Boundary-Value Problems for Differential-Algebraic Equations: A Survey

Abstract

We provide an overview on the state of the art concerning boundary-value problems for differential-algebraic equations. A wide survey material is analyzed, in particular polynomial collocation and shooting methods. Moreover, new developments are presented such as the theory of linear boundary-value problems for arbitrary-index differential-algebraic equations as counterpart of the well-known classical version.

AMS Subject Classification (2010): 34A09, 65L80, 34B05, 34B15

Appendix

1.1 Basics Concerning Regular DAEs

We collect basic facts on the DAE

$$\displaystyle{ f((Dx)'(t),x(t),t) = 0, }$$

(6.1)

which exhibits the involved derivative by means of an extra matrix-valued function D. The function $f: \mathbb{R}^{n} \times \mathcal{D}_{f} \times \mathcal{I}_{f}\longrightarrow \mathbb{R}^{m}$, $\mathcal{D}_{f} \times \mathcal{I}_{f} \subseteq \mathbb{R}^{m} \times \mathbb{R}$ open, is continuous and has continuous partial derivatives f _y and f _x with respect to the first two variables $y \in \mathbb{R}^{n}$, $x \in \mathcal{D}_{f}$. The partial Jacobian f _y (y, x, t) is everywhere singular. The matrix function $D: \mathcal{I}_{f} \rightarrow \mathcal{L}(\mathbb{R}^{m}, \mathbb{R}^{n})$ is continuously differentiable and D(t) has constant rank r on the given interval $\mathcal{I}_{f}$. Then, im D is a $\mathcal{C}^{1}$-subspace in $\mathbb{R}^{n}$. We refer to [86] for proofs, motivation, and more details.

1.1.1 Regular DAEs, Regularity Regions

The DAE (6.1) is assumed to have a properly stated leading term. To simplify matters we further assume the nullspace $\ker f_{y}(y,x,t)$ to be independent of y. Then, the transversality condition (2.3) pointwise induces the continuously differentiable (see [86, Lemma A.20]) border projector $R: \mathcal{D}_{f} \times \mathcal{I}_{f} \rightarrow \mathcal{L}(\mathbb{R}^{n})$ given by

$$\displaystyle{ \mathrm{im}\,R(x,t) =\mathrm{ im}\,D(t),\;\ker R(x,t) =\ker f_{y}(y,x,t),\;(y,x,t) \in \mathbb{R}^{n} \times \mathcal{D}_{ f} \times \mathcal{I}_{f}. }$$

(6.2)

Next we depict the notion of regularity regions of a DAE (6.1). For this aim we introduce admissible matrix function sequences and associated projector functions (cf. [86]). Denote

$$\displaystyle\begin{array}{rcl} A(x^{1},x,t):& =& f_{ y}(D(t)x^{1} + D'(t)x,x,t) \in \mathcal{L}(\mathbb{R}^{n}, \mathbb{R}^{m}), {}\\ B(x^{1},x,t):& =& f_{ x}(D(t)x^{1} + D'(t)x,x,t) \in \mathcal{L}(\mathbb{R}^{m}), {}\\ G_{0}(x^{1},x,t):& =& A(x^{1},x,t)D(t) \in \mathcal{L}(\mathbb{R}^{m}), {}\\ B_{0}(x^{1},x,t):& =& B(x^{1},x,t) \in \mathcal{L}(\mathbb{R}^{m})\quad \text{for }\;x^{1} \in \mathbb{R}^{m},x \in \mathcal{D}_{ f},t \in \mathcal{I}_{f}. {}\\ \end{array}$$

The transversality condition (2.3) implies $\ker G_{0}(x^{1},x,t) =\ker D(t)$. We introduce projector valued functions $Q_{0},P_{0},\varPi _{0} \in \mathcal{C}(\mathcal{I}_{f},\mathcal{L}(\mathbb{R}^{m}))$ such that for all $t \in \mathcal{I}_{f}$

$$\displaystyle{ \mathrm{im}\,Q_{0}(t) = N_{0}(t):=\ker D(t),\quad \varPi _{0}(t):= P_{0}(t):= I - Q_{0}(t). }$$

(6.3)

Since D has constant rank, the orthoprojector function onto N ₀ is as smooth as D. Therefore, as Q ₀ we can choose the orthoprojector function onto N ₀ which is even continuously differentiable. Next we determine the generalized inverse D(x, t)⁻ of D(t) pointwise for all arguments by

$$\displaystyle\begin{array}{rcl} D(x,t)^{-}D(t)D(x,t)^{-}& =& D(x,t)^{-}, {}\\ D(t)D(x,t)^{-}D(t)& =& D(t), {}\\ D(x,t)^{-}D(t)& =& P_{ 0}(t), {}\\ D(t)D(x,t)^{-}& =& R(x,t). {}\\ \end{array}$$

The resulting function D ⁻ is continuous, if P ₀ is continuously differentiable then so also is D ⁻.

Definition 6.1

Let the DAE (6.1) have a properly involved derivative and let $\mathcal{G}\subseteq \mathcal{D}_{f} \times \mathcal{I}_{f}$ be open connected.

