Surveys in Differential-Algebraic Equations I pp 137-172 | Cite as

# Solution Concepts for Linear DAEs: A Survey

## Abstract

This survey aims at giving a comprehensive overview of the solution theory of linear differential-algebraic equations (DAEs). For classical solutions a complete solution characterization is presented including explicit solution formulas similar to the ones known for linear ordinary differential equations (ODEs). The problem of inconsistent initial values is treated and different approaches are discussed. In particular, the common Laplace-transform approach is discussed in the light of more recent distributional solution frameworks.

### Keywords

Differential algebraic equations Descriptor systems Distributional solution theory Laplace transform### Mathematics Subject Classification (2010)

34A09 34A12 34A05 34A25## 1 Introduction

*differential-algebraic equation*(DAE). However, this survey will not treat this most general system description but it will consider its linear counterpart

*E*is square and invertible, the DAE is equivalent to an ordinary differential equation (ODE) of the form

*A*is essential to grasp the whole of the possibilities of solution behaviors. Some features of the solutions of an ODE are highlighted:

*Existence.*For every initial condition

*x*(0)=*x*_{0}, \(x_{0}\in \mathbb{R}^{n}\), and each (locally integrable) inhomogeneity*f*there exists a solution.*Uniqueness.*For any fixed inhomogeneity

*f*the initial value*x*(0) uniquely determines the whole solution; in fact each single value*x*(*t*), \(t\in\mathbb{R}\), determines the solution on the whole time axis.*Inhomogeneity.*The solution is always one degree “smoother” then the inhomogeneity, i.e. if

*f*is differentiable then*x*is at least twice differentiable, in particular, non-smoothness of*f*does not prevent the ODE of having a solution (at least in the sense of Carathéodory).

*regular*DAEs; however, for general DAEs none of these three properties have to hold anymore as the following example shows.

### Example 1.1

*x*

_{2}=−

*f*

_{2}, \(x_{1}=\dot{x}_{2}-f_{1}=-\dot{f}_{2}-f_{1}\) and

*f*

_{3}=0. In particular, not for all initial values or all inhomogeneities there exists a solution. Furthermore,

*x*

_{3}is not restricted at all, hence uniqueness of solutions is not present. Finally,

*x*

_{1}contains the derivative of the inhomogeneity so that the solution is “less smooth” than the inhomogeneity which could lead to non-existence of solutions if the inhomogeneities is not sufficiently smooth.

The aim of this survey is twofold: (1) to present a fairly complete classical solution theory for the DAE (1.1) also for the singular case; (2) to discuss the approaches to treat inconsistent initial values and the corresponding distributional solution concepts. In particular, a rigorous discussion of the so-called Laplace-transform approach to treat inconsistent initial values and its connection to distributional solution concepts is carried out. This is a major difference with the already available survey by Lewis [32], which is not so much concerned with distributional solutions. The focus of Lewis’ survey is more on system theoretic topics like controllability, observability, stability and feedback, which are not treated here.

This survey is structured as follows. In Sect. 2 classical (i.e. differentiable) solutions of (1.1) are studied. It is shown how the Weierstraß and Kronecker canonical form of the matrix pencil \(sE-A\in\mathbb{R}^{m\times n}[s]\) can be used to fully characterize the solutions. Solution formulas which do not need the complete knowledge of the canonical forms will be presented, too. A short overview over the situation for time-varying DAEs is given as well. Inconsistent initial values are the most discussed topics concerning DAEs and different arguments how to treat them have been proposed. One common approach to treat inconsistent values is the application of the Laplace transform to (1.1); the details are explained in Sect. 4. However, the latter approach led to much confusion and therefore a time-domain approach based on distributional solutions was developed and studied by a number of authors, see Sect. 5.

## 2 Classical Solutions

In this section classical solutions of the DAE (1.1) are considered:

### Definition 2.1

(Classical solution)

A classical solution of the DAE (1.1) is any differential function Open image in new window such that \(E\dot{x}(t)=Ax(t)+f(t)\) holds for all \(t\in\mathbb{R}\).

It will turn out that existence of a classical solution in general also depends on the smoothness properties of the inhomogeneity; if not mentioned otherwise it will be assumed therefore in the following that the inhomogeneity *f* is sufficiently smooth, e.g. by assuming that *f* is in fact smooth (i.e. arbitrarily often differentiable).

### 2.1 The Kronecker and Weierstraß Canonical Forms

*E*

_{1},

*A*

_{1}) and (

*E*

_{2},

*A*

_{2}) (via the transformation matrices

*S*and

*T*) the following equivalence holds:

*Kronecker canonical form*(KCF). The solution theory of the DAE (1.1) is a mere application of the KCF. In particular, he does not consider inconsistent initial values or non-smooth inhomogeneities. The existence and representation of the KCF is formulated with the following result.

### Theorem 2.1

(Kronecker canonical form [21, 28])

*For every matrix pencil*\(sE-A\in\mathbb{R}^{m\times n}[s]\)

*there exist invertible matrices*\(S\in\mathbb{C}^{m\times m}\)

*and*\(T\in\mathbb {C}^{n\times n}\)

*such that*,

*for*\(a,b,c,d\in\mathbb{N}\)

*and*

*ε*

_{1},…,

*ε*

_{a},

*ρ*

_{1},…,

*ρ*

_{b},

*σ*

_{1},…,

*σ*

_{c}, \(\eta_{1},\ldots,\eta_{d}\in\mathbb{N}\),

*where*

*The block*-

*diagonal form*(2.1)

*is unique up to reordering of the blocks and is called Kronecker canonical form*(

*KCF*)

*of the matrix pencil*(

*sE*−

*A*).

Note that in the KCF Open image in new window-blocks with *ε*=0 and Open image in new window-blocks with *η*=0 are possible, which results in zero columns (for *ε*=0) and/or zero rows (for *η*=0) in the KCF, see the following example.

### Example 2.1

(KCF of Example 1.1)

- Open image in new window-
**block** - If
*ε*=0 then this simply means that the corresponding variable does not appear in the equations and is therefore free and can be chosen arbitrarily. For*ε*>0 consider the differential equation Open image in new window which equivalently can be written as the ODEHence for any$$\left (\begin{array}{c} \dot{x}_2\\\dot{x}_3\\\vdots\\ \dot{x}_{\varepsilon+1} \end{array} \right ) = \left [\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & & & \\ 1 & \ddots& & \\ & \ddots& \ddots& \\ & & 1 & 0 \end{array} \right ] \left (\begin{array}{c}x_2\\x_3\\\vdots\\ x_{\varepsilon+1} \end{array} \right ) + \left (\begin{array}{c} f_1 \\ f_2 \\ \ldots\\ f_\varepsilon \end{array} \right ) + \left (\begin{array}{c} 1 \\ 0 \\ \ldots\\0 \end{array} \right ) x_1. $$*x*_{1}and any inhomogeneity*f*there exist solutions for*x*_{2},*x*_{3},…,*x*_{ε+1}uniquely determined by the initial values*x*_{2}(0),…,*x*_{3}(0). In particular, for all initial values and all inhomogeneities there exist solutions which are not unique because*x*_{1}can freely be chosen. - Open image in new window-
**block** The differential equation Open image in new window is a standard linear ODE, i.e. it holds that for all initial values and all inhomogeneities a unique solution.

