Criteria for the trivial solution of di erential algebraic equations with small nonlinearities to be
asymptotically stable
Roswitha Marz, Humboldt-Universitat Berlin
Abstract
Di erential algebraic equations consisting of a constant coecient linear part and a small nonlinearity are considered. Conditions that enable linearizations to work well are discussed. In particular, for index-2 di erential algebraic equations there results a kind of Perron-Theorem that sounds as clear as its classical model except for the expensive proofs.
1 Introduction
This paper deals with the question whether the zero-solution of the equation
Ax0(t) +Bx(t) +h(x0(t) x(t) t) = 0 (1.1) is asymptotically stable in the sense of Lyapunov. Equation (1.1) consists of a lin- ear part characterized by the constant matrix-coecients A B 2 L(IRm) and a small nonlinearity described by the function h : D0 1) ! IRm, D IRm IRm open, 02D,
h(0 0 t) = 0 t20 1):
The zero-function solves (1.1) trivially, i.e. the origin represents a stationary solution of (1.1).
The leading coecient matrixA is not necessarily nonsingular, but ifA is so, equation (1.1) represents a regular ordinary dierential equation (ODE). For singular matrices A, there are dierential-algebraic equations (DAEs) under consideration. The matrix pencil fA Bg is assumed to be regular, i.e. the polynomialp() := det(A+B) does not vanish identically. By fA Bg and indfA Bg we denote the nite spectrum and the Kronecker index of the pencil fA Bg, respectively. Recall that fA Bg is the set of the roots of p().
The given functionhis continuous together with its partial Jacobiansh0x0,h0x. Moreover, his small in the following sense. To each" >0, there is a(")>0 such thatjxj("),
jyj("),t 20 1) yield
jh0x0(y x t)j" jh0x(y x t)j": (1.2) 1
Clearly, (1.1) covers the well-understood case of regular explicit ODEs
x0(t) =Bx(t) +g(x(t) t) (1.3) by A = ;I, h(y x t) g(x t). The pencil f;I Bg is always regular, further indf;I Bg = 0, f;I Bg = (B). In this case Perron's Theorem (e.g. 1], 2]) applies immediately. Hence, if fA Bg belongs to IC;, then the trivial solution is asymptotically stable in the sense of Lyapunov.
Does this assertion hold true also in more general cases? If so, to what extend does it hold? Answers should be of great interest, since they constitute the background of further stability considerations via linearization and tracing back linear parts to the constant coecient case.
Although the classical stability results formed by Poincare, Perron and Lyapunov (e.g.
1], 2]) date back more than hundred years, the respective theory for DAEs is rather in its infancy.
For so-called transferable DAEs (1.1), in 3] the stability question is reduced to that for an inherent regular ODE relative to a certain invariant subspace. Unfortunately, this inherent state equation is not attainable in practice. On the other hand, criteria by linearization are expected to enable also practical determinations. For autonomous low index DAEs, stability via linearization is considered e.g. in 4], 5], 6]. Unlike regular ODEs nonautonomous DAEs involve nontrivial new diculties in comparison with autonomous ones. A Lyapunov stability criterion for nonautonomous index-1 DAEs (1.1) is proved in 7]. However, even for autonomous DAEs (1.1) with indfA Bg>1, this index may become an irrelevant detail of (1.1), that is, linearization does not work in those cases (e.g. 4] and Section 2 below).
If the matrix A is nonsingular, then, applying the Implicit Function Theorem, we can transform (1.1) into
x0(t) =;A;1Bx(t) +g(x(t) t): (1.4) The pencil fA Bg is regular, indfA Bg = 0, fA Bg = (;A;1B). Again by stan- dard arguments, fA Bg IC;yields the asymptotical stability of the trivial solution.
Now, let us turn to the more interesting case of A being singular.
To make sure that Ax0(t) in (1.1) may be considered as a somewhat leading term in general, we assume the inclusion
N := kerAkerh0x0(y x t) (y x t)2D0 1) (1.5) to be satised. Note that (1.5) holds for trivial reasons if h(y x t) does not depend at all on its rst argument. Due to condition (1.5) only those components of x0(t) occur in the nonlinear part of (1.1) that are already involved in the leading term Ax0(t).
Next, denote by P 2L(IRm) any projector matrix alongN that isP2 =P, kerP =N. Then Q:=I ;P projects onto the nullspace N, hence A=A(P +Q) =AP.
