Technische Universität Berlin Institut für Mathematik

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1 Technische Universität Berlin Institut für Mathematik A flow-on-manifold formulation of differential-algebraic equations Ann-Kristin Baum Preprint Preprint-Reihe des Instituts für Mathematik Technische Universität Berlin Report July 25

2 A flow-on-manifold formulation of differential-algebraic equations Ann-Kristin Baum July, 25 Abstract We derive a flow formulation of differential-algebraic equations DAEs, implicit differential equations whose dynamics are restricted by algebraic constraints. Using the framework of derivatives arrays and the strangeness-index, we identify the systems that are uniquely solvable on a particular set of initial values and thus possess a flow, the mapping that uniquely relates a given initial value with the solution through this point. The flow allows to study system properties like invariant sets, stability, monotonicity or positivity. For DAEs, the flow further provides insights into the manifold onto which the system is bound to and into the dynamics on this manifold. Using a projection approach to decouple the differential and algebraic components, we give an explicit representation of the flow that is stated in the original coordinate space. This concept allows to study DAEs whose dynamics are restricted to special subsets in the variable space, like a cone or the nonnegative orthant. Keywords: Differential-algebraic equations, flow, flow on surface, Dynamical systems. AMSMOS subject classification: 34A9, 37E35, 37C, 37Cxx. Institut für Mathematik, TU Berlin, Straße des 7. Juni 36, 623 Berlin, Germany. baum@math.tu-berlin.de The author has been supported by the European Research Council through the ERC Advanced Grant "Modeling, Simulation and Control of Multi-Physics Systems".

3 Introduction We consider differential-algebraic equations DAEs F t, x, ẋ =, where F C k I Ω x Ωẋ, R n is defined on open sets I R, Ω x, Ωẋ R n. DAEs model dynamical processes that are constrained by auxiliary algebraic conditions, like e.g. connected joints in multibody systems, connections or loops in networks or balance equations and conservation laws in advection-diffusion equations, see e.g. [6, 9,,, 4, 6, 2, 29, 3, 35, 4, 42] and the references therein. We derive a flow formulation of the DAE by defining a mapping that uniquely relates an initial value with the solution through this point. For ordinary differential equations ODEs ẋ = ft, x, 2 the concept of the flow is well studied [, 23, 24, 45], and allows to study properties of 2 like invariant sets, stability, monotonicity or symmetry, see e.g. [, 23, 24, 45, 2]. Similarly, for differential equations on manifolds there exists the concept of the flow, allowing to study system properties and their preservation in a numerical simulation, see e.g. [22, 9, 2] and the references therein. Under certain smoothness assumptions, DAEs can be considered as differential equations on a manifold, cp. e.g. [9, 29, 42], thus allowing to extend the notion of a flow implicitly to implicit systems. For DAEs in the form, a flow formulation has been considered in [3] to study stability properties. As stability is a coordinate invariant property, in [3] the flow is constructed using variable transformations to separate the differential and algebraic components in. To study coordinate dependent property like the invariance of special sets in the state space, like cones or manifolds, however, we need a flow representation that is stated in the original coordinates. Using the framework of derivatives arrays and the strangeness-index [29], we identify those DAEs that are uniquely solvable on a particular set of initial values. Using a projection approach to decouple the differential and algebraic components without changing the original coordinate system [3], we construct an explicit representation of the flow. Considering the time-derivative of the flow, we obtain an explicit representation for the linearization of solutions of. Specifying our results for linear systems, we generalize Duhamel s formula to DAEs. 2 Preliminaries We consider time or time-state dependent projections, i.e., matrix functions P C k I Ω, R n n, k, that satisfy P 2 t, x = P t, x for every t, x I Ω. Then, the classical properties of constant projections pointwise extend to the function P, cp. [3]. In particular, P R n n is called orthogonal if P is pointwise symmetric, i.e., P T t, x = P t, x on I Ω. The complement P c := I n P of a projection P is again a projection and satisfies rangep c t, x = kerp t, x and kerp c t, x = rangep t, x. In particular, we consider projections that are induced by the Moore-Penrose inverse. For a matrix function E C k Ω, R n n, the Moore-Penrose inverse E + is pointwise defined like for constant matrices, cp. e.g. [4, 2, 8], i.e., E + x := Ex +, where EE + Ex = Ex, E + EE + x = E + x, E + Ex T = E + Ex, EE + x T = EE + x for x Ω.

4 For every matrix Ex R n n, there exists a unique Moore-Penrose inverse [5] and if Ex is nonsingular, then E + x = Ex [44]. If E C l I Ω, R m n and ranket, x = d on I S, where S Ω is an open set, for every t, x I S, then there exist neighborhoods I I, Ux Ω, such that E + C l I Ux, R n m [3, Lemma 2.3]. If E C k I, R m n and ranket = d on I, then E + C k I, R n m [3, Lemma 2.3]. For E C l I Ω, R m n, the product EE + x R m m is the orthogonal projection with rangeee + x = rangeex, keree + x = corangeex and E + Ex R n n is the orthogonal projection with rangee + Ex = cokerex and kere + Ex = kerex, cp. [2, p. 9]. Furthermore, we use the concept of time-varying subsets, in particular time-varying manifolds, as they arise in the analysis of DAEs. For an interval I R and a family {St} t I of subsets St R n, such that there exists St R n for every t I, we call the set S := { } t I {t} St a time-varying subset on I. Extending the standard definitions of charts and coverings, cp. e.g., [3, pp. 5], [32, pp. 97], [29, pp. 98], we can give a time-varying subset the structure of a manifold, cp. [3]. Here, it suffices to introduce time-varying manifolds as time-parameterized level sets as they arise in the analysis of DAEs. Lemma 2.. A time-varying subset S R R n is a time-varying, embedded C k -submanifold with dims = d if and only if for every t, x S, there exist neighborhoods I R, Ux R n and a function G C k I Ux, R n d that satisfies rankdgt, x = rankg x t, x = n d on G and I Ux S = G. Dropping the time-dependancy, Lemma 2. corresponds to the characterization of a C k - submanifold S R n as level set of a submersion, cp. [3, pp. 3],[32, pp. 97],[25, p. ]. Finally, for a locally Lipschitz function f Lip loc CI Ω, Rn defined on an open set I Ω R R n, the ODE ẋ = ft, x 3 is uniquely solvable for every t, x I Ω with solution x Ct, t+, Ω, where t± I or lim t t ± min{distxt, Ω, xt } =, cp. e.g. [5, p. 44] and [, p. 5]. For t, x, t, t+ are called the negative and positive escape time, respectively, and t, t + the maximal interval of existence, cp. [, p. ]. The unique relation between a given initial value and its associated solution motivates the definition of the flow, see e.g. [, p. 33], [5, p. 49]. Lemma 2.2. Consider the ODE 3. If f C Lip loc I Ω, Rn, then there exists a function Φ f : I I Ω R n, t, t, x Φ t f t, x, that satisfies the following properties for every t, x I Ω and t [t, t +. Φ t f t, x = x, Φ t f t, Φ s f t, x = Φ t f t, x, Φ t f t, x = ft, Φ t f t, x. 4a 4b 4c For every t, x C F,µ+, on [t, ˆt +, the solution x of 3 is given by xt = Φt f t, x and Φ f t, x C [t, t +, Rn on I Ω. 2