- Open image in new window-
**block** - Write Open image in new window, then it is easily seen that the differential operator Open image in new window is invertible with inverseIn particular for any smooth inhomogeneity the solution of the differential equation Open image in new window is uniquely given by$$ \biggl(N\frac{\mathrm{d}}{\mathrm{d}t}-I \biggr)^{-1} = -\sum_{i=0}^{\rho-1} N^i \frac{\mathrm{d}}{\mathrm{d}t}^i. $$(2.2)In particular it is not possible to specify the initial values arbitrarily—they are completely determined by the inhomogeneity.$$ x = -\sum _{i=0}^{\rho-1} N^i f^{(i)} = \left (\begin{array}{c} -f_1 \\ -f_2 - \dot{f}_1 \\ \vdots\\ -f_\rho- \dot {f}_{\rho-1} - \cdots- f^{(\rho-1)}_1 \end{array} \right ) . $$(2.3)
- Open image in new window-
**block** - If
*η*=0 then no variable is present and the equation reads 0=*f*, hence not for all inhomogeneities the overall DAE is solvable. If*η*>0 then the solution of the differential equation Open image in new window is given by (2.3) with*ρ*replaced by*η*but only if the inhomogeneity fulfillsIn particular not for all inhomogeneities and not for all initial values solutions exist. However, when solutions exist they are uniquely given by (2.3).$$f_{\eta+1} = \dot{x}_\eta= -\dot{f}_\eta- \ddot{f}_\eta- \cdots- f^{(\eta)}_1. $$

### Corollary 2.2

(Existence and uniqueness of solutions)

*The DAE* (1.1) *has a smooth solution**x**for all smooth inhomogeneities**f**if*, *and only if*, *in the KCF the*Open image in new window-*blocks are not present*. *Any solution**x**of* (1.1) *with fixed inhomogeneity**f**is uniquely determined by the initial value**x*(0) *if*, *and only if*, *in the KCF the*Open image in new window-*blocks are not present*.

The KCF without the Open image in new window and Open image in new window blocks is also called the *Weierstraß canonical form* (WCF) and can be characterized directly in terms of the original matrices. For this the notion of regularity is needed.

### Definition 2.2

(Regularity)

The matrix pencil \(sE-A\in\mathbb{R}^{m\times n}[s]\) is called regular if, and only if, *n*=*m* and det(*sE*−*A*) is not the zero polynomial. The matrix pair (*E*,*A*) and the corresponding DAE (1.1) is called regular whenever *sE*−*A* is regular.

### Theorem 2.3

(Weierstraß canonical form [49])

*The matrix pencil*\(sE-A\in\mathbb{R}^{n\times n}[s]\)

*is regular if*,

*and only if*,

*there exist invertible matrices*\(S,T\in\mathbb{C}^{n\times n}\)

*such that*

*sE*−

*A*

*is transformed into the Weierstraß canonical form*(

*WCF*)

*where*\(J\in\mathbb{C}^{n_{1}\times n_{1}}\), \(N\in\mathbb{C}^{n_{2}\times n_{2}}\),

*n*

_{1}+

*n*

_{2}=

*n*,

*are matrices in Jordan canonical form and*

*N*

*is nilpotent*.

In conclusion, if one aims at similar solution properties as for classical linear ODEs the class of regular DAEs is exactly the one to consider, see also Sects. 2.4 and 2.5. In the classical solution framework there is still a gap between ODEs and regular DAEs because (1.1) does not have solutions for all initial values and not for insufficiently smooth inhomogeneities. However, in a distributional solution framework these two missing properties can also be recaptured, see Sect. 5.

### 2.2 Solution Formulas Based on the Wong Sequences: General Case

*Wong sequences*in the following. It is easily seen that the Wong sequences are nested and terminate after finitely many steps, i.e. Bernhard [6] used the first Wong sequence in his geometrical analysis of (1.1) where the inhomogeneity has the special form

*f*=

*Bu*for some suitable matrix

*B*. Utilizing both Wong sequences Armentano [2] was able to obtain a Kronecker like form. However, his arguments are purely geometrical and it is not apparent how to characterize the solutions of (1.1) because the necessary transformation matrices are not given explicitly. This problem was resolved recently in [4], where the following connection between the Wong sequences and a quasi-Kronecker form was established.

### Theorem 2.4

(Quasi Kronecker form (QKF) [4])

*Consider the DAE*(1.1)

*and the corresponding limits*Open image in new window

*and*Open image in new window

*of the Wong sequences*(2.4).

*Choose any invertible matrices*\([P_{1},R_{1},Q_{1}]\in\mathbb{R}^{n\times n}\)

*and*\([P_{2},R_{2},Q_{2}]\in\mathbb{R}^{m\times m}\)

*such that*

*then*

*T*=[

*P*

_{1},

*R*

_{1},

*Q*

_{1}],

*S*=[

*P*

_{2},

*R*

_{2},

*Q*

_{2}]

^{−1}

*put the matrix pencil*

*sE*−

*A*

*into quasi*-

*Kronecker triangular form*(

*QKTF*):

*where*

*λE*

_{P}−

*A*

_{P}

*has full row rank for all*\(\lambda\in\mathbb {C}\cup \{ \infty\}\),

*sE*

_{R}−

*A*

_{R}

*is regular*,

*and*

*λE*

_{Q}−

*A*

_{Q}

*has full column rank for all*\(\lambda\in\mathbb{C}\cup\{\infty\}\).

*Furthermore*,

*the following generalized Sylvester equations are solvable*:

*and any solutions*

*F*

_{1},

*F*

_{2},

*G*

_{1},

*G*

_{2},

*H*

_{1},

*H*

_{2}

*yield a quasi*-

*Kronecker form*(

*QKF*)

*via*

*where the diagonal block entries are the same as in*(2.5).

