Nonlinear differential-algebraic equations with properly formulated leading term

Nonlinear di erential-algebraic equations with properly formulated leading term

R. Marz

1 Introduction

In BaMa], a uniform theory for investigating linear dierential-algebraic equations (DAEs) and their adjoint equations was proposed. By means of an additional coef- cient matrix it is exactly xed which derivatives of the solutions searched for are actually involved in the equation. Such a DAE is of the form

A(t)(D(t)x(t))⁰ +B(t)x(t) =q(t) t^2I (1.1) where the coecients A(t) andD(t) match well.

As a nonlinear version,

A(x(t) t)(D(t)x(t))⁰+b(x(t) t) = 0 (1.2) can be taken into account rst (e.g. HiMa]). However, with somewhat more con- sistency, we obtain equations of the form

A(x(t) t)(d(x(t) t))⁰+b(x(t) t) = 0 (1.3) which we want to investigate in this paper.

Note that it is just this sort of equations that result in the simulation of electric circuits, which is the origin of DAEs (e.g. EsTi]). In this case, a transition to

A(x(t) t)dx(x(t) t)x⁰(t) +b(x(t) t) +A(x(t) t)dt(x(t) t) = 0 is problematical.

In the next section (^x2) we will determine what we mean by a properly formulated leading term. In ^x3 DAEs with index , ^{2 f}1 2^g will be characterized by alge- braic criteria. ^x4 is devoted to linearization and perturbation theorems. A special structure, which is important for electric circuits for instance, will be analyzed in

x5. Finally, constraint sets will be investigated in ^x6.

2 Properly formulated leading terms and an equivalence theorem

We investigate the equation

A(x(t) t)(d(x(t) t))⁰+b(x(t) t) = 0 (2.1) 1

2 R. Marz with coecient functions A(x t) ² L(^{Rn Rm}) d(x t) ^{2 Rn} and b(x t) ^{2 Rm} x ²

D Rm t ^{2 I R}, that are continuous in their arguments, and which have the continuous partial derivatives Ax dx bx. Denote D(x t) = dx(x t).

De nition 2.1

The leading term in (2:1) is said to be properly formulated if kerA(x t) im D(x t) =^Rn x^2D t^2I (2.2) and if there exists a projector function R ² C¹(^I L(^Rn)) such that R(t)² = R(t), kerA(x t) = kerR(t) im D(x t) = im R(t), and d(x t) = R(t)d(x t) holds for x^2D t ^2I.

Consequently, the matrices A(x t) and D(x t) have constant rank for properly for- mulated leading terms. The subspaces A(x t) and imD(x t) are independent of x and have bases from the class C¹. It holds that A=AR D =RD.

If A(x t) and D(x t) ful l the condition (2.2), but only ker A(x t) is independent of x and smooth, then we have, with a projector PA ² C¹(^I L(^Rn)), the relation A(x t) =A(x t)PA(t) and, hence

A(x(t) t)(d(x(t) t))⁰ =A(x(t) t)(PA(t)d(x(t) t))^0;A(x(t) t)P_A⁰(t)d(x(t) t): Then, with ~A(x t) := A(x t) ~d(x t) := PA(t)d(x t) ~b(x t) := b(x t)^;

;A(x t)P_A⁰(t)d(x t), the equation

A~(x(t) t)(~d(x(t) t))⁰+ ~b(x(t) t) = 0 (2.3) has a proper leading term because of im ~D(x t) = imPA(t). We can proceed anal- ogously if only imD(x t) is independent of x, or if the last condition of De nition 2.1 in not ful lled. Hence, a proper formulation can be obtained if (2.2) holds and if one of the characteristic subspaces is independent of x and comes fromC¹.

De nition 2.2

^{A function x 2 C(I}x ^Rm) ^Ix ^I, is said to be a solution of equation (2.1) if x(t) ^2D t ^{2 I}x and d(x(:) :) ² C¹(^Ix ^Rn), and if equation (2:1) is fullled pointwisely.

Unfortunately, regularity conditions do not de ne a linear function space here in general. In case d(x t) = D(t)x is linear itself (cf. (1.2)), a linear solution space is available by C_D¹ :=^fx²C :Dx²C^1g.

Fortunately, it is relatively simple to transform equation (2.1) into a (1.2) form.

This allows the application of standard-notions and -methods (dierentiability, lin- earization etc.) that are based on linear funtion spaces.

