proof is given in Section 7.3. The explicitation can be also used to characterize solutions of DAEs which are not necessarily internally regular, that is, the restricted DAE Ξ * , given by (6), has non-unique maximal solutions (recall that Ξ * has isomorphic solutions with the original DAE Ξ by Proposition 3.3). We now apply the explicitation method to Ξ * to have the following result. (ii) C 1 -solutions of Ξ * are in one-to-one correspondence with those of any (Q, v)-explicitation Σ * ∈ Expl(Ξ * ) of the form which is a control system without outputs, where Im g * = ker E, g * = (g * 1 , . . . , g * m * ) and v = (v 1 , . . . , v m * ), and v(t) ∈ C 0 .
(iii) If ker E = ker E * is involutive, then Ξ * is ex-equivalent (that is, the original DAE Ξ is inequivalent) to a semi-explicit DAE of the forṁ which can be seen as a control system that is not affine with respect to the control z * 2 .
Proof. We omit the proof since item (i) is clear, and items (ii) and (iii) can be easily deduced by applying, respectively, the results of Proposition 4.5 and that of Theorem 5.3 (see below) to Ξ * .

Driving variable reducing and semi-explicit DAEs
Now we will show by an example that sometimes we can reduce some of driving variables of a (Q, v)-explicitation.
Example 5.1. Consider a DAE Ξ = (E, F ), given by where F 1 : X → R is smooth. By rank E(x) = 1, the explicitation class Expl(Ξ) is not empty. A control system Σ ∈ Expl(Ξ) is: , we can reduce v 2 in a similar way. We are, however, not able to reduce v 1 and v 2 simultaneously.
Before giving the main result of this subsection, we formally define what we mean by "reducing" variables of a control system Σ.
where v 2 = (v 1 2 , . . . , v k 2 ). We will say that Σ can be G red -reduced to the following control system     ẋ where x 2 is a new control and the reduced state x 1 is of dimension n − k. We say that Σ can be fully reduced if G red = G.
Now we connect reducing of control systems with semi-explicit DAEs.
Theorem 5.3. For a DAE Ξ l,n = (E, F ), the following statements are equivalent around a point x p ∈ X: (i) rank E(x) = const. and the distribution ker E(x) is involutive.
(ii) Ξ is locally ex-equivalent to a semi-explicit DAE Ξ SE : .
The proof is given in Section 7.4.
Remark 5.4. (i) Observe that if Ξ is ex-equivalent to Ξ SE , then by rewriting x 2 = w and choosing the output y = F 2 (x 1 , w), we get the following control system Σ w with an input w, The above system Σ w has the same number of variables as Ξ. Thus Σ w is an explicitation without driving variables of Ξ. So there are two kinds of explicitation for nonlinear DAEs, namely, explicitation with, or without, driving variables (the latter is possible if and only if ker E is involutive).
(ii) A linear DAE ∆ = (E, H), given by (3), has always two kinds of explicitations, since the rank of E is always constant and the distribution G = ker E is always involutive. The relations and differences of the two explicitations for linear DAEs are discussed in [32] and Chapter 3 of [33] (note that the explicitation without driving variables for linear DAEs is called the (Q, P )-explicitation there).

Nonlinear generalizations of the Weierstrass form
In this subsection, we will use the explicitation (with driving variables) procedure to transform an internally regular DAE Ξ l,n = (E, F ) with l = n, into normal forms under the external equivalence.
A linear regular DAE is always ex-equivalent (via linear transformations) to the Weierstrass form WF [20], given by WF : where N = diag (N 1 , . . . , N m ), with N i , i = 1, . . . , m being nilpotent matrices of index ρ i , i.e., N j i = 0 for all j = 1, . . . , ρ i − 1 and N ρi i = 0. The following theorem generalizes that result and shows that any internally regular nonlinear DAE (under the assumption that some ranks are constant) is always ex-equivalent to a nonlinear Weierstrass form NWF1 (see (19) below). Note thatφ k in the algorithm of Appendix, defined on W k ⊆ M c k , can be considered as maps on U 0 ⊆ X by takingΦ k =φ k • ϕ k−1 • · · · • ϕ 1 (x). Then for k ≥ 1 , set H k = Φ 1 . . .Φ k T and H 0 is empty.