*sE*_{P}−*A*_{P}: Due to the full rank assumption there exists a unimodular^{1}matrix [*M*_{P}(*s*),*K*_{P}(*s*)] such thatsee e.g. [4, Lem. 3.1]. The solutions$$ (sE_P-A_P) \bigl[M_P(s),K_P(s)\bigr] = [I,0], $$(2.7)*x*_{P}of the DAE \(E_{P}\dot{x}_{P}=A_{P} x_{P} + f_{P}\) are given bywhere \(u:\mathbb{R}\to\mathbb{R}^{n_{P}-m_{P}}\) is an arbitrary (sufficiently smooth) function and where$$x_P = M_P\biggl(\frac{\mathrm{d}}{\mathrm{d}t}\biggr) (f_P) + K_P\biggl(\frac{\mathrm {d}}{\mathrm{d}t}\biggr) (u) $$*m*_{P}×*n*_{P}with*m*_{P}<*n*_{P}is the size of the matrix pencil*sE*_{P}−*A*_{P}. Furthermore, each initial condition \(x_{P}(0)=x_{P}^{0}\) can be achieved by an appropriate choice of*u*.*sE*_{R}−*A*_{R}: The solution behavior for a regular DAE was already discussed at the end of Sect. 2.1, a further discussion is carried out in Sects. 2.4 and 2.5.*sE*_{Q}−*A*_{Q}: Analogous to the*sE*_{P}−*A*_{P}block there exists a unimodular matrix Open image in new window such thatThen \(E_{Q}\dot{x}_{Q}=A_{Q} x_{Q} + f_{Q}\) is solvable if, and only if,$$ \left [\begin{array}{c}M_Q(s)\\K_Q(s) \end{array} \right ] (sE_Q-A_Q) = \left [\begin{array}{c}I\\0 \end{array} \right ] . $$(2.8)and the solution is uniquely determined by$$K_Q\biggl(\frac{\mathrm{d}}{\mathrm{d}t}\biggr) (f_Q) = 0 $$In particular, the initial values cannot be specified as they are already fixed by \(x_{Q}(0)=M_{Q}(\frac{\mathrm{d}}{\mathrm{d}t})(f_{Q})(0)\).$$x_Q = M_Q\biggl(\frac{\mathrm{d}}{\mathrm{d}t}\biggr) (f_Q). $$

In summary, the QKF decouples the corresponding DAE into the underdetermined part (existence but non-uniqueness of solutions), the regular part (existence and uniqueness of solutions) and the overdetermined part (uniqueness of solution but possible non-existence). Furthermore, the above solution characterization can also be carried out directly with the QKTF (2.5), where the analysis for the *sE*_{Q}−*A*_{Q} block remains unchanged, for the regular block the inhomogeneity *f*_{R} is replaced by \(f_{R} + (E_{RQ}\frac{\mathrm {d}}{\mathrm{d}t}- A_{RQ})(x_{Q})\) and for the *sE*_{P}−*A*_{P} block the inhomogeneity *f*_{P} is replaced by \(f_{P} + (E_{PR}\frac{\mathrm{d}}{\mathrm{d}t}-A_{PR})(x_{R}) + (E_{PQ}\frac{\mathrm{d}}{\mathrm{d}t}- A_{PQ})(x_{Q})\).

### Remark 2.1

(Refinement of QKF [3])

*R*

_{1}and

*R*

_{2}in Theorem 2.4 are chosen in the special way \(R_{1}=[R_{1}^{J},R_{1}^{N}]\) and \(R_{2}=[R_{2}^{J},R_{2}^{N}]\) where then a decoupling of the regular part in (2.5) corresponding the WCF is obtained as well. In particular, applying the Wong sequences again to the regular part (see next section) is not necessary for a further analysis.

### 2.3 Existence and Uniqueness of Solutions with Respect to In- and Outputs

*f*in the DAE (1.1) is often generated by a lower dimensional input

*u*, i.e.

*f*=

*Bu*for some suitable matrix

*B*; furthermore, an output

*y*=

*Cx*+

*Du*is introduced to represent the signals of the systems which are available for measurement and/or are of interest. The resulting DAE is then often called

*descriptor system*[52] (other common names are singular systems [8] or generalized state-space system [48])

*E*,

*A*) guarantees existence and uniqueness of solutions for any sufficiently smooth input. However, existence and uniqueness of solutions with respect to the input and output might hold for descriptor systems even when the matrix pair (

*E*,

*A*) is not regular as the following example shows.

### Example 2.2

*u*the unique output

*y*=−

*u*. However, the corresponding matrix pair Open image in new window is not regular.

It is therefore useful to define the notion of *external regularity*.

### Definition 2.3

(External regularity)

The descriptor system (2.9) and the corresponding matrix tuple (*E*,*A*,*B*,*C*,*D*) are called *externally regular* if, only if, for all sufficiently smooth inputs *u* there exist (classical) solutions *x* of (2.9) and the output *y* is uniquely determined by *u* and *x*(0).

With the help of the quasi-Kronecker form it is now possible to prove the following characterization of external regularity.

### Theorem 2.5

(Characterization of external regularity)

*The descriptor system*(2.9)

*is externally regular if*,

*and only if*,

*for infinitely many*\(s\in\mathbb{C}\).

### Proof

*E*,

*A*) is already in QKF (2.6) with corresponding transformation matrices

*S*and

*T*. According to the block size in (2.6) let \(SB=[B_{P}^{\top},B_{R}^{\top},B_{Q}^{\top}]^{\top}\) and

*CT*=[

*C*

_{P},

*C*

_{R},

*C*

_{Q}]. Then (2.10) is equivalent to

*K*

_{Q}(

*s*)

*B*

_{Q}≡0 and

*C*

_{P}

*K*

_{p}(

*s*)≡0. Taking into account the solution characterization given in conclusion to Theorem 2.4 the characterization of external regularity is shown. □

Note that condition (2.10) already appears in the survey paper by Lewis [32] based on arguments in the frequency domain.

### 2.4 Solution Formulas Based on the Wong Sequences: Regular Case

*V*,

*W*] and [

*EV*,

*AW*] are invertible matrices for all basis matrices

*V*and

*W*of Open image in new window and Open image in new window. In fact, any of these invertible matrices yield a transformation which put the matrix pencil

*sE*−

*A*into a quasi-Weierstraß form (QWF):

### Theorem 2.6

(Quasi Weierstraß form (QWF) [2, 5])

*Consider a regular matrix pencil*\(sE-A\in\mathbb{R}^{n\times n}[s]\)

*and the corresponding Wong sequences with limits*Open image in new window

*and*Open image in new window.

*For any full rank matrices*

*V*,

*W*

*with*Open image in new window

*and*Open image in new window

*let*

*T*=[

*V*,

*W*]

*and*

*S*=[

*EV*,

*AW*]

^{−1}.

*Then*

*where*\(J\in\mathbb{R}^{n_{1}\times n_{1}}\), \(n_{1}\in\mathbb{N}\),

*is some matrix and*\(N\in\mathbb{R}^{n_{2}\times n_{2}}\),

*n*

_{2}=

*n*−

*n*

_{1},

*is nilpotent*.

*In particular*, Open image in new window

*is exactly the space of consistent initial values*,

*i*.

*e*.

*for all*Open image in new window

*there exists a unique*(

*classical*)

*solution*

*x*

*of*\(E\dot{x}=Ax\)

*with*

*x*(0)=

*x*

_{0}.

The difference to the WCF from Theorem 2.3 is that *J* and *N* are not assumed to be in Jordan canonical form. Furthermore, the transformation matrices for the QWF can be chosen easily; it is only necessary to calculate the Wong sequences.

