Differential algebraic systems anew.
It is proposed to figure out the leading term in differential algebraic systems more precisely. Low index linear systems with those properly stated leading terms are considered in detail. In particular, it is asked whether a numerical integration method applied to the original system reaches the inherent regular ODE without conservation, i.e., whether the discretization and the decoupling commute in some sense. In general one cannot expect this commutativity so that additional difficulties like strong stepsize restrictions may arise. Moreover, abstract differential algebraic equations in infinite-dimensional Hilbert spaces are introduced, and the index notion is generalized to those equations. In particular, partial differential algebraic equations are considered in this abstract formulation.
Introduction
When dealing with standard linear time-varying coe-cient DAEs
and their adjoint equations
one is confronted with a kind of unsightly dissymmetry. These equations are
of dierent type. In [3], more precise formulations
E(PE
Humboldt University, Institute of Mathematics, D-10099 Berlin, Germany
(iam@mathematik.hu-berlin.de)
are used, where PE denotes a projector function along ker E, say
These formulations show a little bit more symmetry. However, (1.4) is treatet
in [3] via an auxiliary enlarged system that has a correspondig projector
inside the derivative like (1.3). This formal uncompleteness makes us to
treat general equations of the form ([2])
where A; D and B are continuous matrix functions, and A and D are well
matched so that D precisely gures out which derivatives are actually in-
volved. Neither D nor A have to be projectors. The adjoint equation
now ts nicely into this form.
There are remarkable benets of this new
The inherent regular dierential equation of a DAE (1.5) is now uniquely determined
(cf. [2] and Section 2). Nice generalizations of fundamental classical
results (like the relation Y fundamental matrices) are obtained,
but also further symmetries concerning the characteristic subspaces and the
index of (1.5) and (1.6), respectively ([2]). Further, the boundary value
problem resulting from linear-quadratic control problems looks now nice and
transparent (cf. Section 5).
The nonlinear counterparts of (1.5)
and
should be very welcome in applications, e. g. in circuit simulation. In [13],
[14], those nonlinear equations are studied as well as numerical integration
methods. To be more transparent we do not touch at all the nonlinear cases
in the present paper.
On the one hand, the new approach (1.5) allows to maintain all theoretical
and numerical results concerning (1.1) via (1.3), but also those via the
formulation
where RE is a projector function along im E.
On the other hand, a generalization to abstract dierential algebraic equations
in innite dimensional spaces becomes possible (cf. Sections 4, 5). In
particular, this seems to be a useful tool for treating so-called partial dier-
ential algebraic equations.
A really surprising benet of the new approach concerns numerical integration
methods. At this point it should be mentioned that, e. g. in the
early paper [18], the formulation (1.5) was used with the seemingly technical
condition on im D(t) to be time-invariant. Now we understand that this
condition is exactly the one allowing the DAE-decoupling into its essential
parts and the discretization of the DAE to commute in the index-1 case (cf.
[13]). In the index-2 case, if im D(t) is decomposed into certain two further
constant subspaces, the decoupling and the discretization also commute in
some sense. Hence, neither phenomena converting e. g. the implicit Euler
method within the inherent regular ODE into the explicit one ([11], [10])
nor additional stepsize restrictions may arise. Since this is primarily a property
of the DAE itself, we call those equations numerically well formulated
(cf. Section 3, [13], [14]). Obviously, now one should try to have numerically
well formulated equations at the very beginning. If this is not the case,
refactorizations
A ~
D (or factorizations leading to
numerically well formulated versions might be useful (cf. Examples 3.1, 3.2,
3.3 below). In particular, the Euler backward method applied to the famous
index-2 example given in [20] is well-known to get into great di-culties and
to fail. However, in a slight modication (Example 3.1) it works best, and
we understand why this happens.
By the way, the results on the contractivity of standard methods applied to
given (e. g. in [8], [10]) for the case of a constant nullspace ker E, and
those of modied methods (e. g. [12]) for the case of a constant im E are
now completely clear.