We form the natural extension for equation (2.1) with proper leading term

A(x(t) t)(R(t)y(t))⁰+b(x(t) t) = 0 (2.4) y(t)^;d(x(t) t) = 0: (2.5)

Nonlinear dierential-algebraic equations with properly formulated leading term 3 Withx=^;x_y A=^;A0 d(x t)=R(t)y D(x t)=D(t)=(0 R(t)) b(x t)=^;_y^;b(_d^{x t(}_{x t}⁾⁾ we can write (2.4), (2.5) as

A(x(t) t)(D(t)x(t))⁰+b(x(t) t) = 0: (2.6) Due to ker A(x t) = ker A(x t), imD(t) = im R(t) = imD(x t) also (2.6) has a properly formulated leading part with R(t) =R(t) as the corresponding projector.

Now the linear function space C_D¹ =

x=

x y

2C :Dx =Ry ²C¹

(2.7) oers itself as solution space for (2.6). For (2.6) we seek functions x ² C_D¹ whose function values lie in the domain of de nition^Dof the coecients and ful l equation (2.6) pointwisely.

If x(:) is a solution of the original equation (2.1), then the pair x(:) y(:) with y(t) : d(x(t) t) is obviously a solution of the class C_D¹ for (2.6). If, reversely, a pair x(:) y(:) from C_D¹ forms a solution of (2.6), then d(x(:) :) = Rd(x(:) :) = Ry ²C¹ also holds because of Ry ²C¹, andx(:) is a solution of equation (2.1).

Theorem 2.3

Let the leading term of (2:1) be properly formulated.

(i) The the leading term of the extension (2:6) is properly formulated, too.

(ii) The equations(2:1) and (2:6) are equivalent via the relation y(:) =d(x(:) :): The sets

M

0(t) := ^fx^2D:b(x t)² im A(x t)^g and

M

0(t) := ^fx=

x y

2DR

n :b(x t)² im A(x t) y=d(x t)^g

= ^fx^2DRn :x^2M0(t) y=d(x t)^g:

are the geometrical location of the solutions of (2.1) and (2.6), respectively, It always holds that

x(t)^2M0(t) x(t)^2M0(t):

The problem in how far these sets are lled with solutions leads to notions of indices and corresponding solvability statements.

3 Subspaces, matrix chain and index

In this section we de ne characteristic subspaces and matrix chains for (2.1) and (2.6).

Further, let B(y x t) := (A(x t)y)x + bx(x t) for y ^{2 Rn} x ^{2 D} t ^{2 I} and

4 R. Marz B(y x t) := (A(x t)y)⁰_x+bx(x t) fory ^2Rn x^2DRn t ^2I. More precisely, we have B(y x t) =

A(x t)y 0

x+

bx(x t) 0

;D(x t) I

=

B(y x t) 0

;D(x t) I

For the original equation (2.1) we form for x^2D t ^2I y ^2Rn: G⁰(x t) = A(x t)D(x t)

N⁰(x t) = kerG⁰(x t)

S⁰(y x t) = ^fz ^2Rm :B(y x t)z² im G⁰(x t)^g G¹(y x t) = G⁰(x t) +B(y x t)Q⁰(x t)

with a projector Q⁰(x t)²L(^Rm) onto N⁰(x t), P⁰(x t) = I^;Q⁰(x t)

N¹(y x t) = kerG¹(y x t)

S¹(y x t) = ^fz ^2Rm :B(y x t)P⁰(x t)z ² im G¹(y x t)^g:

For (2.6) this yields G⁰(x t) = A(x t)D(x t) etc. for x ^{2 DRn} t ^{2 I} y ^{2 Rn}. N⁰(t) =^RmkerR(t) depending ont only and being smooth is a special feature of (2.6).

Now, the relations among the subspaces of (2.1) and (2.6) are important, because the index will be de ned via these subspaces later on.

Lemma 3.1

Let equation (2:1) have a proper leading term, then

N⁰(t)^\S⁰(y x t) = (N⁰(x t)^\S⁰(y x t))0 (3.1) and

N¹(y x t)^\S¹(y x t) =^n;2RmRn :=Q⁰(x t) =R(t)

D(x t)^;+² N¹(y x t)^\S¹(y x t)^o: ^(3.2) P r o o f. We determine

G⁰(x t) = A(x t)D(x t) =

0 A(x t)

0 0

N⁰(t) =

2R

m

R

n :A(x t) = 0

=^Rm kerR(t)

S⁰(y x t) =

:B(y x t) ² im A(x t) ^;D(x t)+ = 0

=

: ²S⁰(y x t) =d(x t)

Nonlinear dierential-algebraic equations with properly formulated leading term 5

N⁰(t)^\S⁰(y x t) =

:R(t) = 0 =D(x t) ²S⁰(y x t)