The knowledge of the two limiting spaces Open image in new window and Open image in new window is enough to obtain an explicit solution formula similar to the solution formula (1.3) for ODEs as the next result shows. To formulate the explicit solution formula it is necessary to define certain projectors as follows.

### Definition 2.4

(Consistency, differential and impulse projector[43])

*E*,

*A*) and use the same notation as in Theorem 2.6. The

*consistency projector*is given by

*differential projector*is given by

*impulse projector*is given by

*V*and

*W*, because when choosing different basis matrices \(\widetilde{V}\) and \(\widetilde{W}\) it must hold that \(V=\tilde{V}Q\) and \(W=\tilde{W}P\) for some invertible

*P*and

*Q*. Hence

*V*and

*W*is obvious. Furthermore, the differential and impulse projectors are not projectors in the usual sense because they are in general not idempotent.

### Theorem 2.7

(Explicit solution formula based on Wong sequences [47])

*Let*(

*E*,

*A*)

*be a regular matrix pair and use the notation from Definition*2.4.

*Then all solutions of*(1.1)

*are given by*,

*for*\(c\in\mathbb{R}^{n}\),

*In particular*,

*i*.

*e*. \(c\in\mathbb{R}^{n}\)

*implicitly specifies the initial value*(

*but in general*

*x*(0)≠

*c*

*even when*Open image in new window).

The Wong sequences appeared sporadically in the DAE literature: For example, Yip and Sincovec [52] used them to characterize regularity of the matrix pencil, Owens and Debeljkovic [36] characterized the space of consistent initial values via the Wong sequences; they are also included in the text books [1, 29] but not in the text books [7, 8, 9, 14, 30]. In general it seems that the connection between the Wong sequences and the (quasi-)Weierstraß/Kronecker form and their role in the solution characterization is not well known or appreciated in the DAE community (especially in the case of singular matrix pencils).

### 2.5 The Drazin Inverse Solution Formula

Another explicit solution formula was proposed by Campbell et al. [11] already in 1976 and is based on the Drazin inverse.

### Definition 2.5

(Drazin inverse [17])

- 1.
*DM*=*MD*, - 2.
*D*=*DMD*, - 3.
\(\exists\nu\in\mathbb{N}:\ M^{\nu}= M^{\nu+1}D\).

*M*is given by

*T*is such that

*J*is invertible and

*N*is nilpotent. In particular, for invertible

*M*the Drazin inverse is just the classical inverse, i.e.

*M*

^{−1}=

*M*

^{D}.

The following solution formula for the DAE (1.1) based on the Drazin inverse needs commutativity of the matrices *E* and *A*, however, as also regularity is assumed the following result shows that this is not a restriction of generality.

### Lemma 2.8

(Commutativation of (*E*,*A*) [11])

*Assume*(

*E*,

*A*)

*is regular and chose*\(\lambda\in\mathbb{R}\)

*such that*

*λE*−

*A*

*is invertible*.

*Then*

*commute*,

*i*.

*e*.

*the whole equation*(1.1)

*can simply be multiplied from the left with*(

*λE*−

*A*)

^{−1}

*which will not change the solution properties but will guarantee commutativity of the coefficient matrices*.

### Theorem 2.9

(Explicit solution formula based on the Drazin inverse [11])

A direct comparison of the solution formula (2.12) based on the Wong sequences and (2.13) indicates that *E*^{D}*A* plays the role of *A*^{diff}, *E*^{D}*E* plays the role of the consistency projector and *E*^{D} plays the role of the differential projector. However, the connection between the impulse projector and *E*^{imp} to the expressions involving the Drazin inverse of *A* is not immediately clear. The following result justifies the previous observations.

### Lemma 2.10

(Wong sequences and Drazin inverse [5])

*Consider the regular matrix pair*(

*E*,

*A*)

*with*

*EA*=

*AE*

*and use the notation from Theorem*2.6.

*Then*

*A*

^{D}appears and one might therefore think that the occurrence of zero eigenvalues in

*A*plays some special role for the solution. However, this is just an artifact and it turns out that in the expression

*J*

^{D}can be replaced by an arbitrary matrix without changing the result of the solution formula (2.13). One canonical choice is to replace

*J*

^{D}by the zero matrix which yields the impulse projector and which makes the “correction term” (

*I*−

*E*

^{D}

*E*) superfluous.

### 2.6 Time-Varying DAEs

*E*(⋅) and

*A*(⋅) have real analytical entries then a solution characterization similar to Corollary 2.2 holds. In particular, they showed that unique solvability is equivalent to finding time-varying (analytical) transformation matrices

*S*(⋅),

*T*(⋅), such that

*N*(

*t*) is a strictly lower triangular (and hence nilpotent) matrix. In particular, as in the time-invariant case, the DAE decouples into an ODE part and a pure DAE part. It is easily seen that for a strictly lower triangular matrix

*N*(

*t*) also the differential operator \(N(\cdot)\frac{\mathrm{d}}{\mathrm{d}t}\) is nilpotent, hence the inverse operator of \((N(\cdot)\frac{\mathrm{d}}{\mathrm {d}t}- I)\) can be calculated nearly identically as in (2.2):

If the coefficient matrices are not analytical the situation is not so clear anymore and different approaches have been proposed. Most methods have their motivation in numerical simulations and a detailed description and discussion is outside the scope of this survey. The interested reader is referred to the nice survey by Rabier and Rheinboldt [38], and to the text book by Kunkel and Mehrmann [30] as well as the recent monograph by Lamour, März and Tischendorf [31]. However, all these approaches do not allow for discontinuous coefficient matrices. These are studied in [46] and because of the connection to inconsistent initial value problems the problem of discontinuous coefficient matrices is further discussed in Sect. 5.

## 3 Inconsistent Initial Values and Distributional Solutions

After having presented a rather extensive discussion of classical solutions, this section presents an introductory discussion of the problem of inconsistent initial values. From the above derived solution formulas for (1.1) it becomes apparent that *x*(0) cannot be chosen arbitrarily, certain parts of *x*(0) are already fixed by the DAE and the inhomogeneity, cf. Theorem 2.7. In the extreme case that the QWF of (*E*,*A*) only consists of the nilpotent part, the initial value *x*(0) is completely determined by the inhomogeneity and no freedom to choose the initial value is left. However, there are situations where one wants to study the response of a system described by a DAE when an *inconsistent initial value* is given. Examples are electrical circuits which are switched on at a certain time [48]. There have been different approaches to deal with inconsistent initial values, e.g. [12, 18, 35, 37, 39, 42], some of them will be presented in detail in the later sections. All have in common that jumps as well as Dirac impulses may occur in the solutions. The Dirac impulse is a distribution (a generalized function), hence one must enlarge the considered solution space to also include distributions. In fact, also the presence of non-smooth inhomogeneities (or inputs) can lead to distributional solutions. However, the latter do not produce conceptional difficulties as the solution characterization of the previous section basically remains unchanged.