This paper ist organized as follows.
In Section 2 the basic decoupling is described in the nessesary details. With
this, we follow the lines of [2], but now the index-1 and index-2 cases are
put together such that index-1 appers to be a special case. Furthermore,
the matrix functions A and D may be rectangular ones now, i. e., not necessarily
quadratic. In Section 3 we deal with the BDF as a prototype of a
discretization (the same can be done, with some more eort, with Runge-Kutta
methods, cf. [13], [14]), and show the advatages of numerically well
formulated DAEs. A rst attempt to formulate and to apply abstract differential
algebraic equations is presented in Section 4. A linear-quadratic
control problem for a DAE (in abstract formulation) is proved to be solvable
in Section 5. Thereby, a Hamiltonian property turns out to be a further
benet of the numerically well formulated case.
Linear DAEs with properly stated leading
Consider equations
with matrix coe-cients
The coe-cients A and D that determine the leading term of equation (2.1)
are assumed to be well matched in the following sense (cf.[2]). Condition C1
for the case that
Denition 2.1 The ordered pair of continuous matrix functions A and D is
said to be well matched if
im D(t)ker
and these subspaces are spanned by continuously dierentiable bases.
The leading term is properly stated if A and D are well matched.
Obviously, if A and D are well matched, there is a uniquely determined
projector function R(t) 2 L(R n );
Remark 2.1 The decomposition (2.2) implies that, for all t 2 I,
and vice versa. Roughly speaking, D plays the role of an incidence matrix.
It gures out the really involved derivatives of the unknown function.
Remark 2.2 Starting with a standard form equation
E(t)x 0 (t)+F
we may use any factorization matched A and D,
to obtain
In particular, the factorizations can be used if PE (t)
is a projector along the nullspace of E(t), but RE (t) denotes a projector onto
the image of E(t).
Naturally, a solution of equation (2.1) should be a function x 2
that has a continuously dierentiable product Dx, and satises the equation
at all t 2 I. Let
D

denote the respective function space.
Next we introduce certain subspaces and matrices to be used throughout this
paper:
Further, D(t) denotes the re
exive generalized inverse of D(t) with
Note that D(t) is uniquely determined if P 0 (t) is given and vice versa.
Denition 2.2 (cf. [2]): The DAE (2.1) with properly formulated leading
term and nontrivial N 0 (t) has index  2 f1; 2g if N j (t) \S j (t) has constant
dimension  j+1 on I and  j > 0 for j <  ,
Remark 2.3 In terms of the matrices G j (t), the index  case is characterized
by
rank G j+1 (t) =: r
Namely, using the auxiliary matrix
and the re
exive generalized inverse G j determined by
but I G remains nonsingular.
Remark 2.4 In an analogous way, considering a somehow more complex
contruction of the matrices B j we could continue the chain (2.6) for i > 1
dene also an index higher than two. However, we dispense with that
here in favour of more transparency, and since in the sections below we fully
focus on the lower index case.
Turn, for a moment, to the well understood case of Hessenberg size-two DAEs
Letting
I
I
we may rewrite (2.9) as a DAE (2.1) with properly stated leading term. Then
we have
I
Furthermore
is the projection onto im B 12 (t) along ker B 21 (t). In our
context,
are projectors that realize the decomposition
(we have ker here). The second equation in (2.9) gives simply
while the rst one leads to
and
In order to extract a regular explicit ODE with respect to the unknown
component one has to dierentiate
I H and to replace (I H)x 0
Then as an inherent
regular ODE, the equation
results. The solutions of (2.9) are given by the formula
I
Note that there is no need to dierentiate the second equation of (2.9) or
any part of the coe-cients { except for the projection H and the term
q. Thus we obtain solvability for all
supposed H belongs to the class C 1 .