It is easily checked that (1.5) implies the identity
h(y x t)h(Py x t) (1.6) 2
and vice versa. This suggests to reformulate equation (1.1) more precisely as
A(Px)0(t) +Bx(t) +h((Px)0(t) x(t) t) = 0: (1.7) In the following we indicate equation (1.1) as a shorter notation of (1.7). Naturally, now we should ask for solutions of (1.1) that belong to the class
CN1 :=fx()2C : Px()2C1g:
Only those components of the unknown function are expected to be from C1 whose derivatives are really involved in (1.1). For the other components continuity will do.
At this place it should be mentioned that both the class CN1 and the formulation (1.7) are invariant of the special choice of the projector P. For nonsingular A, we have trivially N = kerA = f0g, P = I, thus CN1 = C1. However, if A is singular, imP IRm becomes a proper subspace, and CN1 is a larger class than C1 in fact.
Example:
Consider the two-dimensional systemx01(t) +x1(t) +(t)x1(t)2 = 0 (1.8) x2(t) +(t)x1(t)2+ (t)x2(t)2 = 0 (1.9) with continuous, uniformly bounded on 0 1) scalar functions (), (), (). Obvi- ously, all the above assumptions on h are satised. In particular, (1.5) holds due to h0x0 = 0. Further, we have
A= diag(1 0) B =I det(A+B) = + 1 indfA Bg= 1
and P = diag(1 0) is a possible choice. The respective class CN1 consists of all con- tinuous functions x() = (x1() x2())T, the rst component of which is continuously dierentiable.
System (1.8), (1.9) shows once more that, looking for C1 solutions instead of those from CN1, would necessitate more smoothness of the function h. However, in view of applications, we try for lower smoothness conditions if possible.
It is evident that the regular ODE (1.8) forx1() can be treated again by standard argu- ments. Its zero-solution is stable. The constraint equation (1.9) determines the second component in dependence of the rst one, and x1(t) ! 0 (t ! 1) yields x2(t) ! 0 (t!1).
Obviously, to cover all neighbouring solutions of the trivial one in the complete system (1.8), (1.9) we should vary only the initial data of the rst component. Observe that fA Bg = f;1g IC; and that the trivial solution is asymptotically stable in this
modied sense. 2
The example discussed above demonstrates an important peculiarity of DAEs. One has to deal with constraints like (1.9), but also with so-called hidden ones (cf. x2 below).
Naturally, the initial values x0 :=x(t0) of solutions satisfy all relevant constraints, i.e., x0 is consistent at t0. However, how to state initial value problems? Formulations like
3
x(t0) =x0,x0 2IRm is consistent att0, are nice but unt for practical use. In general, one has no idea on how the constraints look like. On the other hand, simply stating
x(t0) =x0 2IRm
would yield unsolvable problems. In the following, we try to pick up and x the free integration constants involved by means of a certain projector matrix 2L(IRm) that can be computed practically in terms of A, B. We state
x(t0) = x0 x0 2IRm (1.10)
as initial condition.
Note that, in case of nonsingular A, we obtain again = I, of course. In example (1.8), (1.9) the choice of = P = diag(1 0) is convenient. depends on the pencil
fA Bgin general, and on its index in particular.
Denition:
The zero-solution of (1.1) is stable in the sense of Lyapunov if there is a certain projector 2L(IRm) and, for each t0 0(i) a value >0 can be found such that the initial value problem (1.1), (1.10) with
jx0j has a CN1-solution x( x0 t0) dened at least on t0 1), and further (ii) a value %() > 0 to each 0 < can be found so that jx0j %() yields
jx(t x0 t0)j for t t0.
The trivial solution of (1.1) is asymptotically stable in the sense of Lyapunov if it is stable and, for all suciently small jx0j, it holds that
x(t x0 t0);!0 (t!1):
No doubt, this is a straightforward generalization of the classical notion for regular ODEs which is recovered by = I. As mentioned before, a respective stability result for (1.1) with an index-1 pencil fA Bg is given in 7]. It says that fA Bg IC; implies the trivial solution to be asymptotically stable, whereby = P is chosen. In particular, this assertion applies to the special system (1.7), (1.8) and conrms the stability behaviour we discussed before.