=

: ²N⁰(x t)^\S⁰(y x t) = 0

: Furthermore, by Q⁰(t) =

I 0

0 I^;R(t)

P⁰(t) =

0 0 0 R(t)

we also have B(y x t)P⁰(t) =

0 0 0 R(t)

G¹(y x t) =

B(y x t) A(x t)

;D(x t) I^;R(t)

: which implies

S¹(y x t) =

8

>

<

>

:

;

:^;_R⁽⁰_t⁾² im

2

6

4

B(y x t) A(x t)

;D(x t) I^;R(t)

3

7

5 9

>

=

>

= ^n;:R(t) =^;D(x t)+ (I ^;R(t))

with ^{2Rm 2Rn} B(y x t)+A(x t) = 0^o

= ^n;:R(t) =^;D(x t) with ²S⁰(y x t)^o

N¹(y x t) = ^n;:B(y x t)+A(x t) = 0 D(x t) = 0 =R(t)^o

= ^n;: ²N⁰(x t) =R(t) 0 = (A(x t)D(x t)+

+B(y x t)Q⁰(x t))(D(x t)^;+)^o

= ^n;: =Q⁰(x t) =R(t) D(x t)^;+ ²N¹(y x t)^o: Finally, we obtain

=Q⁰(x t) =R(t) =^;D(x t) ²S⁰(y x t) D(x t)^; + ²N¹(y x t) for ^;2N¹(y x t)^\S¹(y x t). Thus,

B(y x t)P⁰(x t)(D(x t)^;+) = B(y x t)P⁰(x t)D(x t)^;

= ^;B(y x t)P⁰(x t) ² im G¹(y x t) because of B(y x t)=G⁰(x t)w, i. e.,

B(y x t)P⁰(x t)= (G⁰(x t) +B(y x t)Q⁰(x t))(P⁰(x t)w^;Q⁰(x t)): ²

Conclusion 3.2

^{For (2}:1) and (2:6) it holds that

dim(N⁰(t)^\S⁰(y x t)) = dim(N⁰(x t)^\S⁰(y x t)) and dim(N¹(y x t)^\S¹(y x t)) = dim(N¹(y x t)^\S¹(y x t)):

6 R. Marz

De nition 3.3

An equation (2:1) with properly formulated leading term is called a DAE of index 1 if

N⁰(x t)^\S⁰(y x t) = 0 for x^2D t^2I y^2Rn or a DAE of index 2 if

dim(N⁰(x t)^\S⁰(y x t)) = const

N¹(y x t)^\S¹(y x t) = 0 for x^2D t ^2I y ^2Rn:

Theorem 3.4

The original equation (2:1) with proper leading term and its natural extension (2:6) have the index ^2f1 2^g simultaneously.

Now we continue the above matrix chain with

G²(y x t) :=G¹(y x t) +B(y x t)P⁰(x t)Q¹(y x t) and G²(y x t) :=G¹(y x t) +B(y x t)P⁰(t)Q¹(y x t)

where Q¹(y x t) ² L(^Rm), Q¹(y x t) ² L(^Rm+n) are projectors onto N¹(y x t) and N¹(y x t), respectively. We will investigate only problems (2.1) with index

2f1 2^ghere, hence,

N¹(y x t)S¹(y x t) =^Rm (3.3) holds on principle, in fact for = 1 with N¹(y x t) = 0, S¹(y x t) = ^Rm trivially, and for= 2 due to GrMa], Theorem A.13. Hence, we may assume thatQ¹(y x t) projects onto N¹(y x t) alongS¹(y x t).

Analogously, letQ¹(y x t) be the projector ontoN¹(y x t) alongS¹(y x t). Simple computation now yield

Lemma 3.5

^{For 2 f}1 2^g it holds that Q¹ =

0 Q⁰Q¹D^; 0 DQ¹D^;

D P¹D^; =DP¹D^; D Q¹D^;=DQ¹D^; and D Q¹G^;12 = (DQ¹G^;12 DQ¹D^;).

4 Linearization and perturbation theorems

First, we consider the equation

A(x(t) t)(D(t)x(t))⁰+b(x(t) t) = 0 (4.1) i.e., the case that D(x t) = D(t)x x ^{2 D} t ^{2 I}. Later on, we apply the obtained results via (2.6) to the general form (2.1).