Assumption 1 of the algorithm of Appendix says that rankF 2 k (z k−1 ) = const. for z k−1 ∈ M k ∩ U k . In (A1) below, we replace it by a stronger rank assumption on a neighborhood U ⊆ X of x p . Theorem 6.1. Consider a DAE Ξ l,n = (E, F ), assume that rank E(x) = const. = q around a point x p . Also assume in the geometric reduction algorithm of Appendix that (A3) l = n and dim M * = dim E(x)T x M * , i.e., r * = n * , for all x ∈ M * around x p .
More specifically, for 1 ≤ i ≤ m, the ρ i × ρ i nilpotent matrices N ρi and the ρ i -dimensional vector-valued functions a i + b iż ρ are of the following form The proof of Theorem 6.1 is given in Section 7.5. This proof is closely related to the zero dynamics algorithm for nonlinear control systems shown in [26] and the construction procedure of the above normal form is not difficult but quite tedious, so in order to avoid reproducing the zero dynamics algorithm, we will use some results directly from [26] with small modifications.
Remark 6.2. (i) Assumption (A2) of Theorem 6.1 is equivalent to Assumption 1 of the geometric reduction algorithm of Appendix. By Theorem 3.12, we know that (A3) of Theorem 6.1 implies that Ξ is internally regular around x p .
(ii) A component-wise expression of the above NWF1 is NWF1 : where a k i , b k i,s , f * and G depend on (z, z * ). (iii) The submanifolds M c k , k ≥ 1, of the algorithm are given by and the maximal invariant submanifold M * is given by Therefore, an equivalent condition for where I k is the ideal generated by z j i , 1 ≤ i ≤ m, 1 ≤ j ≤ k in the ring of smooth functions of z a b and z * c . (iv) We see that all maximal solutions (z(·), z * (·)) are unique and of the form (0, z * (·)), where z * (·) are maximal solutions of the ODEż * = f * (0, z * ) on M * , which agrees with the result of Theorem 3.12(iii). Example 6.3 (continuation of Example 3.14). Consider the DAE Ξ 6,6 = (E, F ) of (10) around the point x p = (0, 1, 0, 0, 0, 1). A control system Σ 6,2,2 ∈ Expl(Ξ) is Σ : It can be observed from Example 3.14 that the assumptions (A1)-(A3) of Theorem 6.1 are satisfied. Now via the following local changes of coordinates defined on U = X = {x ∈ X : x 6 > 0, x 1 = x 6 }: we can bring Σ into the system Σ ′ below, which is of the zero dynamics form (36) as given by Claim 7.1, where the feedback transformation brings the system Σ ′ into the system Σ ′′ above. In order to eliminate z * ṽ 1 inż 2 2 = z 3 2 + z * ṽ 1 + z 2 1 z 2 2 z 3 2 − z 2 1 z 3 2ṽ 2 of Σ ′′ , we define the change of coordinates and the output multiplication ] .
Then the system Σ ′′ becomes Σ : Now we drop all the tildes in the systemΣ for ease of notation. By setting y 1 = y 2 = 0, replacing v 1 =ż 2 1 , v 2 = z 3 2 , and deleting the equationsż 2 1 = v 1 , z 3 2 = v 2 , we get the following DAEΞ fromΣ, Ξ : where The form NWF1 of Theorem 6.1 is related to the zero dynamics of nonlinear control systems.
In the remaining part of this section, we will use the notions of (vector) relative degree and invariant distributions of nonlinear control theory to study when a DAE Ξ is ex-equivalent to a simpler form NWF2 : where N = diag (N 1 , . . . , N m ), with N i ∈ R ρi×ρi , i = 1, . . . , m, being nilpotent matrices of index ρ i .
The NWF2 is a perfect nonlinear counterpart of the linear WF because the nonlinear terms G, a i and b j of NWF1 are absent in NWF1 and f * depends on z * -variables only. We now recall the definitions of (vector) relative degree and (conditional) invariant distributions for nonlinear control systems.
Definition 6.4 (relative degree [26]). A square control system Σ n,m,m = (f, g, h) has a (vector) For a nonlinear control system Σ n,m,p = (f, g, h), define a sequence of distributions S i by Theorem 6.5. For a nonlinear DAE Ξ n,n = (E, F ) (i.e., l = n), assume that rank E(x) = const.
around a point x p ∈ X. Then Ξ is locally ex-equivalent to the NWF2, given by (21), around x p if and only if there exists a control system Σ = Σ n,m,m = (f, g, h) ∈ Expl(Ξ) such that (i) the system Σ has a well-defined relative degree ρ = (ρ 1 , . . . , ρ m ) at x = x p ; (ii) the distributions S i of Σ, defined by (22), are involutive for all 1 ≤ i ≤ n − 1.