For the given level $\kappa \in \mathbb{N}$, we call the sequence $G_{0},\ldots,G_{\kappa }$ an admissible matrix function sequence associated with the DAE (6.1) on the set $\mathcal{G}$, if it is built pointwise for all $(x,t) \in \mathcal{G}$ and all arising $x^{j} \in \mathbb{R}^{m}$ by the rule:

set $G_{0}:= AD,\,B_{0}:= B,\,N_{0}:=\ker G_{0}$,

for i ≥ 1:

$$\displaystyle\begin{array}{rcl} G_{i}:= G_{i-1} + B_{i-1}Q_{i-1},& &{}\end{array}$$

(6.4)

$$\displaystyle\begin{array}{rcl} & & \qquad \qquad \qquad N_{i}:=\ker G_{i},\quad \mathop{N}\limits^{\frown }\!_{i}:= (N_{0} + \cdots + N_{i-1}) \cap N_{i}, \\ & & \qquad \qquad \qquad \text{find a complement }X_{i}\text{ such that }N_{0} + \cdots + N_{i-1} = \mathop{N}\limits^{\frown }\!_{i} \oplus X_{i}, \\ & & \qquad \qquad \qquad \text{choose a projector }Q_{i}\text{ such that }\;\mathrm{im}\,Q_{i} = N_{i}\text{ and }X_{i} \subseteq \ker Q_{i}, \\ & & \qquad \qquad \qquad \text{set }P_{i}:= I - Q_{i},\;\varPi _{i}:=\varPi _{i-1}P_{i}, \\ & & B_{i}:= B_{i-1}P_{i-1} - G_{i}D^{-}(D\varPi _{ i}D^{-})'D\varPi _{ i-1}, {}\end{array}$$

(6.5)

and, additionally,

• (a) the matrix function G _i has constant rank r _i on $\mathbb{R}^{mi} \times \mathcal{G}$, i = 0, …, κ,

• (b) the intersection $\mathop{N}\limits^{\frown }\!_{i}$ has constant dimension $u_{i}:=\dim \mathop{N}\limits^{\frown }\!_{i}$ there,

• (c) the product function Π _i is continuous and $D\varPi _{i}D^{-}$ is continuously differentiable on $\mathbb{R}^{mi} \times \mathcal{G}$, i = 0, …, κ.

The projector functions $Q_{0},\ldots,Q_{\kappa }$ linked with an admissible matrix function sequence are said to be admissible themselves.

An admissible matrix function sequence G ₀, …, G _κ is said to be regular admissible, if

$$\displaystyle{\mathop{N}\limits^{\frown }\!_{i} =\{ 0\}\quad \text{for all}\quad i = 1,\ldots,\kappa.}$$

Then, also the projector functions $Q_{0},\ldots,Q_{\kappa }$ are called regular admissible.

The numbers $\;r_{0} =\mathrm{ rank}\,G_{0},\ldots,r_{\kappa } =\mathrm{ rank}\,G_{\kappa }\;$ and $\;u_{1},\ldots,u_{\kappa }\;$ are named characteristic values of the DAE on $\mathcal{G}$.

To shorten the wording we often speak simply of admissible projector functions having in mind the admissible matrix function sequence built with these admissible projector functions. Admissible projector functions are always cross-linked with their matrix function sequence. Changing a projector function yields a new matrix function sequence.

We refer to [86] for many useful properties of the admissible matrix function sequences. It always holds that

$$\displaystyle{ r_{0} \leq \cdots \leq r_{\kappa -1} \leq r_{\kappa }. }$$

The notion of characteristic values makes sense, since these values are independent of the special choice of admissible projector functions and invariant under regular transformations.

In the case of a linear constant coefficient DAE, the construct simplifies to a sequence of matrices. In particular, the second term in the definition of B _i disappears. It is long-known that a pair {E, F} of m × m matrices E, F is regular with Kronecker index μ exactly if an admissible sequence of matrices starting with $G_{0} = AD = E$, B ₀: = F yields

$$\displaystyle{ r_{0} \leq \cdots \leq r_{\mu -1} <r_{\mu } = m. }$$

(6.6)

Thereby, neither the factorization nor the special choice of admissible projectors matter. The characteristic values describe the structure of the Weierstraß–Kronecker form : we have $l =\sum _{ j=0}^{\mu -1}(m - r_{j})$ and the nilpotent part N contains altogether $s = m - r_{0}$ Jordan blocks, among them $r_{i} - r_{i-1}$ Jordan blocks of order i, i = 1, …, μ, see [86, Corollary 1.32].

For linear DAEs with time-varying coefficients, the term (⋅ )′ in (6.5) means the derivative in time, and all matrix functions are functions in time. In general, the term (⋅ )′ in (6.5) stands for the total derivative in jet variables and then the matrix function G _i depends on the basic variables $(x,t) \in \mathcal{G}$ and, additionally, on the jet variables $x^{1},\ldots,x^{i+1} \in \mathbb{R}^{m}$. Owing to the total derivative $(D\varPi _{i}D^{-})'$ the new variable $x^{i+2} \in \mathbb{R}^{m}$ comes in at this level, see [86, Sect. 3.2].