We x x ² C_D¹(^{I Rm}) with x(t) ^{2 D} t ^{2 I}. Let ^{I I} be compact. For all

Nonlinear dierential-algebraic equations with properly formulated leading term 7 x from a suciently small neighbourhood ^U(x) of x inC_D¹(^{I Rm}) we can de ne the map

F : ^U(x)C_D¹(^{I Rm})^!C(^{I Rm})

F(x) := A(x(:) :)(Dx)⁰(:) +b(x(:) :) x^2U(x) (4.2) and we can write equation (4.1) as

F(x) = 0 where ^F is Frechet-dierentiable. The derivative

Fx(x)²Lb(C_D¹(^{I Rm}) C(^{I Rm})) is given by

Fx(x)⁴x = A(x(:) :)(D⁴x)⁰(:) +B((Dx)⁰(:) x(:) :)⁴x(:)

for ⁴x²C_D¹(^{I Rm}) (4.3)

where we assume natural norms on C_D¹(^{I Rm}) and C(^{I Rm}).

For applying the implicit function theorem (e.g. KaAk]), the property of the image im^Fx(x) is of crucial importance.

With A(t) =A(x(t) t) B(t) =B((D(t)x(t))⁰ x(t) t) t^2I, the linear DAE A(t)(D(t)⁴x(t))⁰ +B(t)⁴x(t) =q(t) t ^2I (4.4) is nothing else but the equation

Fx(x)⁴x=q:

Let S⁰(t) N⁰(t) G¹(t) etc. denote the chain of subspaces and matrices generated for (4.4).

Lemma 4.1

Let the DAE (4:1) have a properly formulatd leading term. Let DP¹D^; DQ¹D^;2C¹(^I L(^Rn)).

(i) The index--property, ^{2 f}1 2^g, transforms itself from (4:1) onto the lin- earization (4:4).

(ii) For = 1, ^Fx(x) is surjective.

(iii) For = 2 it holds that im^Fx(x) = C_DQ¹ 1G^;12(^{I Rm}).

P r o o f. The assumption (i) immediately results from the construction of the subspaces and the matrices in ^x3. The assumptions (ii) and (iii) are concluded from the existence theorems for linear DAEs in BaMa]. ²

Conclusion 4.2

In the index-2 case, im^Fx(x) is a non-closed proper subset in C(^{I Rm}).

8 R. Marz If the DAE (4.1) has a dynamic degree of freedom, it has to be completed by initial or boundary conditions. Since A and D are singular, we cannot expect a degree of freedom mas in the case of regular DAEs, but a lower one. An IVP with the initial condition x(t⁰) = x^{0 2Rm} is not solvable in general. Hence,x⁰ has to be consistent to an extend.

Basing on our experience with linear DAEs ( BaMa]), we impose an initial condition for (4.1) with t^{0 2I} in the form

D(t⁰)(x(t⁰)^;x⁰) = 0 with x^{0 2Rm} (4.5) where ²L(^Rm) will still have to be xed. By means of the mapping

FIV P : ^U(x)C_D¹(^{I Rm})^!C(^{I Rm})LIC

LIC := im D(t⁰)

FIV P(x) := (^Fx D(t⁰)x(t⁰)) x^2U(x) (4.6) we can describe the IVP (4.1), (4.5) in a compact way by

FIV P(x) = (0 D(t⁰)x⁰): Now, the equation

FIV P(x) = (q D(t⁰)x⁰) corresponds to the perturbed IVP

A(x(t) t)(D(t)x(t))⁰ +b(x(t) t) =q(t) t^2I (4.7) D(t⁰)(x(t⁰)^;x⁰) = 0: (4.8)

Theorem 4.3

^{Let (4:}1) be a DAE with proper leading term and index ^{2 f}1 2^g, and let x ²C_D¹(^{I Rm}) be a solution of (4:1) and

DP¹D^; DQ¹D^{; 2}C¹(^I L(^Rn)) :=P¹(t⁰): Further, for x^2U(x), let

D(t)Q ¹(t)G²(t)^;1b(x(t) t) be continuously dierentiable w.r.t. t: (4.9) (i) If = 1, then the IVP (4:7) (4:8) is uniquely solvable on ^I for arbitrary x^{0 2 Rm} with ^jD(t⁰)(x^{0 ;}x(t⁰))^j , and q ² C(^{I Rm}) with ^k q ^k1 , >0 su ciently small. For the solutions x²C_D¹(^{I Rm}) it holds that

kx^;x ^kC^D1 const (^jD(t⁰)(x^0;x(t⁰))^j+^kq^k1):

(ii) If = 2, then the IVP (4:7) (4:8) is uniquely solvable on ^I for arbitrary x^{0 2Rm} with

jD(t⁰)(x^0;x(t⁰))^j and q²C_DQ^{;1 1}_G ²(^{I Rm}) ^kq^k1 +

k(DQ ¹G^;12q)^{0 k1} >0 su ciently small.