5 The characteristic properties 4 reflect the unique solvability of 3 if f is locally Lipschitz on I Ω. Property 4a uniquely relates the flow Φ f with the initial value t, x, property 4b ensures that every solution can be maximally extended on Ω and property 4c claims that Φ t f t, x solves the differential equation 3. For linear ODEs ẋ = Atx + bt =: f A,b t, x, 5 with A CI, R n n and b CI, R n, linearity implies that f A,b C Lip loc I Ω, Rn if f A,b CI R n, R n. The maximal interval of existence is given by t, t+ = Ī, cp. [5, p. 48]. The flow Φ A,b := Φ fa,b is an affine linear transformation of the initial values, whose system matrix is given by the homogeneous flow Φ A induced by f A := f A,, cp., e.g., [46, p. 63], and that generalizes Duhamel s formula [46] to linear systems with time-varying coefficients. Lemma 2.3. Consider the ODE 5 with f A,b CI R n, R n. On I I R n, the flow Φ A,b is given by t Φ t A,b t, x = Φ t At x + Φ t Abs ds, t 6 where Φ A is the homogeneous flow induced by f A. Φ t A t = Φ t A t. The flow Φ A is pointwise invertible with 3 A flow formula for DAEs To define a flow for DAEs, we need a set of initial conditions on which the implicit equation is uniquely solvable and solutions can be maximally extended. There are several approaches to study DAEs like derivative arrays [6, 8, 7], projector chains [6, 33, 34, 43] or a structural analysis [39, 4] that differ in the way they separate the differential and algebraic components and in the regularity assumptions on the system. Related with these approaches are different index concepts, like the differentiation or strangeness index, the tractability index or the structural index, which measure, roughly spoken, the complexity of solving a given DAE in terms of the necessary differentiations. A comparison of the different index concepts is given, e.g., in [, 37]. We follow the concept of derivative arrays and the strangeness index as developed in [26, 27, 28, 29], because it is applicable to a large class of DAEs and provides a suitable framework to construct a flow. 3. Nonlinear differential-algebraic equations For the DAE with sufficiently smooth system function F, the derivative array of size l, l N, the derivative array of size l is the inflated DAE F t, x, ẋ d F F,l t, x, ẋ,..., x l+ dtf t, x, ẋ :=. = 7 d l F t, x, ẋ dt l 3

6 obtained by successive differentiation. Every sufficiently smooth solution of F t, x, ẋ = solves the inflated system 7. Vice versa, if t, x, ẋ,..., x l solves the derivative array 7, then t, x, ẋ also solves F t, x, ẋ =. For a derivative array of suitable size, the idea of the strangeness index is to filter out a set of differential and algebraic equations that uniquely determines the x-part of this solution t, x, ẋ,..., x l. This may include algebraic equations for derivatives of x, so we consider 7 formally as an algebraic equation for the algebraic variable z l := t, x, v,..., v l+ with v k = x k t, k =,..., l +. The algebraic solution set is denoted by F F,l = {z l I R n... R n F l z l = }. 8 To solve the derivative array 7 locally for t, x, ẋ, we make following assertions on the Jacobians M l z l := v,...,v l+ F F,l z l, N l z l := x F F,l z l, 9 containing the partial derivatives of F F,l z l with respect to the variables v,..., v l+ and x, respectively, cp. [29, p. 55]. Hypothesis 3. [29]. Consider F : D R n. Let there exist µ, d, a N, n = d + a, such that F C µ+ D, R n, Fµ and for every z µ, Fµ, there exists a sufficiently small neighborhood Uz µ,, such that the following properties hold.. On Uz µ, F µ, rankm µ z µ = µ + n a and there exists a pointwise orthogonal matrix function Z 2 C µ Uz µ,, R µ+n a with rankz 2 z µ = a and Z T 2 M µz µ =. 2. On Uz µ, F µ, rankz T 2 N µ z µ = a, where N µ = N µ [I n, ], and there exists a pointwise orthogonal matrix function T C µ Uz l,, R n d with rankt z µ = d and Z T 2 N µ T z µ =. 3. On Uz µ, F µ, rankfẋt, x, ẋt z µ = d and there exists an orthogonal matrix Z R n d with rankz = d and rankz T F ẋt z µ = d. The minimal µ s for which F satisfies Hypothesis 3. on D, is called the strangeness index sindex of [29]. If F has s-index µ s and satisfies Hypothesis 3. with µ s +, d, a, we say that has regular s-index µ s [29]. If F has regular s-index µ =, then F is called regular and s-free [29]. If F is s-free, then the Jacobians Fẋ, F x satisfy the assertions of Hypothesis 3., implying that every algebraic equation a solution of satisfies is explicitly contained in. Conversely, if F is of higher index, then there are algebraic equations hidden in the systems and have to be filtered out by differentiation. Numerlcally, s-free systems can be solved with the same accuracy as ODEs, cp. [29, p. 25]. To match the smoothness assumptions of Hypothesis 3., we can reduce the domain of definition D. The set of functions satisfying Hypothesis 3. with integers µ, d, a and µ +, d, a is denoted by C l µ,d,a,reg D, Rn := { F C l D, R n F satisfies Hypothesis 3. with µ, d, a and µ +, d, a }, where l µ +. Initial values that are part of a vector in the algebraic solution set are summarized in the set of consistent initial values C F,µ := { t, x I Ω v,..., v µ+ Ωẋ R n... R n : t, x, v,..., v µ+ F µ }. 4

7 Similarly, tuples t, x, ẋ part of a vector in F µ are summarized in the set of consistent initializations L F,µ := { t, x, v F v 2,..., v µ+ R n... R n : t, x, v, v 2..., v µ+ F µ }. 2 For functions F C µ+ µ,d,a,reg D, Rn and initial values t, x C F,µ+, the DAE is uniquely solvable and the solution is maximally extendable on C F,µ+, cp. [28] and [29, p. 63, p, 67]. Theorem 3.. If F C µ+ µ,d,a,reg D, Rn, then the DAE is uniquely solvable for every t, x C F,µ+. The solution is x C [t, ˆt +, Rn, where ˆt + = sup{t t t, xt C F,µ+ }. The positive escape time ˆt + denotes the time where the derivative array ceases to satisfy the rank assertions of Hypothesis 3., for example because the Jacobians M µ, N µ suffer from a rank drop in t = t +. As a consequence of Theorem 3.2, we consider the initial value problem IVP F t, x, ẋ =, F C µ+ µ,d,a,reg D, Rn 3a xt = x, t, x C F,µ+, 3b and define a flow on the set of consistent initial values C F,µ+. Corollary 3.. Consider the DAE 3a. There exists a function Φ F : C F,µ+ I R n, t, t, x Φ t F t, x, that satisfies the following properties for every t, x C F,µ+ and t [t, ˆt +. Φ t F t, x = x, Φ t F t, Φ s F t, x = Φ t F t, x, F t, Φ t F t, x, Φ t F t, x =. 4a 4b 4c For every t, x C F,µ+, on [t, ˆt +, the solution x of 3 is given by xt = Φt F t, x and Φ F t, x C [t, ˆt +, Rn on C F,µ+. The function Φ F in Corollary 3. is called the flow associated with the DAE 3a. Like for ODEs, the characteristic properties 4 reflect the unique solvability of the IVP 3 and the extendability of solutions on the set C F,µ+. In contrast to the ODE flow Φ f that is defined on the full phase space, the DAE flow Φ F is defined only on the set of consistent initial values. Remark 3.. For a particular problem or a clever formulation of the DAE, the smoothness assumptions of Theorem 3.2 may be significantly relaxed to prove the existence and uniqueness of solutions on a particular set of initial values. Consequently, for these problems, the flow can be defined under less restrictive smoothness assertions. Treating a more general class of problems, however, we have to assume that the system function is sufficiently smooth to set up the full derivative array of size µ +, such that we can show the uniqueness and existence of solutions. To represent the flow and its linearization explicitly, we use the strangeness-free s-free formulation [28, 29], which gives an equivalent formulation of 3 by specifying the same solution. In contrast to 3, however, this surrogate model is s-free and regular at the solution x and the differential and algebraic equations are explicitly given. 5