It should be mentioned that, in the above Lyapunov stability notion, the projector matrix 2L(IRm) can be replaced by any matrixC 2L(IRm) with the only property kerC = ker . This fact can be realized easily by using the relations C = C, = C+C, where C+ 2L(IRm) indicates the Moore-Penrose inverse of C. Hence, in particular, Lyapunov stability does not depend on the special choice of the projector , the only relevant characteristic feature is its nullspace, but that is fully determined by the DAE itself.
One might expect that, in general, fA Bg IC; yields the trivial solution to be asymptotically stable. In Section 2, this tentative, somewhat coarse conjecture, is discussed by means of examples. After that, we derive the main result of the present paper (Theorem 3.3, Section 3), a stability criterion for the index 2 case. Section 4 contains the detailed proofs.
4
2 A tentative conjecture and counterexamples
The good experience with regular ODEs and index-1 DAEs of the form (1.1), which corresponds to matrix pencils fA Bg of index zero or one, gives rise to the tentative conjecture that the origin is an asymptotically stable stationary point iffA Bg IC;. We know this to become true for indfA Bg 1. Thereby, we have ker = kerA. If the nonlinearity in (1.1) disappears, i.e.,h(y x t)0, the above conjecture also holds true. The projector projects along the innite eigenspace of the matrix pencilfA Bg. As shown in 8], if indfA Bg = k, the projector can be constructed by a special matrix chain as =P0P1 Pk;1, where A0 :=A, B0 :=B, Ai+1 :=Ai+Bi(I;Pi), Pi 2L(IRm) projects along kerAi, Bi+1 :=BiPi,i1. Then, equation
Ax0(t) +Bx(t) = 0 can be reduced to
P0P1 Pk;1x0(t) +P0P1 Pk;1A;1k Bx(t) = 0 (I ;P0 Pk;1)x(t) = 0
while fA Bgconsist of exactly those eigenvalues ofP0P1 Pk;1A;1k B whose associ- ated eigenspaces belong to imP0 Pk;1.
Unfortunately, our conjecture is wrong if there are nonlinearities in (1.1), even in the case of indfA Bg= 2.
Example 1
Given the autonomous systemx01;x2 = 0 x1;x32 = 0 x03;x3 = 0 x4;x2+x3 = 0
9
>
>
>
=
>
>
>
(2.1) which can be rewritten in compact form (1.1) by
A=
2
6
6
6
4
1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
3
7
7
7
5 B =
2
6
6
6
4
0 ;1 0 0 1 0 0 0 0 0 ; 0 0 ;1 1 1
3
7
7
7
5 h(y x t) =
2
6
6
6
4
0
;x32 00
3
7
7
7
5: 2 IRis a parameter.
This special function h satises all conditions we agreed upon in Section 1. Choosing P = diagf1 0 1 0g we consider
A1 :=A+B(I;P) =
2
6
6
6
4
1 ;1 0 0 0 0 0 0 0 0 1 0 0 ;1 0 1
3
7
7
7
5
: 5
Since A1 is singular, we know that indfA Bg>1. Further, we realize that
fz 2IR4 :z 2kerA1 BPz 2imA1g=f0g:
Consequently (cf. 9]), the matrix pencil fA Bg has index 2. Furthermore, p() = det(A+B) = ;, thus fA Bg = fg. For < 0, our conjecture promises asymptotical stability for the origin. However, taking a look at the ow-picture in the (x1 x2)-plan we can realize immediately that the solutions move away from the origin.
Hence, our conjecture is wrong.
- 6
x2 x1
2
As we shall see below, the problem with Example 1 is that linearization does not work in this case. The DAE (2.1) does not represent an index-2 DAE although we have indfA Bg= 2. System (2.1) is rather a singular index-1 DAE having a singularity at x2 = 0. In Section 3 below we shall formulate convenient structural conditions that enable linearization and exclude this kind of singularities.
Our next example makes clear that even if linearization works and indfA Bg = 2, additional smoothness and boundedness conditions for h have to be satised.
Example 2
Given the DAEx02+x1 = 0 x2+q(t)x23 = 0 x03;x3 = 0
9
>
=
>
(2.2) which can be described in terms of (1.1) as
A=
0
B
@
0 1 0 0 0 0 0 0 1
1
C
A B =
0
B
@
1 0 0 0 1 0 0 0 ;
1
C
A h(y x t) =
0
B
@
q(t0)x23 0
1
C
A:
In (2.2), < 0 indicates a parameter and q(t) is a continuous, uniformly bounded, scalar function. Again, all conditions for h given in Section 1 are fullled. Moreover, we derive that indfA Bg= 2, fA Bg=fg.