We omit the proof the Theorem 6.5 since it is indicated by Theorem 4.6 and some results from nonlinear control theory, see Remark 6.6(i) below. Remark 6.6. (i) Note that, under conditions (i) and (ii) of Theorem 6.5, using the results in [30], we can transform the system Σ into the following form (called the input-output special form in [30]) via suitable coordinates transformations and feedback transformations, Rewritef * (z * , y) =f * (z * , 0) + γ(z * , y)y for some smooth function γ, then we can always get rid of the y-variables inf * (z * , y) by an output injectionf * →f * − γy = f * , where f * = f * (z * ). Thus the system Σ is always sys-equivalent to the systemΣ below Σ sys ∼Σ : So by Theorem 4.6, the DAE Ξ is ex-equivalent toΞ represented in the NWF2 sinceΣ ∈ Expl(Ξ).
(ii) The linear counterparts of the distributions S i , given by (22), for linear control systems of the Conversely, suppose that dim E(x)T x M = const. =r and F (x) ∈ E(x)T x M locally for all x ∈ M ∩ U . Notice that M is a smooth connected embedded submanifold, thus there exists a smaller neighborhood U 1 of x p and local coordinates ψ(x) = z = (z 1 , z 2 ) on U 1 such that M ∩U 1 = {z 2 = 0}, where z 1 are any complementary coordinates, with dim z 1 =n, dim z 2 = n −n andn = dim M . In the local z-coordinates, the DAE Ξ has the following form  0).
Proof of Propostion 2.7. Let k be the smallest integer such that M c and M c k+1 = M k+1 ∩ U k+1 is a submanifold (by the recursive procedure assumptions) such that dim M c k = dim M c k+1 . Then k * = k is the integer whose existence is asserted. The condition k * ≤ n follows from dim If an consistent point x c ∈ S c ∩ U k * , then x c ∈ M k * +1 . Now we prove the Claim holds. Since x c is consistent, there exists a solution (x(t), u(t)), defined on I, and t 0 ∈ I such that x(t 0 ) = x c . It follows that for all t ∈ I, So F (x(t)) ∈ Im E(x(t)), ∀t ∈ I. Thus by equation (4), we have x(t) ∈ M 1 , ∀t ∈ I. Suppose that for a certain i > 1, we have x(t) ∈ M i−1 , ∀t ∈ I. We then have thatẋ(t) ∈ T x(t) M i−1 , ∀t ∈ I (note that when restricted to U i−1 , the set M i−1 is a submanifold). Thus in U k * ⊆ U i , equation (26) implies F (x(t)) ∈ E(x(t))T x(t) M c i−1 . It follows that x(t) ∈ M i ∩ U i−1 , for any t ∈ I, due to (4). By an induction argument, we conclude that x(t) ∈ M k * +1 ∩ U k * , and, in particular, we have . So, using Proposition 2.3, we conclude that M * is a locally invariant submanifold on U * . To prove that M * is maximal in U * , let M ′ be any invariant submanifold, then any point x 0 ∈ M ′ ∩ U * is consistent, so x 0 ∈ S c ∩ U * , then by the above Claim, (ii) We now prove that M * coincides with the consistent set S c on U * . Since M * ∩ U * is locally invariant, for any point x 0 ∈ M * ∩ U * , there exist at least one solution (x(·), u(·)) on I and t 0 ∈ I such that x(t 0 ) = x 0 , which implies that x 0 is consistent i.e., x 0 ∈ S c . It follows that M * ∩ U * ⊆ S c ∩ U * . Conversely, consider any point x 0 ∈ S c ∩ U * , using again the above Claim,

Proofs of Proposition 3.3 and Theorem 3.12
Proof of Proposition 3.3. At every Step k of the algorithm in Appendix, consider the DAEΞ k = Ξ k−1 = (E k−1 , F k−1 ) andΞ k = (Ê k ,F k ), the latter given by (40). Then we show that the following items are equivalent. (a). z k−1 (·) = ψ −1 k (z k (·),z k (·)) is a solution of Ξ k−1 ; (b). (z k (·),z k (·)) is a solution ofΞ k ; (c).z k (·) = 0 and z k (·) is a solution of Ξ k : 0) and whereÊ 1 k ,F 1 k are defined in (40). SinceΞ k = Ξ k−1 is locally ex-equivalent toΞ k via Q k and ψ k , we have that item (a) and item (b) above are equivalent (see Remark 3.2). The equivalence of item (b) and item (c) follows from the fact that the solutions exists on M c k only and should respect the constraintsz k = 0. Then by the equivalence of (c) and (a), we have, at the first step