Owing to the constant-rank conditions, the terms $D\varPi _{i}D^{-}$ are basically continuous. It may happen, for making these terms continuously differentiable, that the data function f must satisfy additional smoothness requirements. A precise description of this smoothness is much too involved and an overall sufficient condition, say $f \in \mathcal{C}^{m}$, is much too superficial. To indicate that there might be additional smoothness demands we restrict ourselves to the wording f is sufficiently smooth.

The next definition ties regularity to the inequalities (6.6) and so generalizes regularity of matrix pencils for time-varying linear DAEs as well as for nonlinear DAEs. We emphasize that regularity is supported by several constant-rank conditions.

Definition 6.2

Let the DAE (6.1) have a properly involved derivative. Let $\mathcal{G}\subseteq \mathcal{D}_{f} \times \mathcal{I}_{f}$ be an open, connected subset. The DAE (6.1) is said to be

• (1) regular on $\mathcal{G}$ with tractability index 0, if r ₀ = m,

• (2) regular on $\mathcal{G}$ with tractability index μ, if an admissible matrix function sequence exists such that (6.6) is valid on $\mathcal{G}$,

• (3) regular on $\mathcal{G}$, if it is, on $\mathcal{G}$, regular with any index (i.e., case (1) or (2) applies).

The open connected subset $\mathcal{G}$ is called a regularity region or regularity domain.

A point $(\bar{x},\bar{t}) \in \mathcal{D}_{f} \times \mathcal{I}_{f}$ is a regular point if there is a regularity region $\mathcal{G}\ni (\bar{x},\bar{t})$.

If $\mathcal{D}\subseteq \mathcal{D}_{f}$ is an open subset and $\mathcal{I}\subseteq \mathcal{I}_{f}$ is a compact subinterval, then the DAE (6.1) is said to be regular on $\mathcal{D}\times \mathcal{I}$ if there is a regularity region $\mathcal{G}$ such that $\mathcal{D}\times \mathcal{I}\subset \mathcal{G}$.

Example 6.1 (Regularity Regions)

We write the DAE

$$\displaystyle{ \begin{array}{rcl} x'_{1}(t) + x_{1}(t)& =&0, \\ x_{2}(t)x'_{2}(t) - x_{3}(t)& =&0, \\ x_{1}(t)^{2} + x_{2}(t)^{2} - 1 -\gamma (t)& =&0, \end{array} }$$

in the form (6.1), with $n = 2,\;m = k = 3$,

$$\displaystyle{ \begin{array}{c} f(y,x,t) = \left [\begin{array}{*{10}c} y_{1} + x_{1} \\ x_{2}y_{2} - x_{3} \\ x_{1}^{2} + x_{2}^{2} -\gamma (t) - 1 \end{array} \right ],\quad f_{y}(y,x,t) = \left [\begin{array}{*{10}c} 1& 0\\ 0 &x_{ 2} \\ 0& 0 \end{array} \right ], \\ D(t) = \left [\begin{array}{*{10}c} 1&0&0\\ 0 &1 &0 \end{array} \right ],\end{array} }$$

for $y \in \mathbb{R}^{2}$, $x \in \mathcal{D}_{f} = \mathbb{R}^{3}$, $t \in \mathcal{I}_{f} = \mathbb{R}$.

The derivative is properly involved on the open subsets $\mathbb{R}^{2} \times \mathcal{G}_{+}$ and $\mathbb{R}^{2} \times \mathcal{G}_{-}$, $\;\mathcal{G}_{+}:= \{x \in \mathbb{R}^{3}: x_{2}> 0\} \times \mathcal{I}_{f}$, $\;\mathcal{G}_{-}:= \{x \in \mathbb{R}^{3}: x_{2} <0\} \times \mathcal{I}_{f}$. We have there

$$\displaystyle{ G_{0} = AD = \left [\begin{array}{*{10}c} 1& 0 &0\\ 0 &x_{ 2} & 0\\ 0 & 0 &0 \end{array} \right ],\quad B_{0} = \left [\begin{array}{*{10}c} 1 & 0 & 0 \\ 0 & x_{2}^{1} & -1 \\ 2x_{1} & 2x_{2} & 0. \end{array} \right ]. }$$

Letting

$$\displaystyle{ Q_{0} = \left [\begin{array}{*{10}c} 0&0&0\\ 0 &0 &0 \\ 0&0&1 \end{array} \right ],\quad \text{ yields }\quad G_{1} = \left [\begin{array}{*{10}c} 1& 0 & 0\\ 0 &2x_{ 2} & -1\\ 0 & 0 & 0 \end{array} \right ]. }$$