In the following, we will derive an anlogous result for general equations (2.1),
where H and I H correspond to DQ 1 D and DP 1 D .
Lemma 2.1 Given an index-2 DAE (2.1).
denote the projector onto N 1 (t) along S 1
Then, for all t 2 I, the decomposition
holds true, and
are the uniquely determined projectors that realize this decomposition.
If, additionally, D(t)S 1 (t) and D(t)N 1 (t) are spanned by continuously dier-
entiable on I functions, then
Proof.
This assertion can be veried in the same way as [2], Lemma 2.5 and Lemma
2.6 were proved for the case of
In contrast to the index-1 case, im D is once more smoothly decomposed into
the further subspaces DS 1 and DN 1 for index-2 DAEs. Let us mention, that
if we think of index-3, we have to decompose DS 1 again and so on.
Formally, we put the index-1 and index-2 cases together for brevity, as we
have trivially
for index-1 DAEs. Hence, G 2 (t) is nonsingular on I for index-1 and index-2
DAEs, i. e.,
In the following, Q 1 (t) always denotes the projector onto N 1 (t) along S 1 (t).
Recall from [8], Appendix A the respresentation
Relation (2.17) leads to the properties
Further, due to the chain construction we compute
Consequently, scaling (2.1) by G 1
2 leads to
Multiplying we decouple this equation into
the following three ones:
Now, given any solution x
D
By this, we nd the solution representation
where the inherent regular ODE
and K := I Q
is a nonsingular matrix
function.
Clearly, this generalizes the well-known representation described above for
Hessenberg size-two forms (2.9). In the index-1 case (2.23), (2.24) simplify
to
Of course, the regular explicit ODE (2.24) can be formed without assuming
the existence of a solution of the DAE just by giving the coe-cients A; D; B
and the right-hand side q.
Denition 2.3 Given a DAE (2.1) of index  2 f1; 2g. The explicit ODE
(2.24) is said to be the inherent regular ODE of the DAE.
Lemma 2.2 Given an index- DAE (2.1),  2 f1; 2g. Let DP 1
be continuously dierentiable.
(i) Then, the subspaces D(t)S 1 (t) and D(t)N 1 (t) as well as the inherent
regular ODE are uniquely determined by the problem data.
is a time-varying invariant subspace of the inherent regular
ODE, i. e., if a solution belongs to this subspace at a certain point, it
runs within this subspace all the time.
(iii) If D(t)S 1 (t) and D(t)N 1 (t) do not vary with time t, then, solving the
IVP for (2.24) with the initial condition u(t  yields the
same solution as solving this IVP for
Proof.
(i) (cf. [2]) When constructing the ODE (2.24) the only arbitrariness is the
choice of the projector P 0 and the corresponding D . So we start with
two dierent P 0 and ~
and the corresponding D and ~
D . Recall
that then DD
~
~
~
~
~
~
~
~
Consequently, since the projectors DP 1 D ; DQ 1 D do not depend on
the choice of P 0 , their images do not so, either.
Further, due to
~
~
~
~
I; we have
~
~
G 1= D ~
~
~
~
~
~
(ii) Given any solution ~
some
into (2.24) and multiply the resulting identity by
hence, for ~
vanishes identically as the solution of a homogeneous regular initial
value problem.
(iii) (cf. [13]) If D(t)S 1 (t) and D(t)N 1 (t) are constant, we may use constant
projectors subspaces. Then it holds that
In the consequence the term (DP 1 D
disappears.
In comparison with dierent notions used in the standard DAE theory (un-
derlying ODE, essentially underlying ODE, inherent regular ODE), where
certain transforms resp. projectors are not completely determined by the
problem data, the decoupling described above is fully given by the problem.
This is a real advantage. If the nullspace of A(t) is trivial, i. e., im D(t)
has dimension n as it is the case for the Hessenberg size-two DAE (2.9), we
can also speak of an inherent state space system (in minimal coordinates),
supposed D(t)S 1 (t) does not vary with t. Then, the inherent regular ODE
(2.24) is essentially nothing else but an explicit ODE located in the constant
state space DS 1 . However, in the index-2 case D(t)S 1 (t) and D(t)N 1 (t) (cf.