of the algorithm, that (z 1 (·), 0) is a solution of E 1 (z 1 )ż 1 = F 1 (z 1 ), together withz 1 = 0, if and only if z 0 (·) = ψ −1 1 (z 1 (·), 0) is a solution of Ξ 0 = Ξ = (E, F ). In general, by an induction argument, we can prove that (z k (·), 0, . . . , 0) is a Proof of Proposition 4.2. If. Throughout the proof below, we may drop the argument x for the functions f (x), g(x), h(x), . . ., for ease of notation. Suppose that Σ andΣ are equivalent via transformations given by (16). First, Img = Im gβ = ker E 1 = ker E implies thatg is another choice such that Img = ker E. Moreover, we havẽ Σ : Pre-multiplying the differential partẋ = E † 1 F 1 + gα + γF 2 + gβv ofΣ by E 1 , we get (note that ThusΣ is an (I,ṽ)-explicitation of the following DAE: Since the above DAE can be obtained from Ξ viaQ = Q ′ Q, where Q ′ = Iq E1γ 0 η , it proves thatΣ is a (Q,ṽ)-explicitation of Ξ corresponding to the choice of invertible matrixQ = Q ′ Q. Finally, by for the above choice of right inverseẼ † 1 of E 1 . Only if. Suppose thatΣ ∈ Expl(Ξ) viaQ,Ẽ † 1 andg. First, by Img = ker E = Im g, there exists an invertible matrix β such thatg = gβ. Moreover, since E † 1 is a right inverse of E 1 if and only if any solutionẋ of E 1ẋ = w is given by Since ker E 1 = Im g, it follows that (Ẽ † 1 − E † 1 )F 1 = gα for a suitable α. Furthermore, since Q is such that E 1 of QE = E1 0 is of full row rank, any otherQ, such thatẼ 1 ofQE = Ẽ 1 0 is of full row rank, must be of the formQ = Q ′ Q, where Q ′ = Q1 Q2 0 Q4 . Thus viaQ, Ξ is ex-equivalent to The equation on the right-hand side of the above can be expressed (usingẼ † 1 andg) as: Thus the explicitation of Ξ viaQ,Ẽ † 1 andg is Σ : . Therefore, we can see that Σ andΣ are equivalent via the transformations of the form (16).
Proof of Proposition 4.5. Consider the DAE (11) of the (Q, v)-explicitation procedure. Since Qtransformations preserve solutions of Ξ, (11) resulting from a Q-transformation of Ξ has the same solutions as Ξ. Thus we need to prove that (11) and (14) have corresponding solutions for any choices of E † 1 and g. Moreover, the second equation 0 = F 2 (x) of (11) coincides with 0 = h(x) of (14). So we only need to prove that x(t) ∈ C 1 is a solution of E 1 (x)ẋ = F 1 (x) if and only if there , and of the choice of g satisfying Im g(x) = ker E 1 (x).
Pre-multiplying the latter equation by E 1 (x(t)), we get that Then, by taking a smaller neighborhood U , if necessary, we assume that E 1 1 (x) is invertible locally around x p (if not, we permute the components of x such that the first q columns of E 1 (x) is independent). Thus a choice of right inverse of E 1 is . So the maps f and g can be defined as f : Notice that if we choose another right inverseẼ † 1 of E 1 and another matrixg such that Img = ker E 1 , then by Proposition 4.2, we havė We thus conclude that there existsṽ(t) = α(x(t)) + β(x(t))v(t) = α(x(t)) + β(x(t))ẋ 2 (t) such that (x(t),ṽ(t)) solvesẋ =f (x) +g(x)ṽ. Therefore, Ξ has corresponding solutions with any (Q, v)explicitation Σ independently of the choice of Q, E † 1 and g.
Proof of Theorem 4.6. By the assumptions that rank E(x) = const. = q and rankẼ(x) = const. =q around x p andx p , respectively, we have that Ξ andΞ are locally ex-equivalent to respectively, where E 1 (x) andẼ 1 (x) are full row rank matrices and their ranks are q andq, respectively. By Definition 4.1, we have Note that the explicitation system is defined up to a feedback, an output multiplication and a generalized output injection. Any two control systems belonging to Expl(Ξ) are sys-equivalent to each other and so are any two control systems belonging to Expl(Ξ). Thus the choice of an explicitation system makes no difference for the proof of sys-equivalence. Without loss of generality, we will use f (x), g(x), h(x) andf (x),g(x),h(x) given in (28) for the remaining part of this proof.