G ₁ is singular but has constant rank. Since $N_{0} \cap N_{1} =\{ 0\}$ we find a projector function Q ₁ such that $N_{0} \subseteq \ker Q_{1}$. We choose

$$\displaystyle{ Q_{1} = \left [\begin{array}{*{10}c} 0& 0 &0\\ 0 & 1 &0 \\ 0& \frac{1} {x_{2}} & 0 \end{array} \right ],\ P_{1} = \left [\begin{array}{*{10}c} 1& 0 &0\\ 0 & 0 &0 \\ 0&-\frac{1} {x_{2}} & 1 \end{array} \right ],\ \varPi _{1} = \left [\begin{array}{*{10}c} 1&0&0\\ 0 &0 &0 \\ 0&0&0 \end{array} \right ],\ D\varPi _{1}D^{-} = \left [\begin{array}{*{10}c} 1&0 \\ 0&0 \end{array} \right ], }$$

and obtain $B_{1} = B_{0}P_{0}Q_{1}$, and then

$$\displaystyle{ G_{2} = \left [\begin{array}{*{10}c} 1& 0 & 0 \\ 0&2x_{2} + x_{2}^{1} & -1 \\ 0& 2x_{2} & 0 \end{array} \right ]. }$$

The matrix $G_{2} = G_{2}(x^{1},x,t)$ is nonsingular for all arguments (x ¹, x, t) with x ₂ ≠ 0. The admissible matrix function sequence terminates at this level. The open connected subsets $\mathcal{G}_{+}$ and $\mathcal{G}_{-}$ are regularity regions, here both with characteristics r ₀ = 2, r ₁ = 2, r ₂ = 3, and tractability index μ = 2. □ 

For regular DAEs, all intersections $\mathop{N}\limits^{\frown }\!_{i}$ are trivial ones, thus u _i  = 0, i ≥ 1. Namely, because of the inclusions

$$\displaystyle{\mathop{N}\limits^{\frown }\!_{i} \subseteq N_{i} \cap N_{i+1} \subseteq N_{i+1} \cap N_{i+2} \subseteq \ldots \subseteq N_{\mu -1} \cap N_{\mu },}$$

for reaching a nonsingular G _μ , which means N _μ  = { 0}, it is necessary to have $\mathop{N}\limits^{\frown }\!_{i} = \{0\}$, i ≥ 1. This is a useful condition for checking regularity in practice.

Observe that each open connected subset of a regularity region is again a regularity region. A regularity region consist of regular points having uniform characteristics. The union of regularity regions is, if it is connected, a regularity region, too. Further, the nonempty intersection of two regularity regions is also a regularity region. Only regularity regions with uniform characteristics may yield nonempty intersections. Maximal regularity regions are then bordered by so-called critical points. Solutions may cross the borders of maximal regularity regions and undergo there bifurcations etc., see examples in [82, 86, 95]. No doubt, much further research is needed to elucidate these phenomena.

1.1.2 The Structure of Linear DAEs

The general DAE (6.1) captures linear DAEs

$$\displaystyle{ A(t)(Dx)'(t) + B(t)x(t) - q(t) = 0 }$$

(6.7)

as $f(y,x,t):= A(t)y + B(t)x - q(t),\;t \in \mathcal{I}_{f}$. Now, admissible matrix function sequences depend only on time t; and hence, we speak of regularity intervals instead of regions. A regularity interval is open by definition. We say that the linear DAE with properly leading term is regular on the compact interval $[t_{a},t_{e}]$, if there is an accommodating regularity interval, or equivalently, if all points of $[t_{a},t_{e}]$ are regular.

If the linear DAE is regular on the interval $\mathcal{I}$, then it is also regular on each subinterval of $\mathcal{I}$ with the same characteristics. This sounds a triviality; however, there is a continuing profound debate about some related questions, cf. [96, Sect. 4.4].

If the linear DAE (6.7) is regular on the interval $\mathcal{I}$, then (see [86, Sect. 2.4]) it can be decoupled by admissible projector functions into an IERODE

$$\displaystyle{ u' - (D\varPi _{\mu -1}D^{-})'u + D\varPi _{\mu -1}G_{\mu }^{-1}B_{\mu }D^{-}u = D\varPi _{\mu -1}G_{\mu }^{-1}q }$$

(6.8)

and a triangular subsystem of several equations including differentiations

$$\displaystyle\begin{array}{rcl} \left [\begin{array}{cccc} 0&\mathcal{N}_{01} & \cdots & \mathcal{N}_{0,\mu -1} \\ & 0 & \ddots & \vdots\\ & & \ddots &\mathcal{N}_{ \mu -2,\mu -1} \\ & & & 0 \end{array} \right ]\;\;\left [\begin{array}{c} 0 \\ (Dv_{1})'\\ \vdots \\ (Dv_{\mu -1})' \end{array} \right ]& & \\ +\left [\begin{array}{cccc} I &\mathcal{M}_{01} & \cdots & \mathcal{M}_{0,\mu -1} \\ & I & \ddots & \vdots\\ & & \ddots &\mathcal{M}_{ \mu -2,\mu -1} \\ & & & I \end{array} \right ]& \left [\begin{array}{c} v_{0} \\ v_{1}\\ \vdots \\ v_{\mu -1} \end{array} \right ] + \left [\begin{array}{c} \mathcal{H}_{0} \\ \mathcal{H}_{1}\\ \vdots \\ \mathcal{H}_{\mu -1} \end{array} \right ]D^{-}u = \left [\begin{array}{c} \mathcal{L}_{0} \\ \mathcal{L}_{1}\\ \vdots \\ \mathcal{L}_{\mu -1} \end{array} \right ]q.&{}\end{array}$$

(6.9)

The subspace im D Π _μ−1 is an invariant subspace for the IERODE (6.8).