For the solutions x²C_D¹(^{I Rm}) it holds that

k x^;x ^kC^D1 const (^jD(t⁰)(x^0;x(t⁰))^j+^kq^k1 + ^k(DQ¹G^;12q)^{0 k1}):

Nonlinear dierential-algebraic equations with properly formulated leading term 9 (iii) The solution x of the IVP (4:1) (4:8) is continuously dierentiable w.r.t. x⁰.

The sensitivity matrix xx⁰(t) =:X(t)²L(^Rm) satises the IVP A(x(t) t)(D(t)X(t))⁰ +B((D(t)x(t))⁰ x(t) t)X(t) = 0 t^2I

D(t⁰)(X(t⁰)^;I) = 0:

Remark:

If we take into account that P¹(t) = I, Q¹(t) = 0 for = 1, then the smoothness assumptions of Theorem 4.3. are always trivially given in this case. For = 2 the regularity condition (4.9) implies restrictions of the admissible structure of (4.1). The following condition is sucient for (4.9):

D^;2C¹(^I L(^{Rn Rm}))

(x t) := D(t)Q ¹(t)G²(t)^;1b(x t) is continuously dierentiable and

b⁰_x(P⁰(t)x+sQ⁰(t)x t)Q⁰(t)x² im G¹(t) s² 0 1] x^2D t^2I: (4.10) Namely, then_d (x t) (P⁰(t)x t) is true and for x(:) ^{2 U}(x) it holds that

dt(x(t) t) = _dt^d(P⁰(t)x t) = _dt^d(D(t)^;D(t)x(t) t) = x(D(t)^;D(t)x(t) t)

fD(t)^;0D(t)x(t) +D(t)^;(D(t)x(t))^0g +t(D(t)^;D(t)x(t) t). The condition (4.10) is equivalent to

W ⁰(t)b⁰_x(P⁰(t)x+sQ⁰(t)x t)Q⁰(t)x² im W ⁰(t)B ⁰(t)Q⁰(t): (4.11) If the derivative free part of (4.1) is linear in x or if (4.1) is a DAE in Hessenberg form, then (4.11) is given.

P r o o f of Theorem 4.3:

FIV P x(x) is a bijection from C_D¹(^{I Rm}) onto C(^{I Rm})L^IC for = 1.

For = 2, ^FIV P x(x) is injective but not surjective. According to Lemma 4.1, im^FIV P x(x)L^IC =C_DQ¹ 1G^;12(^{I Rm})L^IC =:X.

Equipped with a natural norm, X is a Banach space. We summarize the two cases that = 1 = 2 by using Q¹ = 0 P ¹ =I for = 1.

For the mapping

H(x d q) := ^FIV P(x)^;(q d) x^2U(x) (q d)²X it holds, due to the condition (4.9), that

H(x d q)²X:

With d :=D(t⁰)x(t⁰) we obtain

H(x d 0) = 0 ^Hx(x d 0) = ^FIV P x(x):

Hx(x d 0) is a homeomorphism for = 1 as well as for = 2. Due to the im- plicit function theorem there exists a uniquely determined continuously dierentiable mapping

f :B(d )B(0 )X ^!C_D¹(^{I Rm})

10 R. Marz withf(d 0) = x,^H(f(d q) d q) = 0,^kf(d q)^;f(d 0)^k_CD¹ K(^jd^;d^j+^kq^kX) for ^jd^;d^{j k}q^kX:=^kq^k1 +^k(DQ¹G^;12q)^{0 k1} .

Then the two assumption (i) and (ii) follow because of d=D(t⁰)x⁰.

In particular, f(d 0) is continuously dierentiable w.r.t. d, i.e., the solution x(:) of the IVP (4.1) with D(t⁰)x(t⁰) = d has a continuous derivative w.r.t. d. With d=D(t⁰)x⁰ this impliesxx⁰(t) =xd(t)D(t⁰). As usually, the variational equation (4.10) now results by dierentiation w.r.t. x⁰: ²

If we introduce a perturbation index for (4.1.) analogously to the standard case of DAEs, then the inequalities in (i) and (ii) mean that a DAE (4.1) with index has the perturbation index ^2f1 2^g, too.