*test functions*is given by which is equipped with a certain topology.

^{2}The space of distributions, denoted by \(\mathbb{D}\), is then the dual of the space of test functions, i.e. A large class of ordinary functions, namely locally integrable functions, can be embedded into \(\mathbb{D}\) via the following injective

^{3}homomorphism:

*f*is differentiable, then

*δ*(at

*t*=0). In general, the Dirac impulse

*δ*

_{t}at time \(t\in\mathbb{R}\) is given by

*δ*

_{t}(

*φ*):=

*φ*(

*t*). Furthermore, if

*g*is a piecewise differentiable function with one jump at

*t*=

*t*

_{j}, i.e.

*g*is given as

*g*

_{1}and

*g*

_{2}are differentiable functions and

*α*:

Now it is no problem to consider the DAE (1.1) in a distributional solution space, instead of *x* and *f* being vectors of functions they are now vectors of distributions, i.e. \(x\in\mathbb{D}^{n}\) and \(f\in\mathbb{D}^{m}\) where *m*×*n* is the size of the matrices *E* and *A*. The definition of the matrix vector product remains unchanged^{4} so that (1.1) reads as *m* equations in \(\mathbb{D}\).

*x*(0)=

*x*

_{0}makes no sense. Even when assuming that a pointwise evaluation is well defined for certain distributions, the DAE (1.1) will still not exhibit (distributional) solution with arbitrary initial values. This is easily seen when considering the DAE \(N\dot{x}=x+f\) with nilpotent

*N*. Then also in the distributional solution framework the operator \(N\frac{\mathrm{d}}{\mathrm{d}t}- I:\mathbb{D}\to\mathbb{D}\) is invertible with inverse as in (2.2) and there exists a unique (distributional) solution given by

*x*cannot be assigned arbitrarily (i.e. independently of the inhomogeneity).

*t*=0 and before that the system was governed by different (maybe unknown) rules. This viewpoint was also expressed by Doetsch [16, p. 108] in the context of distributional solutions for ODEs:

So mathematically, there is some given past trajectoryThe concept of “initial value” in the physical science can be understood only when the past, that is, the interval

t<0, has been included in our considerations. This occurs naturally for distributions which, without exception, are defined on the entiret-axis.

*x*

^{0}for

*x*up to the initial time and the DAE (1.1) only holds on the interval [0,∞). This means that a solution of the following

*initial trajectory problem*(ITP) is sought:

*D*

_{I}for some interval \(I\subseteq\mathbb{R}\) and \(D\in\mathbb{D}\) denotes a distributional restriction generalizing the restrictions of functions given by

This problem was resolved especially in older publication [8, 9, 48] by ignoring it and/or by arguing with the Laplace transform (see the next section). Cobb [13] seems to be the first to be aware of this problem and he resolved it by introducing the space of piecewise-continuous distributions; Geerts [22, 23] was the first to use the space of impulsive-smooth distributions (introduced in [27]) as a solution space for DAEs. Seemingly unaware of these two approaches, Tolsa and Salichs [44] developed a distributional solution framework which can be seen as a mixture between the approaches of Cobb and Geerts. The more comprehensive space of piecewise-smooth distributions was later introduced [45] to combine the advantages of the piecewise-continuous and impulsive-smooth distributional solution spaces. The details are discussed in Sect. 5.

Cobb [12] also presented another approach by justifying the impulsive response due to inconsistent initial values via his notion of *limiting solutions*. The idea is to replace the singular matrix *E* in (1.1) by a “disturbed” version *E*_{ε} which is invertible for all *ε*>0 and *E*_{ε}→*E* as *ε*→0. If the solutions of the corresponding initial value ODE problem \(\dot{x}=E_{\varepsilon}^{-1} A x\), *x*(0)=*x*_{0} converges to a distribution, then Cobb calls this the limiting solution. He is then able to show that the limiting solution is unique and equal to the one obtained via the Laplace-transform approach. Campbell [9] extends this result also to the inhomogeneous case.

## 4 Laplace Transform Approaches

*frequency domain*(in contrast to the

*time domain*). In particular, when the input-output mapping is of interest the frequency domain approach significantly simplifies the analysis. The transformation between time and frequency domain is given by the

*Laplace transform*defined via the Laplace integral:

*g*and \(s\in\mathbb{C}\). Note that in general the Laplace integral is not well defined for all \(s\in\mathbb{C}\) and a suitable domain for \(\hat{g}\) must be chosen [16]. If a suitable domain exists, then Open image in new window is called the

*Laplace transform*of

*g*and, in general, Open image in new window denotes the Laplace transform operator. Again note that it is not specified at this point which class of functions have a Laplace transform and which class of functions are obtained as the image of Open image in new window. The main feature of the Laplace transform is the following property, where

*g*is a differentiable function for which

*g*and

*g*′ have Laplace transforms:

*g*is not continuous at

*t*=0 but

*g*(0+) exists and

*g*′ denotes the derivative of

*g*on \(\mathbb {R}\setminus \{ 0\}\), then (4.2) still holds in a slightly altered form:

*g*behaved for

*t*<0 which is a trivial consequence of the definition of the Laplace integral. This observation will play an important role when studying inconsistent initial values.

*E*,

*A*) is regular and

*x*(0+)=0, the latter can be solved easily algebraically:

*G*(

*s*) is a matrix over the field of rational functions and is usually called transfer function. As there are tables of functions and its Laplace transforms it is often possible to find the solutions of descriptor system with given input simply by plugging the Laplace transform of the input in the above formula and lookup the resulting output \(\hat{y}(s)\) to obtain the solution

*y*(

*t*) in the time domain. Furthermore, many important system properties can be deduced from properties (like the zeros and poles) of the transfer function directly.

A first systematic treatment of descriptor systems in the frequency domain was carried out by Rosenbrock [40]. He, however, only considered zero initial values and the input-output behavior. In particular, he was not concerned with a solution theory for general DAEs (1.1) with possible inconsistent values. Furthermore, he restricted attention to inputs which are exponentially bounded (guaranteeing existence of the Laplace transform), hence formally his framework could not deal with arbitrary (sufficiently smooth) inputs.

*g*for which Open image in new window is defined on a suitable domain it holds that

*t*. However, the derivative rule (4.8) is consistent with (4.3); to see this let

*g*be a function being zero on (−∞,0), differentiable on (0,∞) with well defined value

*g*(0+). Denote with

*g*′ the (classical) derivative of

*g*on \(\mathbb{R} \setminus\{0\}\), then (invoking linearity of Open image in new window) which shows equivalence of (4.8) and (4.3). The key observation is that the distributional derivative takes into account the jump at

*t*=0 whereas the classical derivative ignores it, i.e. in the above context

*g*with \(g_{\mathbb{D}}\) (even in [16]), the above distinction is difficult to grasp, in particular for inexperienced readers. As this problem plays an important role when dealing with inconsistent initial values, it is not surprising that researchers from the DAE community who are simply using the Laplace transform as a tool, struggle with the treatment of inconsistent initial values, cf. [16].