This structural decoupling is associated with the decomposition

$$\displaystyle{x = D^{-}u + v_{ 0} + v_{1} + \cdots + v_{\mu -1}.}$$

The coefficients are continuous and explicitly given in terms of an admissible matrix function sequence as

$$\displaystyle\begin{array}{rcl} \mathcal{N}_{01}&:=& -Q_{0}Q_{1}D^{-} {}\\ \mathcal{N}_{0j}&:=& -Q_{0}P_{1}\cdots P_{j-1}Q_{j}D^{-},\qquad \qquad \qquad \quad j = 2,\ldots,\mu -1, {}\\ \mathcal{N}_{i,i+1}&:=& -\varPi _{i-1}Q_{i}Q_{i+1}D^{-}, {}\\ \mathcal{N}_{ij}&:=& -\varPi _{i-1}Q_{i}P_{i+1}\cdots P_{j-1}Q_{j}D^{-},\qquad \quad \quad j = i + 2,\ldots,\mu -1,\;i = 1,\ldots,\mu -2, {}\\ \mathcal{M}_{0j}&:=& Q_{0}P_{1}\cdots P_{\mu -1}\mathcal{M}_{j}D\varPi _{j-1}Q_{j},\qquad \quad \quad j = 1,\ldots,\mu -1, {}\\ \mathcal{M}_{ij}&:=& \varPi _{i-1}Q_{i}P_{i+1}\cdots P_{\mu -1}\mathcal{M}_{j}D\varPi _{j-1}Q_{j},\quad j = i + 1,\ldots,\mu -1,\;i = 1,\ldots,\mu -2, {}\\ \mathcal{L}_{0}&:=& Q_{0}P_{1}\cdots P_{\mu -1}G_{\mu }^{-1}, {}\\ \mathcal{L}_{i}&:=& \varPi _{i-1}Q_{i}P_{i+1}\cdots P_{\mu -1}G_{\mu }^{-1},\qquad \quad \qquad i = 1,\ldots,\mu -2, {}\\ \mathcal{L}_{\mu -1}&:=& \varPi _{\mu -2}Q_{\mu -1}G_{\mu }^{-1}, {}\\ \mathcal{H}_{0}&:=& Q_{0}P_{1}\cdots P_{\mu -1}\mathcal{K}\varPi _{\mu -1}, {}\\ \mathcal{H}_{i}&:=& \varPi _{i-1}Q_{i}P_{i+1}\cdots P_{\mu -1}\mathcal{K}\varPi _{\mu -1},\qquad \quad i = 1,\ldots,\mu -2, {}\\ \mathcal{H}_{\mu -1}&:=& \varPi _{\mu -2}Q_{\mu -1}\mathcal{K}\varPi _{\mu -1}, {}\\ \end{array}$$

with

$$\displaystyle\begin{array}{rcl} \mathcal{K}:= (I -\varPi _{\mu -1})G_{\mu }^{-1}B_{\mu -1}\varPi _{\mu -1} +\sum _{ l=1}^{\mu -1}(I -\varPi _{ l-1})(P_{l} - Q_{l})(D\varPi _{l}D^{-})'D\varPi _{\mu -1},& & {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} \mathcal{M}_{j}:=\sum _{ k=0}^{j-1}(I -\varPi _{ k})\{P_{k}D^{-}(D\varPi _{ k}D^{-})'& -& Q_{ k+1}D^{-}(D\varPi _{ k+1}D^{-})'\}D\varPi _{ j-1}Q_{l}D^{-}, {}\\ l& =& 1,\ldots,\mu -1. {}\\ \end{array}$$

The IERODE is always uncoupled from the second subsystem, but the latter is tied to the IERODE (6.8) if among the coefficients $\mathcal{H}_{0},\ldots,\mathcal{H}_{\mu -1}$ there is at least one which does not vanish. One speaks about a fine decoupling, if $\mathcal{H}_{1} = \cdots = \mathcal{H}_{\mu -1} = 0$, and about a complete decoupling, if $\mathcal{H}_{0} = 0$, additionally. A complete decoupling is given, exactly if the coefficient $\mathcal{K}$ vanishes identically.

If the DAE (6.7) is regular and the original data are sufficiently smooth, then the DAE (6.7) is called fine. Fine DAEs always possess fine and complete decouplings, see [86, Sect. 2.4.3] for the constructive proof. The coefficients of the IERODE as well as the so-called canonical projector function $\varPi _{can} = (I -\mathcal{H}_{0})\varPi _{\mu -1}$ are independent of the special choice of the fine decoupling projector functions.