H(t) in (2.9)) vary with time in general.
Theorem 2.3 Given a DAE (2.1) with index ;  2 f1; 2g,
(i) For any
equation (2.1) is solvable on C 1
D
(ii) The initial value problem for (2.1) with the initial condition
D
(iii) Supposed I is compact, the inequality
becomes true, i. e., the DAE has perturbation index .
(iv) The subspace S ind (t) := K(t)D(t) D(t)S 1
the geometric solution space of the homogeneous equation (2.1) with
ind (t) is lled by solutions and all solutions run within
this subspace S ind
Proof. (cf.
(i) is a simple consequence of (ii).
(ii) Solve the regular ODE (2.24) with the initial condition
Using the solution u
we construct the function x 2 according to formula (2.23).
Obviously,
2 q is continuously
dierentiable, thus x 2 C 1
D we check that x satises (2.1)
indeed.
The homogeneous IVP has only the trivial
solution.
(iii) and (iv) are due to the construction.
Remark 2.5 One could relax the constant rank conditions in the Denitions
2.1 and 2.2 to characterize a class of DAEs (2.1) that have index ;   2,
but undergo certain nondangerous index changes. If the interval I consists
of a number of subintervals where all constant rank conditions are satised,
we realize the decoupling on all these subintervals simulaneously and ask then
for a continuously dierentiable extension of DP 1 D on the whole interval.
If it exists, we may take advantage of (2.23), (2.24) again. However, we have
to consider that some terms may have discontinuities now.
Example 2.1 The system x 0
has the solution x
independently whether the coe-cient function  2 C(I; R)
vanishes. If we put this system in the form (2.1), we choose
further, if (t) does not vanish,
otherwise
Then, on subintervals with (t) 6= 0, we have
, i. e., the index is two there, further
and points where (t) is zero we have G I. Trivially the
problem has index 1 on those intervals. Then G
In both cases we have DP 1
Remark 2.6 In [2] we aimed at a theory to treat both an original DAE and
its adjoint equation in the same way . The adjoint equation of
is now
Both equations have properly stated leading terms at the same time. More-
over, they have index ;  2 f1; 2g, simultaneously. Nice relations between
the fundamental solution matrices are shown in [2].
Remark 2.7 The index notion given by Denition 2.2 resp. Remark 2.3 is
approved for creating a practical index monitor ([6]).
3 Numerical integration methods and
numerically well formulated DAEs
Given a DAE (2.1) with a properly formulated leading term and having index
2g. The natural modication of the k-step BDF applied to (2.1)
is
to provide an approximation x n to x(t n ) at the current step t
Let us use shorter denotations A n := A(t n ) and so on for all given coe-cient
functions and resulting projectors.
Denote the numerical derivative by
we proceed in the same way as we did, in Section 2, with the DAE (2.1)
itself, i. e. we scale by G 1
2n and then use the decoupling projectors. The
resulting system analogous to system (2.20)-(2.22) is, with
Denoting
we derive
nh
(D
A nj D n j x
where A nj :=
further
sjh)ds , so that
Insert (3.7) into (3.4) to obtain the recursion formula for
A nj
Then, by (3.5), (3.6), (3.8), we compute the representation of x n in the form
with
~
On the other hand, if we decoupled the DAE (2.1) rst, applied the same
BDF to the inherent regular ODE (2.24) and then used formula (2.23) to
compute x n , we would obtain
Comparing (3.11), (3.12) and (3.9), (3.10) we observe that, except for the
terms
these formulas coincide, supposed the subspaces D(t)S 1 (t); D(t)N 1 (t) do not
vary with t (cf. Lemma 2.2 above). Namely, in (3.11) and (3.12), we have
then
but in (3.9), (3.10) we use u
and, analogously, A nj
Denition 3.1 We will say that a discretization method and the DAE-decoupling
commute, if for the approximation x n obtained by applying the method to the
original DAE, and the further approximation ^
obtained by applying the
method to the inherent regular ODE and then using formula (2.23), it holds
that
and [DQ 1 G 1
n denotes a numerical approximation of (DQ 1 G 1
Let us stress that in the index-1 case, commutativity means simply x
because of Q
Theorem 3.1 Given an index- DAE (2.1),  2 f1; 2g; which has constant
characteristic subspaces DS 1 and DN 1 . Then the BDF-discretization and the
decoupling commute.