For the extended system (2.4), (2.5) and for (2.6), respectively, it holds withx=^;x_y,

4x=^;44x_y that

Fx(x)⁴x=

A(x(:) :)(R⁴y)⁰(:) +B((Ry)⁰(:) x(:) :)⁴x(:)

4y(:)^;D(x(:) :)⁴x(:)

:

In particular, the equation^Fx(x)⁴x= (q r) withx =^;x_y is nothing else but the linear DAE

A(x(t) t)(R(t)⁴y(t))⁰+B((R(t)y(t))⁰ x(t) t)⁴x(t) = q(t)

4y(t)^;D(x(t) t)⁴x(t) = r(t): (4.12) Now, let x(:) be a solution of the nonlinear equation (2.1) and y = d(x(:) :).

Then x(:) solves the extended form (2.4), (2.5) with y =Ry =d(x(:) :).

With the coecients

A(t) = A(x(t) t) D(t) =D(x(t) t)

B(t) = B((d(x(t) t))⁰ x(t) t) (4.13) (4.12) can be reduced to

A(t)(R(t)⁴y(t))⁰+B(t)⁴x(t) = q(t)

4y(t)^;D(t)⁴x(t) = r(t): (4.14) For r(t) = 0 this yields

A(t)(D(t)⁴x(t))⁰ +B(t)⁴x(t) =q(t) (4.15) which can be regarded as a linearization of the initial equation (2.1).

By Theorem 3.4 and Lemma 4.1 the index ^2f1 2^g is transformed from (2.1) to (4.14) and (4.15).

Now, let N⁰, S ⁰, G¹ etc. be the subspaces and matrices of the chain formed for A,D and B from (4.13).

We investigate the perturbed IVP

A(x(t) t)(d(x(t) t))⁰+b(x(t) t) =q(t) (4.16)

Nonlinear dierential-algebraic equations with properly formulated leading term 11 D(t⁰)D(t⁰)^;(d(x(t⁰) t⁰)^;y⁰) = 0 (4.17)

y^{0 2Rn} q²C(^{I Rm}):

For =I the initial condition (4.17) simpli es to R(t⁰)(d(x(t⁰) t⁰)^;y⁰) = 0, i.e., d(x(t⁰) t⁰) =R(t⁰)y⁰: (4.18)

Theorem 4.4

Let the DAE (2:1) with proper leading part be of index = 1. Let x(:) be the solution of (2.1) on the interval ^I. Further, let = I. Then the IVP (4:16) (4:18) is uniquely solvable with a solution x(:) dened on ^I for q ² C(^{I Rm}), ^k q^k1 , y^{0 2Rn}, ^jR(t⁰)(y^0;d(x(t⁰) t⁰))^j , >0 su ciently small. x(:) depends continuously dierentiably on y⁰. It holds that

kx^;x ^k1 + ^kd(x(:) :)^;d(x(:) :)^kC¹

const(^jR(t⁰)(d(x(t⁰) t⁰)^;d(x(t⁰) t⁰))^j+^kq^k1):

P r o o f of the theorem: We form the extended system for the DAE (4.16) and apply Theorem 4.3 for = 1. ²

In the nature of things DAEs of index 2 require a higher regularity of some compo- nents. Here it is essential in how far we can assume (4.9) for the extended equation (2.6). We have (cf. ^x2)

n(x t) := D(t)Q¹(t)G²(t)^;1b(x t)

= n(x t) +D(t)Q¹(t)D(t)^;(y^;d(x t)) (4.19) with n(x t) :=D(t)Q¹(t)G²(t)^;1b(x t): (4.20) In the case of a linear original equation (2.1) with A(x t)A(t), b(x t)B(t)x+ q(t), d(x t)D(t)xthe expressions (4.19), (4.20) simplify to

n(x t) = D(t)Q¹(t)G²(t)^;1q(t) +D(t)Q¹(t)D(t)^;y n(x t) = D(t)Q¹(t)x+D(t)Q¹(t)G²(t)^;1q(t):

It is natural to demand that DQ¹G^;12 q ² C¹ for index-2 DAEs, and likewise that DQ¹D^{; 2} C¹. Thus, condition (4.9), which requires that n(x(:) :) has to belong to the class C¹ for continuous x(:) y(:) and continuously dierentiable (Ry)(:), is ful lled for (2.1) in the linear case.

Theorem 4.5

Let the DAE (2:1) with proper leading term be of index = 2. Let x ² C(^{I Rm}) be the solution of (2:1), and DP¹D^;, DQ¹D^; be continuously dierentiable. Moreover, let :=P¹(t⁰).

Let n(x(:) :)²C¹(^{I Rn}) for all x²C(^{I Rm}) from a neighbourhood ofx and for y ²C_R¹(^{I Rn}) from a neighbourhood of y =d(x(:) :).