On the other hand, the last two equations of (2.2) yield x3(t) = e tx3(0)
x2(t) = ;q(t)e2 x3(0)2: 6
Considering the rst equation from (2.2), i.e. x1 = ;x02, we learn that continuity of the functionq() is not adequate for this problem. To obtain just a continuous solution component x1 we have to demand that q() is C1. Then we derive
x1(t) = (q0(t) + 2q(t))e2 tx3(0)2: (2.3) An appropriate projector to state the initial condition (1.10) is = diagf0 0 1g. Obviously, =P = diagf0 1 1g would not do in this case.
As far as the asymptotical behaviour of t ! 1 is concerned, there are no problems with the solution components x2(), x3(). However, to make sure that fA Bg IC; yields also x1(t) ! 0 (t ! 1), we have to demand the uniform boundedness of q0(t) additionally. In terms of the function h, our additional assumptions mean thath has continuous partial derivatives h0t,h00tx, too. Further, the relation
h0t(0 0 t) = 0 t20 1) (2.4)
as well as the inequality
jh00tx(0 x t)jcjxj for small jxj
with a certain constant c are given. Considering this additional regularity and small- ness of the function h, now fA Bg = fg IC; implies the zero-solution to be asymptotically stable. This simple fact will be conrmed once more by Theorem 3.3
below. 2
We know the above conjecture to be somewhat coarse. To improve it, one has to add { structural conditions that guarantee linearization to work, but also
{ more regularity and smallness conditions for the nonlinearity h.
3 A positive result for the case
indfA Bg = 2In this part we study equation (1.1) with an index-2 matrix pencil fA Bg. For that case, we shall verify our improved conjecture (cf. Section 2) and give precise formula- tions of all additional assumptions needed, respectively.
For more clarity, we recall the standard assumptions used in Section 1 and indicate them as (A).
Assumption (A):
(i) h:D0 1)!IRm, DIRmIRm open,
02D h(0 0 t) = 0 for t 20 1) h is continuous together with its partial Jacobians h0x0, h0x.
7
(ii) To each small " > 0, a (") > 0 can be found such that jxj ("), jx0j (") yield
h0x0(x0 x t)j" jh0x(x0 x t)j"
uniformly for allt 20 1).
(iii) fA Bg is a regular matrix pencil.
(iv) N := kerAkerh0x0(x0 x t) for all (x0 x t)2D0 1).
Further, let us formulate certain additional smoothness and smallness conditions as suggested by Example 2 in x2.
Assumption (B):
(i) The function ~h dened by ~h(x t) := (I ; AA+)h(0 x t) has also continuous partial derivatives ~h0t, ~h00tx, ~h00xx.
(ii) ~h0t(0 t) = 0 for allt20 1).
(iii) A constant can be found so that, with (") from (A), jxj(") yields
j~h00xt(x t)j" for all t 20 1): (iv) ~h00xx(x t) is uniformly bounded by a constant 0.
For the special DAE (2.2) it holds that AA+ = diagf1 0 1g, ~h(x t) = (0 q(t)x23 0)T. If the function q() and its derivative q0(t) are uniformly bounded as discussed in Ex- ample 2 (x2), then this special ~h fulls (B).
The following matrices and subspaces will be used below (cf. 9]):
N := kerA, S :=fz 2IRm :Bz 2imAg,
Q2L(IRm), Q2 =Q, imQ=N, P :=I;Q, A1 :=A+BQ,
N1 := kerA1, S1 :=fz 2IRm :BPz 2imA1g,
Q1 2L(IRm), Q21 =Q1, imQ1 =N1, P1 :=I;Q1, A2 :=A1+BPQ1,
V 2L(IRm), V2 =V, imV =N \S, U :=I;V.