It is noteworthy that, if $Q_{0},\ldots,Q_{\mu -1}$ generate a complete decoupling for a constant coefficient DAE $Ex'(t) + Fx(t) = 0$, then Π _μ−1 is the spectral projector of the matrix pencil {E, F}. In this way, the projector function Π _μ−1 associated with a complete decoupling of a fine time-varying DAE represents the generalization of the spectral projector.

1.1.3 Linearizations

Given is now a reference function $x_{{\ast}}\in \mathcal{C}_{D}^{1}(\mathcal{I}_{{\ast}}, \mathbb{R}^{m})$ on an individual interval $\mathcal{I}_{{\ast}}\subseteq \mathcal{I}_{f}$, whose values belong to $\mathcal{D}_{f}$. For each such reference function (here not necessarily a solution!) we may consider the linearization of the (6.1) along x _∗, that is, the linearized DAE

$$\displaystyle{ A_{{\ast}}(t)(Dx)'(t) + B_{{\ast}}(t)x(t) = q(t),\quad t \in \mathcal{I}_{{\ast}}, }$$

(6.10)

with coefficients

$$\displaystyle{ A_{{\ast}}(t):= f_{y}((Dx_{{\ast}})'(t),x_{{\ast}}(t),t),\quad B_{{\ast}}(t):= f_{x}((Dx_{{\ast}})'(t),x_{{\ast}}(t),t),\quad t \in \mathcal{I}_{{\ast}}. }$$

The linear DAE (6.10) inherits from the nonlinear DAE (6.1) the properly stated leading term.

We denote by $\mathcal{C}_{ref}^{m}(\mathcal{G})$ the set of all $\mathcal{C}^{m}$ functions x _∗, defined on individual intervals $\mathcal{I}_{x_{{\ast}}}$, and with graph in $\mathcal{G}$, that is, $(x_{{\ast}}(t),t) \in \mathcal{G}$ for $t \in \mathcal{I}_{x_{{\ast}}}$. Clearly, then we also have $x_{{\ast}}\in \mathcal{C}_{D}^{1}(\mathcal{I}_{x_{{\ast}}}, \mathbb{R}^{m})$. By the smoothness of the reference functions x _∗ and the function f we ensure that also the coefficients A _∗ and B _∗ are sufficiently smooth for regularity.

Next we adapt the necessary and sufficient regularity condition from [86, Theorem 3.33] to our somewhat simpler situation.

Theorem 6.1

Let the DAE (6.1) have a properly involved derivative and let f be sufficiently smooth. Let $\mathcal{G}\subseteq \mathcal{D}_{f} \times \mathcal{I}_{f}$ be an open connected set. Then the following statements are valid:

• (1) The DAE (6.1) is regular on $\mathcal{G}$ if the linearized DAE (6.10) along each arbitrary reference function $x_{{\ast}}\in \mathcal{C}_{ref}^{m}(\mathcal{G})$ is regular, and vice versa.

• (2) If the DAE (6.1) is regular on $\mathcal{G}$ with tractability index μ and characteristic values $r_{0} \leq \cdots \leq r_{\mu -1} <r_{\mu } = m$ , then all linearized DAEs (6.10) along reference functions $x_{{\ast}}\in \mathcal{C}_{ref}^{m}(\mathcal{G})$ are regular with uniform index μ and characteristics $r_{0} \leq \cdots \leq r_{\mu -1} <r_{\mu } = m$ .

• (3) If all linearized DAEs (6.10) along reference functions $x_{{\ast}}\in \mathcal{C}_{ref}^{m}(\mathcal{G})$ are regular, then they have uniform index and characteristics, and the nonlinear DAE (6.1) is also regular on $\mathcal{G}$ , with the same index and characteristics.

Corollary 6.2

Let the DAE (6.1) have a properly involved derivative and let f be sufficiently smooth. Let $\mathcal{D}\subseteq \mathcal{D}_{f}$ be an open connected set and $\mathcal{I}\subset \mathcal{I}_{f}$ be a compact interval. Then the following statements are valid:

• (1) The DAE (6.1) is regular on $\mathcal{D}\times \mathcal{I}$ if the linearized DAE (6.10) along each arbitrary reference function $x_{{\ast}}\in \mathcal{C}^{m}(\mathcal{I}, \mathbb{R}^{m})$ with values in $\mathcal{D}$ is regular, and vice versa.

• (2) If the DAE (6.1) is regular on $\mathcal{D}\times \mathcal{I}$ with tractability index μ and characteristic values $r_{0} \leq \cdots \leq r_{\mu -1} <r_{\mu } = m$ , then all linearized DAEs (6.10) along reference functions $x_{{\ast}}\in \mathcal{C}^{m}(\mathcal{I}, \mathbb{R}^{m})$ with values in $\mathcal{D}$ are regular with uniform index μ and characteristics $r_{0} \leq \cdots \leq r_{\mu -1} <r_{\mu } = m$ .