Remark 3.1 Recall once more that one has for the index-1 case.
Trivially, DS 1 and DN 1 are constant i im
being constant is a necessary condition
for DS 1 and DN 1 to be constant.
Remark 3.2 Here, we have chosen the BDF only for brevity. Of course,
similar results can be proved for Runge-Kutta methods. Also, nonlinear DAEs
may be considered ([14], [15]).
Remark 3.3 In particular, Theorem 3.1 generalizes the respective results
obtained for DAEs with constant leading nullspace (cf. (1.3) with P 0
e. g. in [8] for in [10] for 2. On the other hand, Theorem
3.1 covers the situation studied in [12], i. e., an equation (1.1) rewritten as
time-invariant.
Remark 3.4 It is a remarkable benet of commutativity that we can now
prove nice assertions on contractive or dissipative
ows and the respective
discretized versions ([14]). Having modied the respective notions (contrac-
tivity and dissipativity inequalities, adsorbing sets etc.) in such a way that
the standard ODE notions apply to the inherent regular ODE on its invariant
subspace DS 1 , we may use the well-known techniques and results approved in
the regular ODE case.
In particular ([14]), a stiy accurate and algebraically stable Runge-Kutta
method applied to a contractive index-1 DAE yields
for each two sequences of approximations starting with consistent x
Thereby,
is a bound of the canonical version of D corresponding to the
(canonical) projector onto S 0 along N 0 .
It is also shown in [14] that the Euler backward method re
ects the dissipativity
behaviour properly without any stepsize restriction.
Example 3.1 Rewrite the equation (cf. [20])
E(t)x 0 (t)+F
with
as
or as
where
Both reformulated versions (3.15) and (3.16) have properly formulated leading
Now, we observe that we have index-2 DAEs with constant subspaces as supposed
in Theorem 3.1. Thus, for both versions (3.15) and (3.16), the BDF
and the decoupling commute.
In both cases, the Euler backward yields simply
Recall that the Euler backward directly applied to (3.14) gets into great diculties
for jj > j1 g. [4], [20]).
Example 3.2 The standard form DAE E(t)x
has index-1 on I = R for arbitrary values  6= 0; - 6= 1. The solution is
For - < , one would expect the backward Euler method to generate a sequence
fx n;2 g n such that x n;2 ! 0(n !1) without any stepsize restriction.
However, the formula
reads in detail
While for the constant coe-cient case works ne, for - 6= 0, the
numerical solution may explode close to h  1. To realize j1
j1 hj one has to accept strong extra stepsize restriction.
On the other hand, if we apply the Euler method to the reformulation
we arrive at
and things work well. No stepsize restrictions are caused by stability.
Example 3.3 Consider the seemingly harmless Hessenberg index-2 DAE
with
The leading term may be rewritten in a proper form by choosing (cf. (2.9))
~
A ~
With ~
The subspaces
but also
~
~
move with time. We refer to [10] for the sobering eect of the numerical
tests. Recall that this problem was introduced in [10] to demonstrate that just
a Hessenberg form DAE may get into dieculties with extra stepsize restrictions

Observe, on the other hand, that there are more possibilities in factorizing
E. If we nd a further well-matched factorization AD such that
appear to be time-invariant, we are done since Theorem 3.1
applies. Such a much more comfortable factorization is given by ~
D,
The reformulated DAE is
D, and the
subspaces
are constant such that Theorem 3.1 applies in fact.