*x*(0+) does not occur anymore. In particular, if the matrix pair (

*E*,

*A*) is regular, the only solution of (4.9) is given by (4.5) independently of

*x*(0+). In particular, if

*u*=0 the only solution of (4.9) is \(\hat{x}(s)=0\) and \(\hat{y}(s)=0\). Assuming a well defined inverse Laplace transform this implies that the only solution of (2.9) with

*u*=0 is the trivial solution, which is of course not true in general. Altogether the following dilemma occurs.

### Dilemma

(Discrepancy between time domain and frequency domain)

An ad hoc analysis calls for

*distributional solutions*in response to inconsistent initial values. For consistent initial value there exist classical (non-zero) solutions.Using the

*distributional*Laplace transform to analyze the (distributional) solutions of (1.1) or (2.9) reveals that the*only*solution is the trivial one. In particular, no initial values (neither inconsistent nor consistent ones) are taken into account at all.

*D*(0−). Note, however, that, by definition,

*D*(0−)=0 for every \(D\in \bigcup_{k}\mathbb{D}_{\geq0,k}\); hence at this stage it is not clear why this definition makes sense. This problem was also pointed out by Cobb [34]. Nevertheless, a motivation for this choice will be given in Sect. 5.

*x*(0−) and solutions are sought in the space \(( \bigcup_{k} \mathbb{D}_{\geq0,k} )^{n}\), i.e.

*x*is assumed to be zero on (−∞,0). Applying the distributional Laplace transform to (4.11) yields

*x*(0+) but is the virtual initial value for

*x*(0−). If the matrix pair (

*E*,

*A*) is regular, the solution of (4.12) can now be obtained via

*E*is not invertible in general, the rational matrix (

*sE*−

*A*)

^{−1}may contain polynomial entries resulting in polynomial parts in \(\hat{x}\) corresponding to Dirac impulses in the time domain, for details see the end of this section.

*E*,

*A*, and

*B*are known and for suitable inputs

*u*the inverse Laplace transform of \(\hat{x}(s)\) can also be obtained analytically. This is the main advantage of the Laplace transform approach. There are, however, the following major drawbacks:

- 1.
Within the frequency domain it is not possible to motivate the incorporation of the (inconsistent) initial values as in (4.11); in fact, Doetsch [16, p. 108] who seems to have introduced this notion, needs to argue with the help of the distributional derivative and (4.10) within the time domain!

- 2.
The Laplace transform ignores everything that was in the past, i.e. on the interval (−∞,0); this is true for the classical Laplace transform (by definition of the Laplace integral) as well as for the distributional Laplace transform (by only considering distributions which vanish for

*t*<0). Hence the natural viewpoint of an initial trajectory problem (3.2) as also informally advocated by Doetsch cannot possibly be treated with the Laplace transform approach. - 3.
A frequency domain analysis gets useless when the original system is time-varying or nonlinear, whereas (linear) time-domain methods may in principle be extended to also treat time-variance and certain non-linearities. In fact, the piecewise-smoothly distributional solution framework as presented in Sect. 5 can be used without modification for linear time-varying DAEs [16, p. 129] and also for certain non-linear DAEs [12].

- 4.
Making statements about existence and uniqueness of solution with the help of the frequency domain heavily depends on an isomorphism between the time-domain and the frequency domain; there are, however, only a few special isomorphisms between certain special subspaces of the frequency and time domain, no general isomorphism is available, see also the discussion concerning (4.9).

*E*,

*A*) to be well defined, which will therefore be assumed in the following. Applying a coordinate transformation Open image in new window according to the QWF (2.11), the solution in the new coordinates is given by

*N*. Since (

*sI*−

*J*)

^{−1}is a strictly proper rational matrix, the solution for

*v*(resulting from taking the inverse Laplace transform) is the corresponding standard ODE solution (1.3). In particular, \(v(0+)=v_{0}^{-}\) and no Dirac impulses occur in

*v*. Applying the inverse Laplace transformation on the solution formula for \(\hat {w}(s)\), one obtains the solution

*w*=

*w*

_{f}+

*w*

_{i}, where

*w*

_{f}is the response with respect to the inhomogeneity given by

*w*

_{i}consists of Dirac impulses at

*t*=0 produced by the inconsistent initial value:

*w*

_{f}by using the correspondence (4.8), the distributional derivatives of

*f*

_{2}have to be considered. As the (distributional) Laplace transform can only be applied to distributions vanishing on (−∞,0), the inhomogeneity

*f*

_{2}will in general have a jump at

*t*=0, hence

*w*

_{f}will also contain Dirac impulses depending on \(f_{2}^{(i)}(0+)\),

*i*=0,1,…,

*ν*−1. In summary:

### Theorem 4.1

(Solution formula obtained via the Laplace transform approach)

*Consider the regular DAE*(1.1)

*with its “distributional version”*(4.13).

*Let*\(\nu\in\mathbb{N}\)

*be the nilpotency index of*

*N*

*in the QWF*(2.11)

*of the matrix pair*(

*E*,

*A*).

*Assume*\(f:\mathbb{R}\to\mathbb{R}^{n}\)

*is zero on*(−∞,0)

*and*

*ν*−1

*times differentiable on*(0,∞)

*with well defined values*

*f*

^{(i)}(0+),

*i*=0,1,…,

*ν*−1.

*Use the notation from Definition*2.4.

*Then*\(x\in(\bigcup_{k}\mathbb{D}_{\geq0,k})^{n}\)

*given by*(2.12)

*on*(0,∞)

*with*\(c=x_{0}^{-}\)

*and by the impulsive part at*

*t*=0,

*denoted by*

*x*[0],

*is the unique solution of*(4.13)

*obtained via solving*(4.14).

*In particular*,

*hence if*

*f*≡0

*then the consistent reinitialization is given by the consistency projector*

*Π*

_{(E,A)}

*via*

### Proof

## 5 Distributional Solutions

The previous section introduced distributional solutions in order to treat inconsistent initial values with the help of the Laplace transform. This leads to the consideration of the distributional space \(\bigcup_{k}\mathbb{D}_{\geq0,k}\) which contains all distributions which can be written as a (distributional) *k*th derivative, \(k\in\mathbb{N}\), of a continuous function being zero on (−∞,0) and of which a Laplace transform exists. This choice is motivated by the applicability of the Laplace transform and is actually not motivated by dealing with inconsistent initial values. In fact, as was pointed out in the previous section, the Laplace transform ignores by definition/design all what has happened before *t*<0 and is therefore in principle not suitable to treat inconsistent initial values coming from the past. Most researchers in the field agree with the notion that an inconsistent initial is due to a past which was *not* governed by the system description (1.1). One way of formalizing this viewpoint is the ITP (3.2). In general, having a past which obeys different rules then the present means that the overall system description is *time-variant* which gives another reason why the Laplace-transform approach runs into difficulties.