• (3) If all linearized DAEs (6.10) along reference functions $x_{{\ast}}\in \mathcal{C}^{m}(\mathcal{I}, \mathbb{R}^{m})$ with values in $\mathcal{D}$ are regular, then they have uniform index and characteristics, and the nonlinear DAE (6.1) is also regular on $\mathcal{D}\times \mathcal{I}$ , with the same index and characteristics.

Proof

Statement (1) is a consequence of Statements (2) and (3).

Statement (2) follows from the construction of the admissible matrix function sequences. Namely, for each $x_{{\ast}}\in \mathcal{C}^{m}(\mathcal{I}, \mathbb{R}^{m})$, with values in $\mathcal{D}$, we have

$$\displaystyle\begin{array}{rcl} G_{0}(x_{{\ast}}'(t),x_{{\ast}}(t),t)& =:& G_{{\ast}\;0}(t), {}\\ B_{i-1}(x_{{\ast}}^{(i+1)}(t),\cdots \,,x_{ {\ast}}'(t),x_{{\ast}}(t),t)& =:& B_{{\ast}\;i-1}(t), {}\\ G_{i}(x_{{\ast}}^{(i+1)}(t),\cdots \,,x_{ {\ast}}'(t),x_{{\ast}}(t),t)& =:& G_{{\ast}\;i}(t),\;t \in \mathcal{I},\quad i = 1,\ldots,\mu, {}\\ \end{array}$$

which represents an admissible matrix function sequence for the linearized along x _∗ DAE.

Statement (3) is proved along the lines of [86, Theorem 3.33 ] by means of so-called widely orthogonal projector functions. The proof given in [86] also works if one supposes solely compact individual intervals $\mathcal{I}_{x_{{\ast}}}$.

By Lemma 6.3 below, each reference function given on an individual compact interval can be extended to belong to $x_{{\ast}}\in \mathcal{C}^{m}(\mathcal{I}, \mathbb{R}^{m})$, with values in $\mathcal{D}$. □ 

The next assertion is proved in [96].

Lemma 6.3

Let $\mathcal{D}\subseteq \mathbb{R}^{m}$ be an open set and $\mathcal{I}\subset \mathbb{R}$ be a compact interval. Let $\mathcal{I}_{{\ast}}\subset \mathcal{I}$ be a compact subinterval and $s \in \mathbb{N}$ .

Then, for each function $x_{{\ast}}\in \mathcal{C}^{s}(\mathcal{I}_{{\ast}}, \mathbb{R}^{m})$ , with values in $\mathcal{D}$ , there is an extension $\hat{x}_{{\ast}}\in \mathcal{C}^{s}(\mathcal{I}, \mathbb{R}^{m})$ , with values in $\mathcal{D}$ .

1.1.4 Linear Differential-Algebraic Operators

Let the linear DAE (6.7) be regular with tractability index $\mu \in \mathbb{N}$ on the interval $\mathcal{I} = [a,b]$. The function space

$$\displaystyle{ \mathcal{C}_{D}^{1}(\mathcal{I}, \mathbb{R}^{m}) =\{ x \in \mathcal{C}(\mathcal{I}, \mathbb{R}^{m}): Dx \in \mathcal{C}^{1}(\mathcal{I}, \mathbb{R}^{n})\} }$$

equipped with the norm $\|x\|_{\mathcal{C}_{D}^{1}}:=\| x\|_{\infty } +\| (Dx)'\|_{\infty }$ is a Banach space. We consider the regular linear differential-algebraic operator (cf. [96])

$$\displaystyle{ Tx:= A(Dx)' + Bx,\quad x \in \mathcal{C}_{D}^{1}(\mathcal{I}, \mathbb{R}^{m}), }$$

and, supposing accurately stated boundary conditions in the sense of Definition 2.3, the composed operator

$$\displaystyle{ \mathcal{T} x:= (\,Tx,\,G_{a}x(a) + G_{b}x(b)\,),\quad x \in \mathcal{C}_{D}^{1}(\mathcal{I}, \mathbb{R}^{m}), }$$

so that the equations Tx = q and $\mathcal{T} x = (q,\gamma )$ represent the DAE and the BVP, respectively.

We consider different image spaces Y and $Y \times \mathbb{R}^{l}$ for the operators T and $\mathcal{T}$. The natural one is

$$\displaystyle{ Y = \mathcal{C}(\mathcal{I}, \mathbb{R}^{m}). }$$

T and $\mathcal{T}$ are bounded in this setting:

$$\displaystyle{ \|Tx\|_{\infty }\leq (\|A\|_{\infty }\|(Dx)'\|_{\infty } +\| b\|_{\infty }\|x\|_{\infty }) \leq k\|x\|_{\mathcal{C}_{D}^{1}},\quad x \in \mathcal{C}_{D}^{1}(\mathcal{I}, \mathbb{R}^{m}). }$$