8 Theorem 3.2. [28, 29] Consider the IVP 3 and let x C [t, ˆt +, Rn be its solution. There exist functions ˆF C [t, ˆt + Ûx Ûẋ, Rd and ˆF 2 C [t, ˆt + Ûx, Ra defined on neighborhoods of x, such that on [t, ˆt +, the function x is also the unique solution of ˆF t, x, ẋ =, xt = x, 5a ˆF 2 t, x =. 5b In particular, ˆF = [ ˆF T, ˆF T 2 ]T C,d,a,reg [t, ˆt + Ûx Ûẋ L µ+, R n. Remark 3.2. The functions ˆF, ˆF 2 are obtained from the derivative array by choosing a suitable parameterization of the algebraic solution set F F,µ along the solution x of, cp. [28] and [29, p. 63, p. 67], and are defined until x leaves the algebraic solution set F F,µ. For a given µ and a consistent initial value t, x C F,µ+, the functions ˆF, ˆF 2 are specified along the solution x up to nonsingular transformations, cp. [2, Thm. 4.2.]. Remark 3.3. The assertions of Hypothesis 3. can be checked numerically along a numerical solution z of by computing the derivative array, e.g., by automatic differentiation [7], and SVDs for the Jacobians M µ z, N µ z [5, 29]. Similarly, the construction of the s-free formulation can be incorporated in the numerical simulation, see [29, Ch. 6]. As the Jacobians M µ z, N µ z only approximate M µ z, N µ z and the computed values µ, d, a are based on numerical rank decisions, these values only indicate the true values µ, d, a. In cases of doubt a higher value of µ should be chosen to ensure that all hidden constraints are explicitly given, see [29, p. 28], [36]. To compute a consistent initial value z Fµ+, one can either use a fixpoint iteration on the derivative array, the Gauss-Newton method [28], or decompose the variables with a time-varying transformation, cp. [3]. The latter, in particular, may be very costly, however, in some cases it may be the only way to construct the needed starting point for the remodeling procedure. We use the s-free formulation to compute the solution of 3. On [t, ˆt + Ûx Ûẋ, the s-free formulation ˆF induces the state-dependent space decomposition R n = coker ˆFẋz ker ˆFẋz. 6 To implement the decomposition 6, we pursue the projection approach considered in [3]. As we use the flow formula to study positive DAEs, cp. [3], i.e., systems for which every solution starting with a componentwise nonnegative initial value stays componentwise nonnegative for all its lifetime, we wish to avoid the change of coordinates occurring when using variable transformations. To realize the partitioning 6, we consider the Moore-Penrose projections P MP z := ˆF +ẋ ˆFẋ z, P MP z := I n P MP z 7 that are pointwise defined on [t, ˆt + Ûx Ûẋ. On [t, ˆt + Ûx Ûẋ L F,µ+, the Moore-Penrose projections P MP, PMP associated with IVP 3 satisfy the following properties. Lemma 3.. Consider the IVP 3 and let x C [t, ˆt +, Rn be its solution. Along x, there exist neighborhoods U PMP x, U PMP ẋ R n, such that P MP C [t, ˆt + U P MP x U PMP ẋ, R n n. On [t, ˆt + U P MP x U PMP ẋ L F,µ+, rankp MP z = d and P MP is independent of the chosen remodeling ˆF. 6

9 Proof. We first prove the proposed properties if P MP is evaluated on the solution x and its derivative ẋ. On t, x, ẋ, the remodeling ˆF is s-free, cp. Theorem 3.2, implying that rank ˆFẋt, x, ẋ = d and thus rankp MP t, x, ẋ = d on [t, ˆt +, cp. [2, p. 9]. Then, there exist neighborhoods U PMP x, U PMP ẋ of x, such that P MP C [t, ˆt + U P MP x U PMP ẋ, R n, cp. [3, Lemma 2.2]. Furthermore, on t, x, ẋ, the remodeling ˆF is specified up to nonsingular transformations, cp. Remark 3.2, i.e., if U R d d, U 2 C 2 Uz µ,, R a a are pointwise orthogonal matrix functions and U = diag U, U 2, then F = U T ˆF also satisfies the assertions of Theorem 3.2 and F ẋ + t, x, ẋ = ˆFẋt, x, ẋuz µ for z µ = t, x, v,..., v µ+ F F,l, cp. [3, Lemma 2.4]. On [t, ˆt +, then it follows that F + ẋ Fẋt, x, ẋ = ˆFẋt, x, ẋu T Uz µ ˆFẋt, x, ẋ = ˆF + ẋ ˆFẋt, x, ẋ, implying that the Moore-Penrose projections provided by ˆF and F agree on t, x, ẋ. As the remodeling ˆF C 2 [t, ˆt + Ûx Ûẋ, Rn is s-free and regular on [t, ˆt + Ûx Ûẋ L F,µ+, cp. Theorem 3.2, it yields a s-free formulation for every IVP 3 with initial condition t, x [t, ˆt + Ûx C F,µ+. Repeating the given arguments for the solution x associated with an initial condition t, x [t, ˆt + U P MP x C F,µ+, we have proved that the given assertions are satisfied pointwise on [t, ˆt + U P MP x U PMP ẋ L F,µ+. For every z [t, ˆt + Ûx Ûẋ, the projections P MP, PMP induce a variable decomposition P MP zx coker ˆFẋz and PMP zx ker ˆFẋz for x R n. For the solution x of 3, we consider the space decomposition 6 along t, x, ẋ and set x d := P MP t, x, ẋx, x a := P MP t, x, ẋx. 8 Solving the s-free formulation 5 for ẋ d, x a, we obtain a differential equation for x d, while the components x a are fixed algebraically. Theorem 3.3. Consider the IVP 3. Let ˆF C 2 [t, ˆt + Ûx Ûẋ, Rn be an s-free remodeling and P MP C [t, ˆt + U P MP x U PMP ẋ, R n n the associated Moore-Penrose projection. Let x C [t, ˆt +, Rn be the solution of 3 and let x d, x a be given by 8 with z = t, x, ẋ. On I, via the components x d, x a, x is the unique solution of ẋ d = h MP t, x d, x d t = P MP z x, 9a x a = g MP t, x d. 9b The functions g MP C [t, ˆt + Ux a, Ux d and h MP C [t, ˆt + Ux d, R n are uniquely defined by 3 as the implicit solution of ˆF t, xd + x a, h MP t, x d + ġ MP t, x d, =, ˆF 2 t, xd + g MP t, x d =. 2a 2b Proof. Let x C [t, ˆt +, Rn solve 3. First, we show that there exists t t, t +, such that, on I := [t, t, x d, x a solve ẋ d = h MP, t, x d, x d t = P MP z x, 2a x a = g MP, t, x d, 2b 7