### 5.1 The Problem of Distributional Restrictions

Treating the ITP (3.2) in a distributional solution framework is, however, also not straightforward, because (as already mentioned above) the distributional restriction used in (3.2) is not well defined.

### Lemma 5.1

(Bad distribution [46])

*Let*

*D*

*be the*(

*distributional*,

*i*.

*e*.

*weak*

^{∗})

*limit of the distributions*:

*Then the restriction*(

*in the sense of*[33])

*of*

*D*

*to the interval*[0,∞)

*is not a well*-

*defined distribution*.

### Proof

*D*

_{[0,∞)}to a test function

*φ*which is identically one on [0,1] yields

*D*

_{[0,∞)}is not a well defined distribution. □

### Remark 5.1

(Restriction to open intervals)

*open*intervals. However, it should be mentioned here that nevertheless the

*equation*

*F*

_{I}=

*G*

_{I}makes sense for arbitrary distributions \(F,G\in\mathbb{D}\) and any open interval \(I\subseteq \mathbb{R}\) by defining: In fact, this definition is consistent with the restriction-definition to be established in the following for a special class of distributions [47]. Nevertheless, restricting the second equation in the ITP (3.2) to the

*closed*interval [0,∞) is essential. Taking an open restriction in both equations of (3.2) would imply that the past and the present are decoupled so that the initial trajectory would not influence the future trajectory. To be more precise: Any (distributional) solution

*x*of (3.2) will exhibit a jump at

*t*=0 in response to an inconsistent value

*x*

_{0}(0−), but the derivative of this jump appears as a Dirac impulse in the expression \(E\dot{x}\). While the restriction to the open interval (0,∞) would neglect this Dirac impulse, the restriction to the closed interval [0,∞) keeps the Dirac impulse in the second equation of the ITP (3.2) and hence the past can influence the present.

### 5.2 Cobb’s Space of Piecewise-Continuous Distributions

*D*[

*t*] of a distribution

*D*at time \(t\in\mathbb{R}\) which can be viewed as a restriction to the interval [

*t*,

*t*]. To this end, Cobb first defined the space of piecewise-continuous distributions given by where Open image in new window denotes the space of piecewise-continuous functions, in particular, for any Open image in new window the values

*g*(

*t*+) and

*g*(

*t*−) are well defined for all \(t\in\mathbb{R}\).

### Definition 5.1

(Cobb’s distributional restriction [45])

*D*coincides with \(g_{\mathbb{D}}\) on each interval (

*t*

_{i},

*t*

_{i+1}), \(i\in\mathbb{Z}\). For any \(\tau\in\mathbb{R}\) choose

*ε*>0 such that (

*τ*−

*ε*,

*τ*)⊆(

*t*

_{i},

*t*

_{i+1}) for some \(i\in \mathbb{Z}\). Then the restriction of

*D*to the interval [

*τ*,∞) is defined via

*φ*=

*φ*

_{τ}+

*φ*

^{ε}with \(\operatorname{supp}\varphi_{\tau}\subseteq(-\infty,\tau]\) and \(\operatorname{supp}\varphi^{\varepsilon}\subseteq[\tau-\varepsilon ,\infty)\).

*φ*

^{ε}, hence

*D*

_{[τ,∞)}is a well defined (continuous) operator on Open image in new window and therefore a distribution. In fact, Open image in new window with

*g*

_{[τ,∞)}as the corresponding piecewise-continuous function. The restriction to the closed interval (−∞,

*τ*] is defined analogously, and the restriction to arbitrary intervals can be defined as follows, \(s,t\in \mathbb{R} \cup\{\infty\}\):

the right sided evaluation:

*D*(*t*+):=*g*(*t*+),the left sided evaluation:

*D*(*t*−)=*g*(*t*−),the impulsive part:

*D*[*t*]:=*D*_{[t,t]}.

### Lemma 5.2

(Derivative of a restriction [45, Prop. 2.2.10])

*Let*Open image in new window

*and assume*Open image in new window

*as well*.

*Then*,

*for any*\(\tau\in\mathbb{R}\),

Note that Cobb did not include the assumption Open image in new window in his result; however, without this assumption the restriction of *D*′ to some interval is not defined, because in general *D*′ is not a piecewise-continuous distributions anymore (actually Cobb claims that the result is “obvious”; this is quite often a hint that there might be something wrong).

### Remark 5.2

(A distributional motivation of Doetsch’s past-aware derivative)

Lemma 5.2 now gives a justification of the past-aware derivative (4.10) as propagated by Doetsch, because *D*_{[0,∞)} as well as (*D*′)_{[0,∞)} are elements of the space \(\bigcup_{k} \mathbb{D}_{\geq0,k}\), however, *D* can still be non-zero on (−∞,0) and *D*(0−)≠0 in general.

A connection between (consistent) distributional solution of (1.1) and the solutions of “distributional” DAEs (4.13) was established in [13], a clearer connection, also allowing for inconsistent initial values, will be formulated in the context of piecewise-smooth distributions (see Sect. 5.4).

### 5.3 Impulsive-Smooth Distributions as Solution Space

*δ*′:

*p*=

*δ*′ and

*δ*is the unit with respect to convolution and hence denoted by one. The (time-domain) equation is now algebraically identically to the one obtained by the Laplace transformation approach without the need to think about problems like the existence of the Laplace transform and domain of convergence. In particular, existence and uniqueness results directly apply because no isomorphism between different solution spaces is needed. Nevertheless, the definition of Open image in new window still assumes that all involved variables are identically zero on (−∞,0), hence speaking of inconsistent initial values is conceptionally as difficult as for the Laplace transform approach. In summary, viewing

*x*

_{0}in (5.1) as the initial value for

*x*(0−) cannot be motivated within the impulsive-smooth distributional framework, because, by definition,

*x*(0−)=0.

*x*are sought in Open image in new window. In fact, the following result holds, which finally gives a satisfying and rigorous motivation for the incorporation of the (inconsistent) initial value as in (4.13).

### Theorem 5.3

(Equivalent description of the ITP (3.2))

*Consider the ITP*(3.2)

*within the impulsive*-

*smooth distributional solution framework with fixed initial trajectory*Open image in new window

*and inhomogeneity*Open image in new window.