As we could see above, the problem whether discretization and decoupling
commute depends primarily on how the DAE is formulated.
Denition 3.2 A DAE (2.1) with index 1 is said to be numerically well
formulated if im D(t) is time-invariant (cf. [13]). A DAE (2.1) with index-
2 is said to be numerically well formulated if D(t)S 1 (t) and D(t)N 1 ) are
constant.
This new possibility to rearrange subspaces for better numerical properties
is a surprise. One should think further on how to exploit this idea best.
Fortunately, e. g. in circuit simulation, the relevant subspaces could be
shown to be constant for a large class of problems ([7], [5]), hence there is no
need for reformulations.
Special questions concerning necessary reformulations will be discussed in
[14],

Abstract

DAEs
Nowadays, e. g. in circuit simulation and simulation of multibody dynam-
ics, there is a remarkable interest in complex systems that consist of coupled
systems of DAEs and PDEs (cf. [21], [9]). There are also certain proposals
for treating so-called PDAEs (= partial dierential algebraic equations, e. g.
[17]). Thereby, one of the questions to be considered is how to formulate
initial and boundary conditions in an appropriate way.
In the following we try to show the usefulness of the decoupling procedure
given in Section 2 for the nite dimensional case, now in an abstract modi-
cation. Being able to understand those abstract DAEs, a more systematical
constructive and numerical treatment can be started on.
In this section we deal with abstract DAEs
are linear operators
acting in the real Hilbert spaces X; Y; Z.
For all t 2 I, let A(t) and D(t) be bounded and normally solvable. More-
over, let A(:) and D(:) depend continuously (in the norm sense) on t. As
bounded maps, A(t) and D(t) have nullspaces ker A(t) and ker D(t), respec-
tively, which are closed linear manifolds, i. e., subspaces. Due to the normal
solvability, im A(t) and im D(t) are subspaces (e. g. [1]), too.
For all t 2 I, let the operator B(t) be dened on a dense subset DB  X,
and let B(:)x 2 C(I; Y ) for all x 2 DB .
A continuous path x 2 C(I; X) with is called a solution
of (4.1) if Dx 2 C 1 (I; Z) and equation (4.1) is satised pointwise.
Denition 4.1 The leading term of (4.1) is properly stated if the operators
A(t) and D(t); t 2 I, are well matched in the following sense:
ker A(t)  im
(ii) The projector R(t) 2 L b (Z) that realizes this decomposition of Z (i. e.,
depends continuously
dierentiably on t.
Remark 4.1 Since R(t) depends smoothly on t, the subspaces im D(t) are
isomorphic for dierent values of t. For the same reason, the nullspaces
ker A(t) are also isomorphic.
Let L b (X) denote the linear space of linear bounded maps on X etc. For
bounded maps, we alway assume their denition region to be the respective
whole space.
Let clM denote the closure of the set M in the respective space.
Next we generalize the matrix- and subspace-chain given in (2.6) by introducing
the following further linear maps, linear manifolds and subspaces for
cl ker G 0 (t),
cl im G 0 (t),
cl ker G 1 (t),
cl im G 1 (t),
By construction, the operators G 1 (t) and G 2 (t) are at least densely dened
in X. In our context, operator products (in particular those with projectors)
are often dened on a larger region by trivial reasons. We will always use
the maximal trivial extensions.
denote the re
exive generalized inverse of D(t) such that
Denition 4.2 Equation (4.1) with a properly stated leading term is said to
be
(i) an abstract index-1 DAE if, for all t 2 I; dim(im W 0 (t)) > 0 and
injective and densely solvable,
(ii) an abstract index-2 DAE if, for all t 2 I;
depends continuously on t, and
injective and densely solvable.