10 where g MP, C I Ux d,, Ux a, and h MP, C I Ux d,, R n are locally defined on neighborhoods of x d,, x a, as implicit solutions of 2. Exploiting the uniqueness and smoothness of the s-free formulation and its Moore-Penrose projection, we can smoothly extend g MP,, h MP, to functions defined on the full interval [t, ˆt +. To solve the algebraic equation 5b for the components x a, we show that along x, ˆF2, P MP satisfy the assertions of the projection-based Implicit Function Theorem, cp. [3, Thm. 3.], with Q = I a, i.e., ˆF2,x PMP + ˆF2,x PMP z = P MP z, 22a ˆF2,x PMP ˆF2,x PMP +z = Ia 22b is satisfied pointwise on [t, ˆt + U P MP x U PMP ẋ L F,µ+. Since rankp MP = d on [t, ˆt + U P MP x U PMP ẋ L F,µ+, cp. Lemma 3., there exist neighborhoods Ũx, Ũẋ and a pointwise orthogonal function T = [T, T 2 ] C [t, ˆt + Ũx Ũẋ, Rn n with spant z = coker ˆFẋz, spant 2 z = ker ˆFẋz, such that ˆF2,x PMP [ ] z = ˆFx,2 T 2 T T z, cp. [3, Lemma 2.2]. To compute ˆF 2,x P MP +, we show that ˆF x,2 T 2 is pointwise nonsingular. As ˆF C,d,a,reg [t, ˆt + Ûx Ûẋ L µ+, R n, on [t, ˆt + Ûx Ûẋ L µ+, Hypothesis 3. implies that rank ˆF,ẋ z = d, rank ˆF 2,x z = n d and ker ˆF,ẋ z ker ˆF 2,x z = {}. Hence, R n \ coker ˆF,ẋ z coker ˆF 2,x z = {}, implying that, on [t, ˆt + Ûx Ûẋ L µ+, there exists a partitioning R n = coker ˆF,ẋ z coker ˆF 2,x z. With ker ˆF,ẋ z coker ˆF 2,x z, it follows that ˆF 2,x ker ˆF,ẋ z is pointwise nonsingular on [t, ˆt + Ûx Ûẋ L µ+. By the choice of T 2, then the Moore-Penrose inverse is given by [ ] ˆF2,x PMP +z = T z ˆF x,2 T 2, 23 z cp. [3, Lemma 2.3]. Hence, condition 22 is satisfied pointwise on t, x, ẋ. Repeating these arguments on [t, ˆt + U P MP x U PMP ẋ L F,µ+, we have proved the assertion. As the remodeling ˆF C 2 [t, ˆt + Ûx Ûẋ, Rn yields a s-free formulation for every IVP 3 with initial condition t, x [t, ˆt + Ûx C F,µ+, we can repeat the given arguments for every solution x associated with an initial conditions t, x [t, ˆt + U P MP x C F,µ+. This proves that condition 22 is satisfied pointwise on [t, ˆt + U P MP x U PMP ẋ L F,µ+. With this observation, we can solve the algebraic equation 5b for the components x a using the projection-based Implicit Function Theorem, cp. [3]. Setting y d := P MP zy, y a := P MP zy for y R n and x d, := P MP z x and x a, := P MP z x, where z = t, x, ẋ, there exist neighborhoods I [t, ˆt +, Ux d,, Ux a, R n and a function g MP, C I Ux d,, Ux a,, such that t, y solves 5b if and only if t, y d I Ux d, and y a = g MP, t, y d. Choosing I sufficiently small such that P MP zx Ux d, on I, then x a solves 2b on I. To solve the differential equation 5a for the derivatives ẋ d, we again use the projection-based Implicit Function Theorem modified for the application to implicit differential equations, cp. [2, Lem. 3..3]. Due to the properties of the Moore-Penrose inverse, ˆFẋP MP z = ˆFẋz is satisfied pointwise on z = t, x, ẋ, implying that ˆF,ẋ P MP z = ˆF,ẋ z as ˆFẋ = [ ˆF T,ẋ, ]T. Then, ˆF,ẋ P MP + ˆF,ẋ P MP z = PMP z, 24 ˆF,ẋ P MP ˆF,ẋ P MP +z = Id, 25 8

11 is satisfied pointwise on [t, ˆt + U P MP x U PMP ẋ L F,µ+ and by [2, Lem. 3..3], choosing I sufficiently small, there exist neighborhoods Uẋ d,, Uẋ a, R n and a function h C I Ux d, Ux a, Uẋ a,, Uẋ d,, such that y C I, R n solves 5a on I if and only if the components y d := P MP zy, y a := P MP zy satisfy t, y d, y a, ẏ a I Ux d, Ux a, Uẋ a, and ẋ d = ht, x d, x a, ẋ a, where the function h solves ˆF t, xd + x a, ht, x d, x a, ẋ a + ẋ a =. As ˆFẋP MP z = on [t, ˆt + Ûx Ûẋ L F,µ+, neither ˆF nor the implicit function h depend on the particular value of ẋ a and we set ht, x d, x a = ht, x d, x a, ẋ a,. For the solution x, this implies that ẋ d = ht, x d, x a for t, x d, x a I Ux d, Ux a,. Replacing x a and ẋ a, using equation 2b, we get that ht, x d := h t, x d, gt, x, ġ MP t, x. Choosing I sufficiently small, such that x d, x a, ẋ a Ux d, Ux a, Uẋ a, for t I, we find that x solves 2a. Now, we show that the implicit functions g MP,, h MP, can be extended onto the full interval [t, ˆt +. We set x := xt, where I = [t, t. Then, t, x C F,µ+ and the IVP F t, x, ẋ = xt = x 26 is uniquely solvable with x C [t, t +, Rn. In particular, 26 can be remodeled along x using the ˆF C 2 [t, ˆt + Ûx Ûẋ, Rn serving as s-free remodeling of 3. Then, the Moore-Penrose projections induced by 3 and 26 as well as the differential and algebraic components x d, x a agree. However, we assume that the implicit functions solving ˆF t, x, ẋ = in the neighborhood of t, x are given by g MP, C I Ux d,, Ux a, and h MP, C I Ux d,, Uẋ d,. The domains of definition of g MP,i, h MP,i, i =, 2, are open and we can assume without loss of generality, that there exists a nonempty interval I I I such that x d t Ux d, := Ux d, Ux d, on I. On I, both g MP, and g MP, specify the components x a, implying that x a t = g MP, t, x d t = g MP, t, x d t. 27 Since g MP i C I i Ux d,i, Ux a,i for i =,, and g MP i C I Ux d,, R n in particular, the identity 27 implies that the composition { g MP, t, x d t, t [t, t m, g MP t, x d := 28 g MP, t, x a t, t [t m, t,r, satisfies g MP C I I Ux d, Ux d,, U ex, where U ex Ux a, Ux a, and t,r = sup I. Similarly, on I, the components ẋ d are equally and uniquely specified by the functions h MP, and h MP,, implying that ẋ d t = h MP, t, x d t = h MP, t, x d t. 29 Since h MP i C I i Ux d,i, R n for i =, and h MP,, h MP, C I Ux d,, R n in particular, then 29 implies that the composition { h MP, t, x d t, t [t, t m, h MP t, x d := 3 h MP, t, x d t, t [t m, t,r, 9