The operator T is surjective exactly if the index μ equals one. Otherwise im T is a proper nonclosed subset in $\mathcal{C}(\mathcal{I}, \mathbb{R}^{m})$, see [86, Sect. 3.9.1], also Appendix 6.1.2. More precisely, one obtains

$$\displaystyle\begin{array}{rcl} \mathrm{im}\,T& =& \{q \in \mathcal{C}(\mathcal{I}, \mathbb{R}^{m}): v_{\mu -1}:= \mathcal{L}_{\mu -1}q,\;Dv_{\mu -1} \in \mathcal{C}^{1}(\mathcal{I}, \mathbb{R}^{n}),\;\text{ for }\;j =\mu -2,\ldots,1: {}\\ & & v_{j}:= \mathcal{L}_{j}q +\sum _{ i=j+1}^{\mu -1}\mathcal{M}_{ j,i}v_{i} +\sum _{ i=j+1}^{\mu -1}\mathcal{N}_{ j,i}(Dv_{i})',\;Dv_{j} \in \mathcal{C}^{1}(\mathcal{I}, \mathbb{R}^{n})\} =: \mathcal{C}^{\mathrm{ind}\,\mu }(\mathcal{I}, \mathbb{R}^{m}).{}\\ \end{array}$$

If μ = 1, then $\mathcal{T}$ acts bijectively between Banach spaces so that the inverse $\mathcal{T}^{-1}$ is also bounded and the BVP $\mathcal{T} x = (q,\gamma )$ is well-posed.

If μ > 1, then the BVP $\mathcal{T} x = (q,\gamma )$ is essentially ill-posed in this natural setting because of the nonclosed image of T.

Let μ > 1. In an advanced setting we choose

$$\displaystyle{ Y = \mathcal{C}^{\mathrm{ind}\,\mu }(\mathcal{I}, \mathbb{R}^{m}) }$$

and by introducing the norm $\|q\|_{\mathrm{ind}\,\,\mu }:=\| q\|_{\infty } +\| (Dv_{\mu -1})'\|_{\infty } + \cdots +\| (Dv_{1})'\|_{\infty }$ we obtain again a Banach space. Regarding the structure of the DAE (cf. Sect. 6.1.2) one knows the operators t and $\mathcal{T}$ to be bounded again. Namely, we derive for each arbitrary $x \in \mathcal{C}_{D}^{1}(\mathcal{I}, \mathbb{R}^{m})$ that

$$\displaystyle{ \|Tx\|_{\mathrm{ind}\,\,\mu }:=\| Tx\|_{\infty } +\| (D\varPi _{\mu -2}Q_{\mu -1}x)'\|_{\infty } + \cdots +\| (D\varPi _{0}Q_{1}x)'\|_{\infty }. }$$

Taking into account that

$$\displaystyle{ (D\varPi _{\mu -2}Q_{\mu -1}x)' = (D\varPi _{\mu -2}Q_{\mu -1}D^{-})'Dx + D\varPi _{\mu -2}Q_{\mu -1}D^{-}(Dx)' }$$

etc. one achieves the required inequality $\|Tx\|_{\mathrm{ind}\,\,\mu } \leq k_{\mathrm{ind}\,\,\mu }\|x\|_{\mathcal{C}_{D}^{1}}$.

In this advanced setting, as a bounded bijection acting in Banach spaces, $\mathcal{T}$ has a bounded inverse and the BVP is well-posed. This sounds fine, but it is quite illusory. The advanced image space $\mathcal{C}^{\mathrm{ind}\,\mu }(\mathcal{I}, \mathbb{R}^{m})$ as well as its norm $\|.\|_{\mathrm{ind}\,\,\mu }$ strongly depend on the special coefficients A, D, B. To describe them, one has to be aware of the full special structure of the given DAE. Except for the index-2 case, there seems to be no way to practice this formal well-posedness.

Furthermore, the higher the index the stronger the topology given by the norm $\|.\|_{\mathrm{ind}\,\,\mu }$, see [86, Sect. 3.9.1], [96, Sect. 2]. It seems to be impossible to capture errors in practical computational procedures using these norms.

1.2 List of Symbols and Abbreviations

$\mathcal{L}(X,Y )$	Set of linear operators from X to Y
$\mathcal{L}(X)$	$= \mathcal{L}(X,X)$
$\mathcal{L}(\mathbb{R}^{m}, \mathbb{R}^{n})$	is identified with $\mathbb{R}^{n\times m}$
K ∗	Transposed matrix
K −	Generalized inverse
K +	Orthogonal generalized (Moore–Penrose) inverse
dom K	Definition domain of the map K
$\ker K$	Nullspace (kernel) of the operator K
im K	Image (range) of the operator K
ind {E, F}	Kronecker index of the matrix pair {E, F}
$\langle \cdot,\cdot \rangle$	Scalar product in $\mathbb{R}^{m}$
(⋅ , ⋅ )	Scalar product in function spaces
| ⋅ |	Vector and matrix norms
$\|\cdot \|$	Norms on function spaces, operator norms
DAE	Differential-algebraic equation
ODE	Ordinary differential equation
IVP	Initial value problem
BVP	Boundary value problem
IERODE	Inherent explicit ODE
LSS	Least squares solution
TPBVP	Two-point BVP