It is well-known that the pencil fA Bg has index 2 if and only if A2 is nonsingular but A, A1 are singular (e.g. 9]). Moreover, if fA Bg has index 2, we can use the decomposition IRm =N1S1. Hence, a convenient choice of the projectorQ1 is given by this decomposition. In the following we agree to have, more precisely, imQ1 =N1, kerQ1 =S1, and consequently (e.g. 3], 9]),
Q1 =Q1A;12 BP =Q1A;12 B Q1Q= 0: (3.1) 8
The relations (3.1) lead to
(PP1)2 =PP1 (PQ1)2 =PQ1
A;12 A =P1P A;12 B =A;12 BPP1+Q1+Q:
Scaling equation (1.1) by A;12 yields
(PP1x)0(t);QQ1(PQ1x)0(t) +A;12 BPP1x(t) +Q1x(t) +Qx(t)
+A;12 h((PP1x)0(t) + (PQ1x)0(t) x(t) t) = 0: (3.2) We decompose
x=Px+Qx=PP1x+PQ1x+Qx=:u+v+w and decouple (3.2) into the system
u0(t) +PP1A;12 Bu(t) +PP1A;12 h(u0(t) +v0(t) u(t) +v(t) +w(t) t) = 0 (3.3) v(t) +PQ1A;12 h(u0(t) +v0(t) u(t) +v(t) +w(t) t) = 0 (3.4)
;QQ1v0(t) +w(t) +PQ1A;12 Bu(t)
+QP1A;12 h(u0(t) +v0(t) u(t) +v(t) +w(t) t) = 0: (3.5) Because of imQQ1 =N \S the latter equation (3.5) splits up further into
Uw(t) +UQA;12 Bu(t) +UQA;12 h(u0(t) +v0(t) u(t) +v(t) +w(t) t) = 0 (3.6)
;QQ1v0(t) +V w(t) +V QP1A;12 Bu(t)
+V QP1A;12 h(u0(t) +v0(t) u(t) +v(t) +w(t) t) = 0: (3.7) If the nonlinearity h disappears, equation (3.3) simplies to a regular explicit linear ODE for u() that has the invariant subspace imPP1. (3.4) realizes v(t) = 0, hence it results that x(t) = u(t) +w(t) = (I ;PQ1A;12 B)u(t) = (I;PQ1A;12 BPP1)PP1u(t).
Note that can:= (I;PQ1A;12 BPP1)PP1 is also a projector. It holds that ker can = kerPP1 = N N1. Further, im can represents the nite eigenspace of the matrix pencil (cf. 8]), while the vectors Uw correspond to that part of the innite eigenspace that has simple structure. The vectors V w and v form the respective part for Jordan blocks of order 2.
In 10] we nd the relation
imA= ker((PQ1+UQ)A;12 ) (3.8) which will become very helpful to realize the appropriate structural conditions below.
Lemma 3.1
Given (A), indfA Bg= 2. Additionally, letimh0x0(x0 x t)imA for (x0 x t)2D0 1): (3.9) Then the identity
(PQ1+UQ)A;12 h(y x t)(PQ1+UQ)A;12 (I;AA+)h(0 x t) (3.10) is valid.
9
Proof:
Due to (3.9) we have(PQ1+UQ)A;12 (h(x y t);h(0 x t)) =
1
Z
0
(PQ1+UQ)A;12 h0x0(sy x t)yds= 0: On the other hand, (3.8) implies
(PQ1+UQ)A;12 = (PQ1+UQ)A;12 (I;AA+):
2
Note that condition (3.9) further species the possible structure of (1.1). By Lemma 3.1, the equations (3.4) and (3.6) are much simpler now, namely
v(t) +PQ1A;12 h~(u(t) +v(t) +w(t) t) = 0 Uw(t) +UQA;12 Bu(t) +UQA;12 h~(u(t) +v(t) +w(t) t) = 0: We put them together compactly to
y(t) +UQA;12 BPP1z(t) + (PQ1 +UQ)A;12 ~h(y(t) + (PP1+V Q)z(t) t) = 0 (3.11) where
y:=v+Uw z :=u+V w: (3.12)
Clearly, if x() satises the original DAE (1.1), then (3.11) is satised by y() = PQ1x() +UQx() and z() =PP1x() +V Qx().
Equation (3.11) suggests to realize y as a function of z and t by applying the Implicit Function Theorem.
Lemma 3.2
Let (A), (B) as well as (3.9) be given, indfA Bg = 2. Then, for suf- ciently small " > 0 and the corresponding (") > 0 from (A), there is a uniquely determined function f :D(")0 1)!IRm,D(") :=nz 2IRm : (1 + 2jUQA;12 BPj)j(PP1+V Q)zj 12(")o satisfying
(i) f(z t) + (PQ1+UQ)A;12 ~h(f(z t) + (PP1+V Q)z t) +UQA;12 BPP1z = 0, z 2D("), t20 1).