12 satisfies h MP C I I Ux d, Ux d,, R n. Repeating this continuation process along x, we can successively extend g MP, h MP onto [t, ˆt +. It remains to show that g MP, h MP do not depend on the choice of the s-free formulation. If ˆF t, x, ẋ =, xt = x, F t, x, ẋ =, xt = x 3 are two s-free formulations of 3 then there exist pointwise orthogonal matrix functions U R d d, U 2 C 2 Uz µ,, R a a, such that F t, x, ẋ = U T z µ ˆF t, x, ẋ, 32 where z µ = t, x, v,..., v µ+, cp. Remark 3.2. Both systems 3 can be remodeled as described above, i.e., there exist implicitly defined functions g MP, g MP C I Ux d,, Ux a, and h MP, h MP, C I Ux d,, Uẋ d, satisfying ˆF t, x d + g MP t, x d, h MP t, x d + ġ MP t, x d =, F t, x d + g MP t, x d, h MP, t, x d + g MP t, x d =. 33a 33b Along x, the Moore-Penrose projections induced by ˆF and F agree, cp. Lemma 3. and the differential and algebraic variables x d, x a coincide. Regarding relation 32 and noting that U is pointwise orthogonal, the functions h MP,, g MP solving 33b also satisfy 33a. As the functions g MP, h MP are the unique solutions of the implicit equation ˆF t, x d + g MP t, x d, ht, x d =, cp. [3, Thm. 3.] and [2, Lem. 3..3], it follows that h MP = h MP and g MP = g MP. Let x C [t, ˆt +, Rn be the solution of 3 and let 9 be constructed along x. Let y C [t, ˆt +, Rn solve 9 on [t, ˆt + via the components y d and y a. We prove that x = y on [t, ˆt +. If y a = g MP t, y d on [t, ˆt +, thenˆf2t, y = 34 on [t, ˆt + by the construction of g MP. If, in addition, ẏ d = ht, y d on [t, ˆt +, then ˆF t, y, ẏ = ˆF t, y, h MP t, y d + ġ MP t, y d. By the construction of h MP, noting that PMP zġ MP t, y d = P MP zy d + y a + ġ MP t, y d due to ẏ a = PMP zẏ a P MP zy a, this equation reads ˆF t, y, ẏ = ˆF t, y, ĥ t, y d, g MP t, y d, ġ MP t, y d, + P MP z y d, + y a, + P MP zy d + y a + ġ MP t, x d = ˆF t, y, ĥ t, y d, g MP t, y d, ġ MP t, y d, + P MP z y d, + y a, + P MP zġ MP t, y d. Using that ẏ a = P MP zẏ a P MP zy a and rangep MP z = ker ˆFẋz, we find that ˆFẋa t, y, ẏ = ˆFẋP MP t, y, ẏ =. As ˆFẋ = [ ˆF,ẋ T, ]T, it follows that ˆF is independent of ẋ a. By the definition of ĥ, then = ˆF t, y, ĥ t, y d, g MP t, y d, ġ MP t, x d, + P MP z x d, + x a, + P MP z ġ MP t, y d = ˆF t, y, ẏ. 35

13 In combination, 34 and 35 imply that y solves 5 on [t, ˆt +. As F Cµ+2 µ,d,a,reg D, Rn n and t, x C F,µ+, then y solves the original problem 3 and since the solution is unique, it follows that x = y on [t, ˆt +. We call 9 and the functions h MP, g MP the Moore-Penrose remodeling of the IVP 3. Remark 3.4. In Theorem 3.3, we decouple the differential and algebraic variables using that the differential and algebraic equations 5a, 5b are explicitly given in the s-free formulation. To remodel a general s-free DAE F t, x, ẋ =, 36 we can filter out the differential and algebraic equations using the Moore-Penrose projections Q MP z = FẋzF ẋ + z, Q MP = I n Q MP. If 36 is s-free, then and only then Q MP ˆFx PMP + Q MP ˆFx PMP z = P MP z, 37a Q MP ˆFx PMP Q MP ˆFx PMP +z = Q MP z 37b is satisfied pointwise on F and we can remodel the DAE 36 as in Theorem 3.3, solving Q MP F t, x, ẋ =, QF t, x, ẋ = for ẋ d, x a, respectively, cp. [2, Thm. 4.3.]. Condition 22 is satisfied if and only if the matrix ˆF x,2 T 2 z is nonsingular, i.e., if and only if the remodeling ˆF is s-free, cp. Hypothesis 3.. Thus, condition 22 allows to check if the computed remodeling ˆF indeed is s-free. Similarly, the DAE 36 is s-free if and only if S 2 ˆFx,2 T 2 z, where PMP = T 2T2 T and Q MP = S 2S2 T, is nonsingular. Solving the decoupled system 9, we find that the differential components x d are evolved by the flow Φ hmp induced by the function h MP, while the algebraic components x a are coupled to this evolution by the function g MP. In combination, we obtain an additively composed solution formula of 3 consisting of a dynamic part related with Φ hmp and a constrained part specified by g MP. Lemma 3.2. Consider the IVP 3 and let x C [t, ˆt +, Rn be its solution. Set z := t, x, ẋt. On [t, ˆt +, the solution x is given by xt = Φ t h MP t, P MP z x + gmp t, Φ t hmp t, P MP z x, 38 where P C [t, ˆt + U P MP x U PMP ẋ, R n n, g MP C [t, ˆt + Ux a, Ux d and h MP C [t, ˆt + Ux d, R n are the Moore-Penrose projection and remodeling induced by 3 and Φ hmp is the flow associated with h MP. Proof. Along the solution x, we can decouple the IVP 3 as decoupled system 9 for the components x d, x a. With h MP C [t, ˆt + Ux d, R n, the ODE 9a induces the flow Φ hmp, such that, on [t, ˆt +, xdt = Φ t h MP t, P MP z x. Inserting 39a into the algebraic equation 9b, we obtain that x a t = g MP t, Φ t h MP t, P MP z x. 39a 39b With x = x d + x a, we have proven the representation 38. Noting that P MP, g MP, h MP are C - functions, we have verified that the representation 38 is continuously differentiable on [t, ˆt +.