If also Y is a bounded map that is continuous with respect to
t in norm sense, then the assertions in Section 2, in particular Theorem 2.3,
can be modied to hold true for the abstract DAE in a straightforward way.
The more challenging problem is an only densely dened B(t). Hence, we
are going to study three dierent cases of this type.
Case 1: A coupled system of a PDE and Fredholm integral equations (on a
special application of this type I was kindly informed by Hermann Brunner).
Given a linear Fredholm integral operator
a linear dierential operator
4 with c  0, and linear bounded coupling operators
s . The system to be considered is
Using the corresponding matrix representations for A; D; B; with
s  L
2 we rewrite (4.2) in the form
(4.1). Namely, we have
K L
is dened on
s .
Clearly, it holds that N
I
I
are dened on X (as trivial extensions of bounded maps).
G 1 is a bijection such that this abstract DAE has index 1.
we nd DG 1
dened on _
Each solution of the DAE is given by the expression
where u(t) is a solution of the abstract regular dierential equation
Obviously, one has to state an appropriate initial condition for (4.3), i. e.,
Case 2: A special linear constant coe-cient PDAE discussed in [17].
Consider the PDAE4 1
Suppose 3.
How should we formulate boundary and initial conditions?
Rewrite (4.4) in abstract form with x(t) := w(; t).
Choose
L
L
L
and use
again matrix representaions for our coe-cients, i. e.,
0 a
is the Laplacian. Let us start with
The operator
is dened on X and bounded.
Obviously, ker G is a nontrivial
closed subspaces and im G
L
0 is closed.
is dened on
is a subspace, and obviously
0 a
5 represents a projector onto N 1 along S 1 , thus
. The map
is dened on L
C
L
8 further
L
C
densely solvable.
The injectivity of G 2 is immediately checked, therefore, this DAE has index
2.
The inherent regular dierential equation is now
with
and
dened on L
dened on C
L
i. e., (4.6) is in fact nothing else but
Now it becomes clear that, for (4.6), the initial condition
as well as boundary conditions should be
given. We take homogeneous Dirichlet conditions and put them into the
denition region of B, i. e., we restart our procedure with
instead of (4.5) or, in spite of more general solvability, with
Using (4.8), G 2 is dened on L
L
6 but G 1
2 on
L
Consequently, admissible right-hand sides are q(t)
with
1(Computing
0 a
we nd the solution representation (cf. (2.23)
A q(t) +@ 0a
c3
q(t) a
while x 1 (t) solves (4.7). No further initial or boundary conditions should be
given. With (4.8) we obtain the unique solvability of the IVP
Case 3: A PDAE and a DAE coupled by a restriction operator.
Consider the system
~
uj
@
~
Assume the linear restriction map
R m to be bounded
and to depend continuously on t.
Rewrite the system (4.9), (4.10) as an abstract DAE for
~ u(; t)
~
and choose
A :=
A
A ~
R
Restrik ~
where DB := _
is dened on X,
im ~
Supposing that the operator Restrik(t) maps into im ~
holds that
~
is the projector onto N 1 along S 1 , G
In the general case the projector Q 1 onto N 1 along S 1 is more di-cult to
construct. Obviously, we have N 1 \
G 2 is a bijection if ~
G 2 is so.
Consequently, the coupled system (4.9) (4.10) interpreted as an abstract DAE
has the same index as the DAE (4.10).
5 Linear-quadratic control problems
Now, the quadratic cost functional
is to be minimized on solutions of the DAE
which satisfy the initial condition
real Hilbert spaces,
Let all these operators depend continuously in the norm sense on t.
Assume further:
(ii) V and the maps
are positive semidenite.
(iii) A(t) and D(t) are normally solvable and well matched for t 2 I.
Admissible controls are those functions u 2 C(I; U) for which a solution
D (I; X) of the IVP (5.2), (5.3) exists.