(ii) f is continuous and has continuous partial derivatives fz0, fzz00, ft0, fzt00. (iii) It holds that
f(0 t) = 0 ft0(0 t) = 0 fz0(0 t) = ;UQA;12 BPP1
(PQ1+UQ)f(z t) =f(z t) =f((PP1+V Q)z t) z 2D(") t 20 1):
10
For the proof we refer to Section 4 below.
Lemma 3.2 enables us to rewrite (3.11) locally equivalently as y(t) = f(u(t) +V w(t) t)
and, in more detail, as
v(t) = Py(t) =Pf(u(t) +V w(t) t) (3.13) V w(t) =Qy(t) =Qf(u(t) +V w(t) t):
Considering once more equation (3.5) we know that v0(t) is needed. Our concept of CN1-solvability means in terms of the decomposition used that u() = PP1x(), v() = PQ1x() are from C1 and w() =Qx() is just continuous. The idea is now to further specify the structure of (1.1) by supposing that
Pf(z t) =Pf(PP1z t) z 2D(") t 20 1) (3.14) or, equivalently, Pfz0(z t)V Q 0. In other words, fz0(z t) is forced to map N N1 into N. By this we meet the natural smoothness of the solution, and we are allowed then to dierentiate equation (3.13) with respect to t. We derive
v0(t) =Pfz0(u(t) t)u0(t) +Pft0(u(t) t): (3.15) Now, expressions for v(t), v0(t), Uw(t) in terms of u(t), u0(t), V w(t) are available.
Inserting them into the equations (3.3) and (3.7), there results a system u0(t) = ;PP1A;12 Bu(t) +'(u0(t) u(t) V w(t) t) V w(t) = (u0(t) u(t) V w(t) t)
that could be transformed locally into a system that reads
u0(t) = ;PP1A;12 Bu(t) +g(u(t) t) (3.16)
V w(t) = k(u(t) t): (3.17)
Together with
v(t) +Uw(t) =f(u(t) +V w(t) t) (3.18) provided by Lemma 3.2, we arrive at a local decoupling of (1.1). If x( x0 t0) solves the initial value problem for (1.1) and the initial condition
PP1x(t0) = PP1x0 x0 2IRm (3.19) with suciently small jPP1x0j, then u() = PP1x( x0 t0) satises the regular ODE (3.16), but also
u(t0) =PP1x0:
Moreover, the componentsv() =PQ1x( x0 t0) andw() = Qx( x0 t0) satisfy (3.17), (3.18). The resulting idea is to use such decouplings to construct all neighbouring solutions of the zero-solution.
11
Let us recall once more the structural conditions used for (1.1) and denote them by (C).
Assumption (C):
(i) imh0x0(x0 x t)imA, (x0 x t)2D0 1).
(ii) fz0(z t) maps N N1 intoN for z 2D("), t20 1).
Now, we are ready to formulate our main result that sounds as clear as its classical model.
Theorem 3.3
Given the Assumptions(A), (B), (C) and indfA Bg=2,fA BgIC;. Then the trivial solution of (1.1) is asymptotically stable in the sense of Lyapunov with :=PP1.The proof will be carried out in Section 4.
Remarks:
1) Theorem 3.3 generalizes the results for autonomous DAEs in 2]. This is not as trivial as one might think when coming from the regular ODE case.
2) The nullspace ker(PP1) = N N1 is nothing else but the innite eigenspace of the pencil fA Bg. Instead of PP1 for stating the initial conditions we can use any matrixC that has the kernel N N1.
3) Concerning condition (C), it is somewhat dicult to check its second part in practice. The function y=f(z t) is only implicitly given by the equation
y+UQA;12 BPP1z+ (PQ1+UQ)A;12 ~h(y+ (PP1+V Q)z t) = 0: (3.20) We close this section by providing sucient criteria for (C)(ii) to be valid, which are given in terms of the original data of (1.1). For dierent index-2 DAEs those criteria are proposed in 4] and 10], respectively.
Lemma 3.4
Let (A), (B) as well as (C)(i) be given, indfA Bg= 2. Then each of the following 4 conditions implies condition (C)(ii) to be satised:(i) ~h(x t) = ~h(Px t) for (0 x t)2D0 1).
(ii) ~h(x t);~h((PP1+PQ1+UQ)x t)2imA for (0 x t)2D0 1).
(iii) ~h(x t);~h(Px t)2imA1, (0 x t)2D0 1).