14 The solution formula 38 is defined for every consistent initial value and on the full interval of existence of the associated solution. Thus, it gives rise to an explicit representation of the flow Φ F. Theorem 3.4. Consider the DAE 3a. For every t, x C F,µ+ and z = t, x, v L µ+, on [t, ˆt + the flow Φ F is given by Φ t F t, x = Φ t h MP t, P MP z x + gmp t, Φ t hmp t, P MP z x, 4 where P C [t, ˆt + U P MP x U PMP ẋ, R n n, g MP C [t, ˆt + Ux a, Ux d and h MP C [t, ˆt + Ux d, R n are the Moore-Penrose projection and remodeling induced by 3 and Φ hmp is the flow associated with h MP. Furthermore, the flow Φ F satisfies P MP t, Φ t F t, x, Φ t F t, x Φ t F t, x = Φ t h MP t, P MP z x, 4a P MP t, Φ t F t, x, Φ t F t, x Φ t F t, x = g MP t, Φ t hmp t, P MP z x. 4b Proof. By definition, the function Φ F t, x agrees with the unique solution of the IVP 3 for every t, x C F,µ+, cp. Corollary 3.. Using formula 38, we have verified the representation 4. As a consequence of the construction of h MP, on [t, ˆt + the associated flow Φ h MP satisfies Φ t h MP t, P MP z x coker ˆFẋ t, Φ t F t, x, Φ t F t, x, cp. [2, Theorem 3.2.2]. Hence, P MP t, Φ t F t, x, Φ t F t, x Φ t h MP t, x =, 42 on [t, ˆt +. Similarly, g MP t, x d ker ˆFẋt, x, ẋ, cp. [3, Thm. 3.], implying that on [t, ˆt + P MP t, Φ t F t, x, Φ t F t, x g MP t, Φ t h t, P MP z x =, 43. From 42 and 43, we conclude that Φ F t, x satisfies 4 on [t, ˆt +. The flow formula 4 reflects the two flavors of a DAE: Parts of the solution are evolved by a flow, while the other part is coupled to this evolution via an algebraic relation. For the overall solution, this results in a dynamic evolution which is constrained to a flat subset in the state space. Locally, this constraint can be represented as a time-varying manifold. Lemma 3.3. Consider the IVP 3 and let ˆF = [ ˆF T, ˆF T 2 ]T C 2 [t, ˆt + Ûx Ûẋ, Rn be an s-free remodeling. For every t, x C F,µ+, the following assertions are true. i The set M F t, x := dimm F t, x = d. ˆF 2 is a time-varying, embedded C 2 -submanifold with ii The set M F t, x is independent of the chosen remodeling ˆF. iii The set of consistent initial values satisfies [t, ˆt + Ûx C F,µ+ MF t, x. iv The flow Φ F satisfies Φ t F t, x M F t, x t for t [t, ˆt +. 2

15 Proof. i, iv For the remodeling ˆF C 2 [t, ˆt + Ûx Ûẋ, Rn, we have that rankd ˆF 2 = rank ˆF 2,x = n d on C ˆF2 I Ux as ˆF is s-free, cp. Hypothesis 3.. Hence, the algebraic solution set ˆF 2 a time-varying submanifold embedded in R n with dim ˆF 2 = d, cp. Lemma 2.. As the function Φ F t, x solves the s-free remodeling, it follows that Φ t F t, x M F t, x t on [t, ˆt +. ii Given µ and an initial value t, x C F,µ+, the remodeling ˆF 2 is specified up to nonsingular transformations of the matrix Z 2, cp. Remark 3.2. These transformations do not alter the solution set ˆF 2, hence M F t, x is independent of the choice of ˆF. iii The remodeling ˆF serves as s-free remodeling for every initial condition t, x [t, ˆt + Ûx Ûẋ L F,µ+. Hence, [t, ˆt + Ûx C F,µ+ M F t, x. The projection properties 4 allow to access the differential and algebraic solution components x d and x a by projecting with P MP t, x, ẋ and PMP t, x, ẋ, respectively. Analyzing system properties like stability or positivity, this allows to specify the condition on the differential and algebraic solution components, cp. [2, ch. 5]. The representation 4 is uniquely defined by 3a as the Moore-Penrose projection P MP and remodeling g MP, h MP are independent of the chosen remodeling ˆF. Remark 3.5. The non-autonomous DAE can be autonomized by setting F aut z, ż =, [ ] ṫ F aut z, ż :=, z := ft, x, ẋ [ ] t, ż := x 44a [ṫ ], 44b ẋ cp. [29, p. 59]. If F C µ+ µ,d,a D, Rn, then F aut C µ+ µ,d+,a Ω x Ωẋ, R n+, cp. [29, p. 59], and the flows Φ F and Φ Faut associated with F and F aut are related by [ ] Φ t t F aut t = Φ t F t. 45, x 3.2 Explicit remodeling using constant Moore-Penrose projections The decoupled system 9 is constructed by decomposing the variables along the solution x, yielding a smooth decomposition of the differential and algebraic components on the full interval of existence. To explicitly compute the remodeling 9 and the flow formula 4, however, we need to consider the explicit variable decomposition x d = P MP t, x, v x, x a = P MP t, x, v x 46 induced by evaluating the Moore-Penrose projections in a consistent initialization t, x, v L µ+. Lemma 3.4. Consider the IVP 3. Let ˆF C 2 [t, ˆt + Ûx Ûẋ, Rn be an s-free remodeling and P MP C [t, ˆt + U P MP x U PMP ẋ, R n the associated Moore-Penrose projection. Let v R n be such that z := t, x, v L F,µ+ and consider the variable decomposition 46. Then, there exists t t, t +, such that, on I := [t, t, the function x C I, R n solves 3 if and only if its components x d, x a solve the decoupled system ẋ d = h MP, t, x d, x d t = P MP z x, 47a x a = g MP, t, x d. 47b 3

16 The functions g MP, C I Ux d,, Ux a, and h MP, C I Ux d,, R n are uniquely defined by 3 as the implicit solutions of ˆF t, xd + x a, h MP, t, x d + ġ MP, t, x d, =, ˆF 2 t, xd + g MP, t, x d = 48a 48b and satisfy g MP, t, x d kerp MP z and h MP, t, x d kerp MP z + P MP P z. Proof. The assertion follows using similar arguments proving Theorem 3.3. If z := t, x, v [t, ˆt + U P MP x U PMP ẋ L F,µ+, we have shown that P z, ˆF 2 z satisfy condition 22. By the projection-based Implicit Function Theorem [3, Thm. 3.], then there exist neighborhoods I, Ux d,, Ux a, and a function g MP, C I Ux d,, Ux a, such that t, x solves ˆF 2 t, x = if and only if t, x d Ĩ Ũx and x a = g MP, t, x d, where x d = P MP z x and x a = PMP z x. In particular, g MP, t, x d kerp MP z. Similarly, condition 24 is satisfied in z, and following the steps in the proof of Theorem 3.3, we can construct a function h MP, C I Ux d,, R n such that t, x, ẋ solves ˆF t, x, ẋ = if and only if ẋ d = h MP, t, x d. In particular, h MP, t, x d kerpmp z + P MP P z f. Hence, for t t, t + sufficiently small, the function x C I, R n solves 3 on I := [t, t if and only if its components x d, x a solve the decoupled system 47. Using the local in time Moore-Penrose remodeling 47, we can explicitly compute the solution x of 3 and hence the flow by proceeding piecewise along x. Corollary 3.2. Consider the DAE 3a. For every t, x C F,µ+ and z := t, x, v L F,µ+, there exists t t, t +, such that, on I := [t, t, Φ t F t, x = Φ t h MP, t, P MP z x + gmp, t, Φ t hmp, t, P MP z x, 49 where g MP, C I Ux d,, Ux a, and h MP, C I Ux d,, R n are induced by P MP z and Φ hmp, is the flow associated with h MP,. On I, the flow Φ F satisfies P MP z Φ t F t, x = Φ t h MP, t, P MP z x, P MP z Φ t F t, x = g MP, t, Φ t h MP, t, P MP z x. 5a 5b In a numerical solution, the projection properties 5 allow to check the consistency of the numerical solution x by projecting with P MP z and P MP z, respectively, and checkin the relation x,n,a = gt N, x,n,d. As the projections P MP z, P MP z are constant, this test is independent of the numerical solution. 3.3 Linear differential-algebraic equations For linear systems F E,A,b t, x, ẋ := Etẋ Atx bt = 5 with E, A C l I, R n n and b C l I, R n, the derivative array 7 is linear in the state z l and the block matrices M l, N l are defined globally on I R n independent of a particular initial value 4