*Then*Open image in new window

*solves the ITP*(3.2)

*if*,

*and only if*, \(z:=x-x^{0}_{(-\infty,0)} = x_{[0,\infty)}\)

*solves*

### Proof

*x*be a solution of the ITP (3.2) and let

*z*=

*x*

_{[0,∞)}. Then, clearly,

*z*

_{(−∞,0)}=0. Furthermore,

*z*=

*x*

_{[0,∞)}is indeed a solution of (5.2). On the other hand, let

*z*be a solution of (5.2) and define \(x:=z+x^{0}_{(-\infty,0)}\). Then, clearly, \(x_{(-\infty,0)}=x^{0}_{(-\infty,0)}\). Furthermore,

### Remark 5.3

- 1.If (5.2) is considered within the one-sided impulsive-smooth distributional framework, i.e. Open image in new window and Open image in new window then (5.2) simplifies to$$ E\dot{z} = Az + f + E x^0(0-)\delta. $$(5.3)
- 2.
Comparing the result of Theorem 5.3 with the result of Cobb [27] reveals three main differences: (1) Cobb only states one direction and not the equivalence, (2) instead of the ITP (3.2) Cobb just considers the original DAE (1.1), hence his result concerns only consistent solutions, (3) Cobb assumes that (5.3) has a unique solution.

- 3.
Regularity of the matrix pair (

*E*,*A*) is not assumed; in particular, neither is it assumed that for all inhomogeneities*f*there exist solutions to (3.2) and (5.2), nor is it assumed that solutions of (3.2) and (5.2) are uniquely given for fixed initial trajectory and fixed inhomogeneity. However, due to the established equivalence all existence and uniqueness results obtained for (5.3) carry over to the ITP (3.2).

*f*

_{ITP}, that

- 1.
The second equation of (5.4) suggest that the DAE (1.1) is valid globally (just with a different inhomogeneity), which conflicts with the intuition that an inconsistent initial value is due to the fact that the system description (1.1) is only valid on [0,∞) and not in the past.

- 2.
In (5.4) the past trajectory of

*x*is formally determined by two equations which could in general be conflicting (depending on the choice of*f*_{ITP}). - 3.
When studying an autonomous system (i.e. without the presence of an inhomogeneity), the formulation (5.4) formally leaves the class of autonomous systems.

*x*is an arbitrary distribution and

*f*as well as

*x*

^{0}are such that

*f*

_{ITP}is well defined. In fact, Rabier and Rheinboldt [37] do consider arbitrary distributions \(x\in\mathbb{D}^{n}\) and show that under certain regularity assumptions the solutions are in fact impulsive-smooth.

### 5.4 Piecewise-Smooth Distributions as Solution Space

- 1.
Open image in new window is not closed under differentiation.

- 2.
Open image in new window does not allow non-smooth inhomogeneities away from

*t*=0.

*t*

_{i}→±∞ as

*i*→±∞ and Open image in new window is such that \(D_{(t_{i},t_{i+1})}\) is induced by the corresponding restriction of a smooth function. A similar idea is proposed in [37], however, in both cases the resulting distributional space is not studied in detail. A more detailed treatment can be found in [37, Thm. 4.1] where, in the spirit of Cobb’s definition, the space of piecewise-smooth distributions is defined as follows: where Open image in new window is a piecewise-smooth function if, and only if, there exists a strictly ordered locally finite set \(\{s_{i}\in \mathbb{R} | i\in\mathbb{Z} \}\) and Open image in new window, \(i\in\mathbb{Z}\), such that \(f=\sum_{i\in\mathbb{Z}} {f_{i}}_{[s_{i},s_{i+1})}\). Clearly, and the space of piecewise-smooth distributions resolves each of the above mentioned drawbacks of the piecewise-continuous and impulsive-smooth distributions.

*E*(⋅) and

*A*(⋅) are smooth it is no problem to use any of the above distributional solution concepts because the product of a smooth function with any distribution is well defined so that (5.5) makes sense as an equation of distributions. In the discussion of the drawbacks of the Laplace transform approach it was already mentioned that an inconsistent initial value could be seen as the results from the presence of a time-varying system. In fact, the ITP (3.2) can be reformulated as the following time-varying DAE [37]:

*E*

_{1},

*A*

_{1}),…,(

*E*

_{P},

*A*

_{P}) are constant matrices.

It turns out that for the space of piecewise-smooth distributions a (non-commutative) multiplication can be defined, named *Fuchssteiner multiplication* after [45, Thm. 3.1.7], which in particular defines the multiplication of a piecewise-smooth function with a piecewise-smooth distribution. Hence (5.5) makes sense even for coefficient matrices which are only piecewise-smooth.

### Remark 5.4

(The square of the Dirac impulse)

- 1.
In the context of impulsive-smooth distributions [37] convolution is viewed as a multiplication and the Dirac impulse is the unit element for that multiplication. Hence

*δ*^{2}=*δ*in this framework. - 2.The Fuchssteiner multiplication for piecewise-smooth distributions yields$$\delta^2=0. $$
- 3.
It is well known that a commutative and associative multiplication which generalizes the multiplication of functions to distributions is not possible in general, but when enlarging the space of distributions the square of the Dirac impulse is well defined (but not a classical distribution). In the context of DAEs this approach was considered in [47], where the square of the Dirac impulse occurs in the analysis of the connection energy (the product of the voltage and current).

Within the framework of piecewise-smooth distributions it is now possible to show [19, 20] that the ITP (3.2) is uniquely solvable for all initial trajectories and all inhomogeneities if, and only if, the matrix pair (*E*,*A*) is regular. In particular, the impulses and jumps derived in this framework [22, 23, 27] are identical to (4.15) and (4.16) obtained via the Laplace transform approach.

## 6 Conclusion

The role of the Wong sequences of the matrix pair (*E*,*A*) for characterizing the (classical) solutions was highlighted. In particular, explicit solution formulas where given which are similar to the ones obtained for linear ODEs. The quasi-Kronecker form (QKF) and quasi-Weierstraß form (QWF) play a prominent role. For time-varying DAEs with analytical coefficients a time-varying QWF is available, however, time-varying Wong sequences and their connection to a time-varying QWF (or even QKF) have not been studied yet. The problem of inconsistent initial values was discussed and it was shown how the Laplace transform was used to treat this problem. However, it is argued that the Laplace transform approach cannot justify the notion of an inconsistent initial value. With the help of certain distributional solution spaces the notion of inconsistent initial values can be treated in a satisfying way and it also justifies the Laplace transform approach.

## Footnotes

- 1.
A polynomial matrix is called unimodular if it is invertible and its inverse is again a polynomial matrix.

- 2.
The topology is such that a sequence \((\varphi_{k})_{k\in\mathbb{N}}\) of test functions converges to zero if, and only if, (1) the supports of all

*φ*_{k}are contained within one common compact set \(K\subseteq\mathbb{R}\) and (2) for all \(i\in\mathbb{N}\), \(\varphi_{k}^{(i)}\) converges uniformly to zero as*k*→∞. - 3.
Two locally integrable functions which only differ on a set of measure zero are identified with each other.

- 4.
Some authors [30, 38] use a different definition for the matrix vector product which is due to the different viewpoint of a distributional vector

*x*as a map from Open image in new window to \(\mathbb{R}\) instead of a map from Open image in new window to \(\mathbb{R}^{n}\). The latter seems the more natural approach in view of applying it to (1.1), but it seems that both approaches are equivalent at least with respect to the solution theory of DAEs.

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