12 R. Marz (i) For

q²C_{D Q}¹ 1G^;12(^{I Rm}) ^kq^k1+^k(DQ¹G^;12q)^0k1 y^{0 2Rn j}DP¹D^;(y^0;d(x(t⁰) t⁰))^j

> 0 su ciently small, the IVP (4:16) (4:17) is uniquely solvable and for the solution x²C(^{I Rm}) it holds that

kx^;x^k1+^kd(x(:) :)^;d(x(:) :)^kC¹

const ^fkq^k1+^k(DQ¹G^;12q)^0k1 + ^jD(t⁰)D(t⁰)^;(d(x(t⁰) t⁰)^;d(x(t⁰) t⁰))^jg: (ii) The solution x(:) of the IVP depends contiuously dierentiably on y⁰.

P r o o f. We use the extended form (2.6) for (2.1) and write the IVP (4.16), (4.17) in the following way (cf. ^xx1,2)

A(x (t) t)(D(t)x(t))⁰+b(x(t) t) =q(t) (4.21) D(t⁰)P¹(t⁰)(x(t⁰)^;x⁰) = 0 (4.22) with D(t⁰)P¹(t⁰) = (0 DP¹D^;) andq(t) = ^;q(0t). For (2.6), the condition (4.9) is given by the assumption.

q ² C_DQ¹

1G^;12(^{I Rm+n}) holds if and only if q ² C_{D Q}¹

1G^;12(^{I Rm}). Consequently, Theorem 4.3 yields the assertion. ²

We formulate a further perturbation theorem, whose assumptions are possibly easier to be checked.

Theorem 4.6

Let the DAE (2:1) with proper leading term be of index = 2. Let x ²C(^{I Rm}) be the solution of (2:1), and letDP¹D^;, DQ¹D^; as well asDx be continuously dierentiable. Let :=P¹(t⁰).

For all x ² C_D¹ (^{I Rm}) from a neighbourhood of x let n(x(:) :) and d(x(:) :) be continuously dierentiable. Then the assumptions from Theorem 4:5 remain true, where the solutions of the IVP (4:16) (4:17) are even from C_D¹ (^{I Rm}).

P r o o f. We check immediately whether the linear index-2 DAE (4.12) has solutions

4x²C_D¹ (^{I Rm}),⁴y²C_R¹(^{I Rn}) for right-hand sidesq ²C_{D Q}¹ 1G^;12(^{I Rm}) r² C_R¹(^{I Rn}). With X :=C_D¹ (^{I Rm})C_R¹(^{I Rm}) and Y :=C_{D Q}¹ 1G^;12(^{I Rm}) C_R¹(^{I Rn}) we obtain ^Fx(x)²Lb(X Y) im^Fx(x) =Y.

Then the related mappping ^FIV P x(x) acts bijectively between the spaces X and Y LIC. By assumption it holds that x =^;x_y ²X.

The regularity conditions forn(x t) andd(x t) ensure that ^F(x)²Y is always true for x ² X from a neighbourhood of x. We can further argue analogously to the proof of Theorem 4.3, where the mapping^H(x d q) :=^FIV P(x)^;(q d) operates in the spaces XLIC Y and Y LIC now. ²

The additional regularity conditions in case of DAEs of index = 2, which shall

Nonlinear dierential-algebraic equations with properly formulated leading term 13 guarantee certain properties of the mappings (the images ^F(x) have to lie in the

"right" space), imply restrictions on the admissible structure.

An interesting special class of DAEs, which is important for applications, consists of DAEs for which N⁰(x t) does not depend on x, i.e.,

kerD(x t) =N⁰(t) x^2D t^2I (4.23) and P⁰(t) is continuously dierentiable w.r.t. t. Quite often N⁰(x t) is even inde- pendent of x and t.

Then it holds that

d(x t) =d(P⁰(t)x t) x^2D t^2I (4.24) and further dx(x t) =dx(P⁰(t)x t), in particular,

D(t) =dx(P⁰(t)x(t) t):

Lemma 4.7

^{Let (2:}1) be a DAE with proper leading term. Let (4:3) be valid and d ² C¹(^{DI Rn}), P^{0 2} C¹(^I L(^Rm)). Let x ² C(^{I Rm}) be a solution of the DAE (2:1).

(i) Then P⁰x D D^; and Dx are continuously dierentiable on ^I. (ii) d(x(:) :) is continuously dierentiable on ^I for all x²C_D¹ (^{I Rm}).

P r o o f. Provided that P⁰x is continuously dierentiable, then D(t) = dx(P⁰(t)x(t) t) is so, too. As a reexive generalized inverse with C¹-projectors P⁰(t) = D(t)^;D(t),R(t) =D(t)D(t)^; alsoD(t)^;is continuously dierentiable.