17 t, x, cp. [29, p. 8]. For sufficiently smooth functions F E,A,b, the assertions of Hypothesis 3. are satisfied globally on R n, cp. [29, p. 8], i.e., F E,A,b Cµ,d,a,reg l D, Rn if F E,A,b C l D, R n, where D = I R n R n. If F E,A,b C µ+ µ,d,a,reg D, Rn, then 5 is uniquely solvable for every initial value t, x C E,A,b,µ, where C E,A,b,µ := C F,µ, and the solution is defined on the full interval I. The s-free formulation ˆFÊ, Â,ˆb of 5 is globally defined on D and independent of the initial value, cp. [29, p. 9, ]. A function x C I, R n solves 5 on I if and only if x solves ] [Ê ˆFÊ, Â,ˆb t, x, ẋ = ẋ ] [Â Â 2 ] [ˆb x =. 52 ˆb2 The Jacobian ˆFẋ = Ê is independent of the state x and hence the Moore-Penrose projection P MP t = Ê+ Êt is globally defined on D. The remodeling h MP, g MP is explicitly given as affine linear transformations that are globally defined on I R n. Theorem 3.5. Consider the DAE 5 with F E,A,b C µ+ µ,d,a,reg D, Rn. Let ˆFÊ, Â,ˆb C,d,a,reg D, Rn be an s-free remodeling and P MP C I, R n n the associated Moore-Penrose projection. A function x C I, R n solves 5 with xt = x, t, x C E,A,b,µ, if and only if the components x d, x a solve ẋ d = h MP t, x, x a = g MP t, x d, 53a 53b where h MP t, x := D d tx + b d t and g MP t, x d := D a tx d b a t with D d := Ê+ Â + P MP P MP, b d := Ê+ˆb Ê + Â + P MP b a, 54a D a := Â2P MP + Â 2 P MP, b a := Â2P MP +ˆb. 54b In particular, h MP CI cokerê, Rn and g MP C I cokerê, kerê. Exploiting the linearity, we can specify the solution formula 38 and construct a globally defined representation of the flow Φ E,A,b := Φ FE,A,b. Theorem 3.6. Consider the DAE 5 with F E,A,b C µ+ µ,d,a,reg D, Rn. Let P MP and D d, D a, b d, b a be the associated Moore-Penrose projection and remodeling and P MP = I n D a P MP. On C E,A,b,µ I, the flow Φ E,A,b is given by t Φ t E,A,b t, x = Φ t E,At, x + Φ t E,A b d s ds ba t, 55 t with the homogeneous flow Φ t E,A t = P MP t Φ t D d P MP t. For every t I, the homogeneous flow Φ E,A possesses the semi inverse Φ t E,A t ginv = Φ t E,A t satisfying Φ t E,A tφ t E,A t = P MP t and Φ t E,A t Φ t E,A t = P MP t. Like for ODEs, the flow Φ FE,A,b is an affine linear transformation composed of the homogeneous flow Φ E,A and an inhomogeneous part induced by b. For constrained systems, however, only the parts of the initial value and the inhomogeneity lying in cokere are dynamically evolved, while the components in kere are fixed by an algebraic relation. Formula 55 generalizes 5

18 Duhamel s formula to linear constrained systems with sufficiently smooth coefficients. As the projections P MP, PMP are linear in the state x, the projection properties 4 allow to access the differential and the algebraic solution components independently of a a given solution. Thus, we can check the consistency of the dynamic and algebraic approximations x d, and x a, of a numerical solution x x exactly by projecting onto cokerê and cokerê, respectively. The semi-inverse Φ t E,A t ginv = Φ t E,A t allows to recover the initial value x from a given solution Φ t E,A t for every time t I. For linear problems, the solution manifold is a time-varying linear subspace that coincides with the set of consistent initial values C E,A,b,µ. Lemma 3.5. Consider the DAE 5 with F E,A,b C µ+ µ,d,a,reg D, Rn. The set of consistent initial values C E,A,b,µ is a time-varying, affine linear C -subspace on I and M E,A,b t, x = C E,A,b,µ for every t, x C E,A,b,µ. The function P MP, ba t, x = P MP t x b a t, is an affine linear projection onto C E,A,b,µ. Proof. An initial value t, x I R n is consistent if and only if ˆFÊ, Â,ˆb;2 t, x = Â2t x ˆF ˆb2 t =, cp. [29, p. ]. Hence, C E,A,b,µ =, implying that C Ê,Â,ˆb;2 E,A,b,µ = M E,A,b t, x. In particular, as ˆFÊ, Â,ˆb;2 is an affine linear function, its algebraic solution set is a time-varying, affine linear subspace on I, cp. [3, Lem. 2.6] and [2, Rem ]. By construction of the function g MP, C E,A,b,µ = ˆF further implies that t, x C Ê,Â,ˆb;2 E,A,b,µ if and only if PMP t x = D a P MP t x b a t, i.e., if and only if x = P MP t b a t. Hence, C E,A,b,µ = rangep MP, ba. Noting that P MP D a = and D a P MP = D a, cp. 54, we verify that P MP = I n D a P MP is idempotent and hence P MP, ba t, x is an affine projection onto C E,A,b,µ. Hence, we can validate the consistency of a numerical solution x using the projection P MP, ba. Remark 3.6. For constant coefficients E, A R n n and b C µ+ I, R n, the Moore-Penrose remodeling is given by D d := Ê+ ÂP MP, D a := Â2P MP + Â 2, b d := Ê+ ˆb Âb a, b a := Â2P MP +ˆb2. The homogeneous flow Φ E,A reads Φ t E,A t := P MP e D dt t P MP. 3.4 Linearization of the flow To study properties of the DAE 3a like invariant sets, stability or positivity, we need the linearization of its solutions. For the ODE 2, the linearization of a solution x in a point t, x is explicitly given by the function f, i.e., xt = x + t t ft, x + Ot t 2 if f C I Ω, R n. For the DAE 3a, the derivative ẋ of a solution is specified implicitly only. Having a flow Φ F, however, that coincides with the solutions, we can define a vector field T F : C F,µ+ R n that assigns the derivative Φ t F t, x to every t, x C F,µ+, i.e., T F t, x := Φ t F t, x. For F C µ+2 D, R n, the linearization of the solution in t, x is given by xt = x + t t T F t, x + Ot t 2. We call T F t, x the tangent field of Φ F. Using the explicit representation 47 of the flow Φ F, we can explicitly compute T F. 6