Consider the boundary value problem
The system (5.4) is a DAE with a properly statet leading term. D(t)  and
A(t)  are normally solvable and well matched at the same time as A(t) and
are so.
Theorem 5.1 If the triple x  2 C 1
A  (I; Y ); u  2 C(I; U)
solves the BVP (5.4), (5.5), then u  is an optimal control for the problem
(5.1)-(5.3).
Proof. By straightforward calculations we prove that the expression
vanishes. Then the assertion follows immediately from
R x(t) x  (t)
u(t) u  (t)
x(t) x  (t)
u(t) u  (t)
dt
In Theorem 5.1, no assumption on the index of the DAE (5.2) is made. How-
ever, for obtaining a theoretically and practically solvable BVP (5.4), (5.5),
additional conditions ensuring the index-1 property of (5.4) should be satised
denote the orthoprojectors onto ker AD and along im AD, respectively
(cf. Section 4).
Denote
Lemma 5.2 The DAE (5.4) has index 1 i, for t 2 I,
Proof. Put ~
The respective maps for (5.4) rewritten as ~
are
~
~
~
~
Y is a bijection i
acts bijectively on
ker D(t)  ker A(t)   U onto ker A(t)   ker D(t)  U .
Further, ~
is bijective at the same time as G 1 (t) is so (cf. Remark 2.3).
Remark 5.1 In [16], a linear-quadratic control problem of a similar form is
studied. Instead of the DAE (5.2) the equation
is considered. Because of the somehow incomplete leading term, the resulting
system corresponding to (5.4) looks less ne and transparent.
Remark 5.2 Since the operator F has the same structure as its counterpart
in [16], all su-cient conditions for bijectivity proved in [16] hold true also
here.
Lemma 5.3 Let, for t 2 I,
be bijective. Then the inherent regular dierential equation of (5.4) is of the
Dx
A
R
Dx
A
are positive semidenite.
Proof.
This assertion can be proved following the lines of [16], Theorem 1.
Obviously, (5.8) is a non-negative Hamiltonian system
RR but this is exactly the case if (5.4) is numerically well
formulated. For Hamiltonian systems, we know the respective BVPs to be
solvable.
Theorem 5.4 Let F(t) in (5.7) be bijective, R 0
is at least one optimal control for the problem (5.1), (5.2), (5.3).
If Z is nite dimensional, there is exactly one optimal control of (5.1), (5.2),
(5.3).
Proof.
It remains to show uniqueness in case of nite dimensional Z, but this can
be done in the same way as [16], Theorem 4, is proved.


--R

Theorie der linearen Opera- toren im Hilbertraum


Numerical solution of initial value problems in di












Indexes and special discretization methods for linear partial di



Simulation gekoppelter Systeme von partiellen und di
--TR
Difference methods for the numerical solution of time-varying singular systems of differential equation
Stability of computational methods for constrained dynamics systems
On Asymptotics in Case of Linear Index-2 Differential-Algebraic Equations
Runge-Kutta methods for DAEs. A new approach
A Differentiation Index for Partial Differential-Algebraic Equations
Analyzing the stability behaviour of solutions and their approximations in case of index-2 differential-algebraic systems

--CTR
G. A. Kurina, Linear-Quadratic Discrete Optimal Control Problems for Descriptor Systems in Hilbert Space, Journal of Dynamical and Control Systems, v.10 n.3, p.365-375, July 2004
I. Higueras , R. Mrz , C. Tischendorf, Stability preserving integration of index-1 DAEs, Applied Numerical Mathematics, v.45 n.2-3, p.175-200, May
I. Higueras , R. Mrz , C. Tischendorf, Stability preserving integration of index-2 DAEs, Applied Numerical Mathematics, v.45 n.2-3, p.201-229, May
numerical integration methods;differential algebraic equations;abstract differential algebraic equations;partial differential algebraic equations