If. Suppose Σ sys ∼Σ in a neighborhood U of x p . By Definition 4.3, there exists a diffeomorphism x = ψ(x) and β : U → GL(m, R) such thatg • ψ = ∂ψ ∂x gβ, which implies ker(Ẽ • ψ) = span{g 1 , ...,g m } • ψ = span ∂ψ ∂x g 1 , ..., ∂ψ ∂x g m = ∂ψ ∂x ker E and q =q (since dim kerẼ =m = m = dimẼ). We can deduce from the above equation that there exists Q 1 : U → GL(q, R) such thatẼ Subsequently, byf • ψ = ∂ψ ∂x (f + γh + gα) of Definition 4.3, we have Pre-multiply the above equation Then byh • ψ = ηh of Definition 4.3, we immediately get Now combining (29), (30) and (31), we conclude that Ξ ′ andΞ ′ are ex-equivalent viax = ψ(x) and Only if. Suppose that locally Ξ ex ∼Ξ around x p . It follows that locally Ξ ′ ex ∼Ξ ′ around x p , which implies that q =q. Assume that they are ex-equivalent via Q : U → GL(l, R) andx = ψ(x) defined on a neighborhood U of x p . Let Q = Q1 Q2 Q3 Q4 , where Q 1 , Q 2 , Q 3 and Q 4 are matrix-valued functions of sizes q × q, q × m, p × q and p × p, respectively. Then by Q1 Q2 Q3 Q4 E1 0 = Ẽ 1•ψ 0 ∂ψ ∂x , we can deduce that Q 3 = 0 and Q 1 , Q 4 are invertible matrices. Then we have which impliesẼ Thus by Im g(x) = ker E(x) = ker E 1 (x) and Img(x) = kerẼ(x) = kerẼ 1 (x), and using (32), we haveg • ψ = ∂ψ ∂x gβ (33) for some β : U → GL(m, R). Moreover, there exists α : U → R m such that In addition, we haveh Finally, it can be seen from (33) Proof of Proposition 4.8. We first show that the sequence of submanifolds M c k of the geometric reduction method of the DAE Ξ and the sequence N c k of the zero dynamics algorithm of any control system Σ = (f, g, h) ∈ Expl(Ξ) locally coincide. Suppose that rank E(x) = const. = q in a neighborhood U 1 of x p . Then there always exists an invertible matrix Q(x) defined on U 1 such that F2(x) . Recall, see Remark 4.7, that N k of the zero dynamics algorithm are well-defined for any Σ ∈ Expl(Ξ) and that N k are the same for all control systems belonging to Expl(Ξ). So the choice of an explicitation system makes no difference for N k . We may choose a control system Σ = (f, g, h) ∈ Expl(Ξ), given . By the definition of M 1 (see (4)) and For k > 1, suppose M c k−1 = N c k−1 . Then by (4) and (17), we have x-coordinates, the distribution .., m. Now letg be a matrix whose columns consist ofg i , for i = 1, ..., m. It follows that rankg(x) = m aroundx 0 = ψ(x 0 ). By dx 1 = G ⊥ , we have dx 1 ,g i = 0, for i = 1, . . . , m.
(ii) There are two differences between system (36) and the zero dynamics form of Proof of Claim 7.1. We will prove that assumptions (A1), (A2), (A3) of Theorem 6.1 correspond to the rank conditions (i), (ii), (iii) of Proposition 6.1.3 in [26]. By the assumption of Theorem 6.1 that rank E(x) = const. around x p , we have Expl(Ξ) is not empty. Now, in order to compare the two algorithms (the geometric reduction algorithm of Appendix for Ξ and the zero dynamics algorithm in [26] for Σ ∈ Expl(Ξ)), we use the same notations as in the algorithm of Appendix.
Step k (k > 1): By the proof of Proposition 4.8, we have N c where H k−1 = (ψ 0 , . . . , ψ k−1 ). By the zero dynamic algorithms, N k consists of all x ∈ N c k−1 such that Then by assumption (A2) of Theorem 6.1, we can deduce that for all x ∈ M c k−1 around x p . Now by dim ker E(x) = const. around x p (implied by rank E(x) = const.), we get dim span{g 1 , . . . , g m }(x) = const.
locally around x p . By (37) and (38), we get rank L g H k−1 (x) = const. for all x ∈ M c k−1 around x p (condition (ii) of Proposition 6.1.3 in [26]).