19 Lemma 3.6. Consider the DAE 3a with flow Φ F. For t, x C F,µ+, let v R n be such that z = t, x, v L F,µ+. Let P MP C [t, ˆt + U P MP x U PMP ẋ, R n be the Moore-Penrose projection induced by 3 and h MP, C I Ux d,, R n, g MP, C I Ux d,, Ux a, the Moore-Penrose remodeling obtained using P MP z. Then, the tangent field T F is given by T F t, x = h MP, t, P MP z x + ġmp, t, P MP z x. 56 Proof. For t, x C F,µ+ and v R n such that z = t, x, v L F,µ+, there exists an interval I on which the solution is represented using the Moore-Penrose remodeling g MP, C I Ux d,, Ux a, and h MP, C I Ux d,, R n induced by P MP z, cp. Lemma 3.2. Considering the time derivative of formula 49 and evaluating in t = t, we obtain formula 56. For linear problems, using the flow formula 55, formula 56 can be specified as affine linear transformation. Corollary 3.3. Consider the DAE 5 with F E,A,b C µ+ µ,d,a,reg D, Rn. Let P and D d, D a, b d, b a be the Moore-Penrose projection and remodeling induced by 5 and P MP = I n D a P MP. On I R n, the tangent field of 5 is given by T E,A,b t, x = T E,A t x + P MP b d ḃa t, 57 where the homogeneous tangent field T E,A is given by T E,A t = ṖMP + P MP D d P MP t. Remark 3.7. For constant coefficients E, A R n n and b C µ+ I, R n, the tangent field is given by T E,A,b t, x = T E,A tx + P MP b d ḃa t on CE,A,b,µ, with the homogeneous tangent field T E,A t = P MP Ê + ÂP MP t. If F t, x, ẋ = ẋ ft, x with f C Lip loc I Ω x, R n, then Φ F = Φ f and Φ t F t, x = ft, x on I Ω. Thus, the tangent field T F coincides with the system function f. We use the tangent field to study properties like flow invariance, stability and positivity of the DAE 3a in [2]. 4 Examples We illustrate the remodeling by the Moore-Penrose projection and the computation of the flow for a linear and a nonlinear DAE. For details of the computation, see [2, Ex. 4.5., Ex ]. Example 4.. We consider F E,A,b C I, R n with E =, A = t t , b = I :=, and b, b 2, b 3 R. We first show that F E,A,b is s-free and already in the remodeled form 52 with Ê = e T, Â = e T A, Â 2 = [e 2, e 3 ] T A and ˆf = e T b, ˆf2 = [e 2, e 3 ] T b. On I, b b 2 b 3, 7

20 ranke = and rank[e 2, e 3 ] T A = 2, so it remains to prove condition 22, cp. Remark 3.4. With E + =. the Moore-Penrose projections are given by P MP =, P MP =. 58 Then, [ Â 2 PMP ] = A 22 A 22, Â2PMP + = and we verify by direct computation that P MP and A satisfy condition 22. Now, we compute the Moore-Penrose remodeling g MP and h MP with D a = D d =, b a = The variables are partitioned according to x d = x +x 2 x +x 2 b 2 b 2 b 3, x a =,, b d = b + b 2 2 3/2 x x 2 x x 2 x In conclusion, the Moore-Penrose remodeling 53 consists of the ODE ẋ d, ẋ d,2 = ẋ d, x d, x d,2 x d,3 + b + and the algebraic relation x a, x d, x a,2 = x d,2 x a,3 x d,3 b 2 2 3/2 b 2 b 2 b 3. 6a 6b 8

21 To compute the flow Φ Dd,b d, we note that [e, e 2, ] T P MP = [e, e 2, ] T P MP and [,, e 3 ] T E + A+ P MP = on I, such that we can simplify the system matrix according to D d = P MP E + A + P MP P MP = E + A + P MP P MP. Thus, ODE 6a according to D d = P MP E + A + P MP P MP = E + A + P 2 MP P MP = 2 Noting that t t 2s+2 ds = 2 ln[ t +2 ] and t t 2s+2s+ ds = 2 ln[ t +2 t + ], cp. [38, p. 96, 98], we get that exp t t 2s+2 ds = t +2 and exp t t 2s+2s+ ds = t +2 t + and the flow Φ D d restricted to C E,A is given by Φ t D d P MP t = t +2 P MP t. t +2 t + With Φ t E,A t = P MP tφ t D d P t, then the homogeneous flow of the DAE is given by Φ t E,At = t + t /2 3/2 t +2 t + t /2 3/2 t +2 t + t +2 3/2 + t +2 t + t /2 3/2 t + t +2 t + t /2 3/2 t + t +2 t + + t +2 3/2 t + t +2 From this formula, we can compute inhomogeneous flow according to Φ t E,A,f t, x = Φ t E,At x + t t Φ t E,Ab d s ds b a t. To compute the set of consistent initial values C E,A,b,µ, we compute the projection and find that, cp. Lemma 3.5, C E,A,b,µ = x R 3 P MP =, x, t +x 2, t +2 t +x, t +x 2, t +2 t +x, +x 2, t + + x,3 = b 2 b 2 Example 4.2. Consider F C D, R 3, D =, R 3 R 3, with F t, x, ẋ = x 2 + ẋ +ẋ 2 x +x 2 x x x 2 3 x +x 2 b 3. + x +x

22 The algebraic solution set is given by F = { t, x, v D x = t + x 2 + 2t + 2, x 3 = v +v 2 = 2 2t + 2 x2 x x 2 + 2t +, } 2 2t +. To prove that F is s-free and already in the remodeled form 5 with ˆF = e T F, ˆF 2 = [e 2, e 3 ] T F, we note that the Jacobians Fẋt, x, ẋ = F x t, x, ẋ =, x +x x +x x x 2 x x x 3 satisfy rank ˆF,ẋ t, x, ẋ =, rank ˆF 2,x t, x, ẋ = 2 on F and the solvability condition 22, cp. Remark 3.4. Comparing Fẋ and the matrix E of Example 4., we find that the Moore- Penrose projections induced by F are given by 58. Noting that ˆF 2,x P MP = [ x x 2 x x 2 ] 2 2, ˆF 2,x P c + = 2x 3, x x 2 x x 2 x +x 2 2x 3 x x 2 2x 3 we verify that ˆF2,x, P c satisfy condition 22. The variables x d, x a are given as in 59. To compute the Moore-Penrose remodeling g MP, we solve [ ] x 2 ˆF 2 t, x d + x a, ẋ d + ẋ a = a 2 x 2 = a3 x d,2, for x a and noting that x a2 = t + x a, we obtain that g MP = {x R 3 x d,2 > } and U gmp C I U gmp, R 3, where g MP t, x d = [ 2 2t + + xd,2 ] T. Then, the set of consistent initial values is given by C F = { t, x I R 3 x = t + x 2 + 2t + 2, x 3 = To compute the function h MP, we solve ˆF t, x d + x a, ẋ d + ẋ a = ẋ d,2 + x 2 d,2 = + x 2 + } 2t +. for ẋ d, and noting that x d = t + x d2, we get that h MP C I R 3, R 3 with h MP t, x d = [ ] T x2 d, + x d, 2 x 2 d,2. 2

23 On I U PMP, U PMP = {x R 3 x +x 2 > }, then the Moore-Penrose remodeling given by On I R 3, the flow Φ hmp ẋ d = x 2 d, + x d, 2 x 2 d,2 is given by Φ t h MP t, x d = and with x d,, = t + x d,2, and x d,2, = Φ t F t, x = 2,, x a = 2t xd,2 2t 2 + t +2 2t 2 t + + x d,,, t t + x d,2, t +x, +x 2, t +2, we get the DAE flow t t +2 2t +. 2t 2 + t +2 2t 2 t + + t + t +x, +x 2, t +x, +x 2, + t t + t t + t +x, +x 2, t +2 2

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