Furthermore, Dx = DP⁰x belongs to the class C¹. Then x ² C_D¹ (^{I Rm}) implies

d(x(:) :) =d(D(:)^;D(:)x(:) :)²C¹(^{I Rn}):

It remains to show that P⁰x is actually C¹. Therefore, we investigate the function

K(x y t) :=y^;d(x t) x^2D y^2Rn t^2I:

We have ^K(x(t) y(t) t) = 0, and ^Kx(x(t) y(t) t) = ^;D(t) acts bijectively between imP⁰(t) and imR(t).

For each t ^{2 I} the equation ^K(x y t) = 0 provides a solution function (: t) with ^K((y t) y t) = 0, (y t) = P⁰(t)(y t), (y t) = (R(t)y t), P⁰(t)x(t) = (y(t) t). Then the regularity of P⁰x results from that of , since y(t) is contin- uously dierentiable. ²

For a better understanding let us remark that, in case of d(x t)D(t)x+(t), the equation y^;D(t)x^;(t) = 0 leads to y = R(t)y and P⁰(t)x = D(t)^;y^;D(t)^;, i.e., (y t) =D(t)^;y^;D(t)^;(t).

For nonlinear DAEs (2.1) satisfying the assumptions of Lemma 4.7, i.e., DAEs with a smooth functiondand an only time-dependent smooth subspaceN⁰, it is convenient to work with the function space C_P¹⁰(^{I Rm}) = C_D¹ (^{I Rm}). (2.1) can be left in the original form or we may take

(d(x(t) t))⁰ =dx(P⁰(t)x(t) t)(P⁰(t)x(t))⁰ +dt(P⁰(t)x(t) t):

14 R. Marz

5 Special systems of circuit simulation

The modi ed nodal analysis (MNA) used in industrial simulation packages gener- ates, for large classes of circuits, systems of the form (cf. EsTi])

Ac(q(A^>_ce(t) t)⁰ +bc(e(t) jL(t) jV(t)) = 0 ((jL(t) t))⁰ +bL(e(t) jL(t) jV(t)) = 0 bV(e(t) jL(t) jV(t)) = 0

9

=

(5.1) where e(t), jL(t), jV(t) denote the nodal potentials and the currents of inductances and the voltage sources, respectively. The functions q(z t) and (w t) are continu- ously dierentiable and the Jacobians qz(z t),w(w t) are positive de nit. With

x:=

0

@

jeL

jV

1

A A~:=

0

@

Ac 0 0 I 0 0

1

A d~(x t) :=

q(A^>_ce t (jL t)

M(x t) :=

qz(A^>_ce t) 0 0 w(jL t)

~b(x t) :=

0

@

bc(e jL jV t) bL(e jL jV t) bV(e jL jV t

1

A

it holds that

D~ :=dx=MA^~> G⁰ := ~AD^~ = ~AMA^~>

imG⁰ = im ~A kerG⁰ = ker ~D= ker ~A^>:

The matrix G⁰(x t) has a nullspace N⁰ that is independent of x and t. Obviously, (5.1) is nothing else but

A~(~d(x(t) t))⁰+ ~b(x(t) t) = 0: (5.2) If im ~D(x t) is independent of x, then (5.2) has a proper leading term. Because of the constant nullspace N⁰, Lemma 4.7 is relevant.

If im ~D(x t) changes with x, we can put ~A = ~AP_A^~ in (5.2) and shift the constant projector P_A^~ with ker P_A^~ = ker ~A below the derivative. The DAE

A~(P_A^~d^~(x(t) t))⁰+ ~b(x(t) t) = 0 (5.3) has a proper leading term with constant subspace N⁰ and ~R = P_A^~. Here, too, Lemma 4.7 may be applied. Moreover, P_A^~D^~(x t) is also constant now.

On the other hand, we can dierentiate

A~(~d(x(t) t))⁰ = ~A^fM(x(t) t)( ~A^>x(t))⁰ +dt(x(t) t)^g and investigate the equation

AM~ (x(t) t)( ~A^>x(t))⁰+ ~b(x(t) t) + ~Adt(x(t) t) = 0: (5.4) If ker( ~AM(x t)) is independent of x, (5.4) represents a DAE with proper leading term. If ker( ~AM(x t)) depends on x, we proceed to

AM~ (x(t) t)P_A^>~( ~A^>x(t))⁰+ ~b(x(t) t) + ~Adt(x(t) t) = 0: (5.5)