Since rank L g H k−1 (x) = const., there exists a basis matrix R k−1 (x) of the annihilator of the image of L g H k−1 (x), that is R k−1 (x)L g H k−1 (x) = 0. Thus N c k can be defined by Notice that by the geometric reduction algorithm, we have By N c k = M c k and the fact that ranks of the differential of (H k−1 (x),F 2 k (x)) are constant for all x around x p (assumption (A1) of Theorem 6.1), it follows that the rank of the differential of Finally, by N * = {x : H k * (x) = 0}, it follows that the matrix L g H k * (x p ) has rank m (condition (iii) of Proposition 6.1.3 in [26]).
Proof of Theorem 6.1. Observe that by assumption (A3) and Theorem 3.12(iii), we have that Ξ is internally regular. Then by Claim 7.1, we have x p is a regular point of the zero dynamics algorithm for any control system Σ ∈ Expl(Ξ). Thus there exist local coordinates (z, z * ) such that Σ is in the form (36) around x p . Notice that the matrix β = (β 1 , . . . , β m ) is invertible at x p and the functions σ k i | N c k = 0 for 1 ≤ i ≤ m, 1 ≤ k ≤ ρ i − 1, which implies σ k i ∈ I k , where I k is the ideal generated by z j i , 1 ≤ i ≤ m, 1 ≤ j ≤ k in the ring of smooth functions of z a b and z * c . Then for system (36), using the feedback transformationṽ = α + βv, where α = (α 1 , . . . , α m ), we get . . , m, wheref * = f * −ḡβ −1 α,G * = g * β −1 , and where a k i = −σ k i β −1 α, b k i = σ k i β −1 , for 1 ≤ i ≤ m, 1 ≤ k ≤ ρ i − 1 and by σ k i ∈ I k , we have a k i , b k i,s ∈ I k . Recall from (36) that the functions δ j i,s ≡ 0 for 1 ≤ j < ρ s , 1 ≤ s ≤ i − 1. Then if the function δ j i,s = 0, j = ρs + k, for a certain 1 ≤s ≤ i − 1 and a certain 0 ≤ k ≤ ρ i − 1 − ρs, we show that, via suitable changes of coordinates and output multiplications, the nonzero function δ k+ρs i of (39) into y i =z 1 i + δ ρs i,s z 1 s . We define a new outputỹ i = y i − δ ρs i,s z 1 s = y i − δ ρs i,s ys (which is actually an output multiplication of the formỹ i = η i y) such that the first equation of (39) becomesỹ i =z 1 i . Repeat the above construction to eliminate all nonzero functions δ j i,s for j ≥ ρ s , 1 ≤ s ≤ i − 1. Then system (39) becomes the following control system Σ : where a k i , b k i,s ∈ I k for 1 ≤ k ≤ ρ i − 1. It is clear that Σ sys ∼Σ (we used coordinates changes, feedback transformations and output multiplications to transform Σ intoΣ). Then consider the last row of every subsystem ofΣ, which isż ρi i =ṽ i . By deleting this equation in every subsystem and setting y i = 0 for i = 1, . . . , m, and replacing the vectorṽ byż ρ , we transformΣ into a DAEΞ below. It is straightforward to see thatΣ ∈ Expl(Ξ). −G * (z, z * )ż ρ +ż * =f * (z, z * ) .

Conclusions
In this paper, we first revise the geometric reduction method for the existence of nonlinear DAE solutions, and then we define the notions of internal and external equivalence, their differences are discussed by analyzing their relations with solutions. We show that the internal regularity (existence and uniqueness of solutions) of a DAE is equivalent to the fact that the DAE is internally equivalent to an ODE (without free variables) on its maximal invariant submanifold. A procedure named explicitation with driving variables is proposed to connect nonlinear DAEs with nonlinear control systems. We show that the external equivalence for two DAEs is the same as the system equivalence for their explicitation systems. Moreover, we show that Ξ is externally equivalent to a semi-explicit DAE if and only if the distribution defined by ker E(x) is of constant rank and involutive. If so, the driving variables of a control system Σ ∈ Expl(Ξ) can be fully reduced. Finally, two nonlinear generalizations of the Weierstrass form WF are proposed based on the explicitation method and the notions as zero dynamics, relative degree and invariant distributions of nonlinear control theory.