(iv) fz 2 IRm : Bz +h0x(y x t)z 2 imAg\N = fz 2 IRm : Bz 2 imAg\N for (y x t)2D0 1).
12
The proof is given in Section 4.
Although criterion (iv) related to certain subspaces looks somewhat strange, it seems to be very useful in practice, e.g. in circuit simulation (11]).
Systems in Hessenberg form are often of special interest, that is
x01+B11x1+B12x2+g1(x1 x2 t) = 0 (3.21) B21x1 +g2(x1 t) = 0: (3.22) System (3.21), (3.22) is a Hessenberg form equation of size 2 if B21B12 is assumed to be nonsingular. With A= diag(I 0),AA+= diag(I 0) we nd
~h(x t) =
0
g2(x1 t)
!
further N =nzz122IRm :z1 = 0o and thus ~h(x t) = ~h(Px t).
Applying Lemma 3.4(i) we conclude the following assertion.
Corollary 3.5
Condition (C) is valid in case of Hessenberg form equations (1.1) of size 2.4 Proofs
Proof of Lemma 3.2:
Let the conditions (A) and (B) be satised, further indfA Bg = 2. The structural condition (C)(i) (which is the same as relation (3.9)) is also assumed to be valid. Since A;12 is a constant nonsingular matrix, the smallness conditions (A)(ii), (B)(iii) apply also toA;12 h,A;12 ~h. Hence, there is a(")>0 to each small" >0 such thatjxj("),
jyj("),t 20 1) yield
jA;12 h0x0(y x t)j "
jA;12 h0x(y x t)j "
jA;12 ~h00xt(y x t)j ": (4.1) Because of
A;12 h(y x t) = A;12 fh(y x t);h(0 0 t)g
=
1
Z
0
fA;12 h0x0(sy sx t)y+A;12 h0x(sy sx t)xgds and
A;12 ~h0t(x t) =A;12 f~h0t(x t);h~0t(0 t)g=
1
Z
0
A;12 ~h00xt(sx t)xds 13
from (4.1) it follows immediately that
jA;12 h(y x t)j"(jyj+jxj) (4.2) and
jA;12 ~h0t(x t)j"jxj (4.3) hold true for jxj("),jyj("),t 20 1). Consider the function (cf. (3.11))
F(y z t) := ;(PQ1+UQ)A;12 h~(y+ (PP1+V Q)z t);UQA;12 BPP1z
= ;(PQ1+UQ)A;12 h(0 y+ (PP1+V Q)z t);UQA;12 BP(PP1+V Q)z mapping fromIRmIRmIRintoIRm. Denote 1 :=jPQ1+UQj and choose" small enough to realize 21" <1. Then, F is well-dened on B(")D0(")0 1), where
B(") := ny2IRm :jyj 12(")g
D
0(") := nz 2IRm :jPP1z+V Qzj 12(")o: More precisely, for all y y2B("),z 2D0("),t 20 1), we have
F(0 0 t) = 0
jF(y z t);F(y z t)j1"jy;yj
jF(y z t)j1"jy+ (PP1+V Q)zj+jUQA;12 BPjjPP1z+V Qzj
12jyj+ 12(1+2jUQA;12 BPj)jPP1z+V Qzj: Form the set D(")D0("),
D(") :=nz 2IRm : (1 + 2jUQA;12 BPj)jPP1z+V Qzj 12(")o
such that for each xed z 2 D("), t 2 0 1), F( z t) maps the closed ball B(") into itself. Since F( z t) is contractive with "1 < 12 due to Banach's Fixed Point Theorem, there is a uniquely determined function
f :D(")0 1);!B(")IRm such that, for all z 2D("), t20 1),
f(z t)F(f(z t) z t) (4.4)
f(0 t) = 0 jf(z t)j 1 2(")
(PQ1+UQ)f(z t) =f(z t) =f((PP1+V Q)z t) hold true.
By the Implicit Function Theorem, the smoothness of F is passed on to the function 14
f. SinceF is continuous together with its partial derivatives Fy0, Fz0,Ft0, Fyy00,Fyz00, Fyt00, Fzz00, Fzt00 (cf. (A), (B)), the implicitly given functionf is also continuous and possesses continuous partial derivatives fz0, ft0, fzz00, fzt00. Further with
Fy0(y z t) =;(PQ1+UQ)A;12 ~h0x(y+ (PP1+V Q)z t) we have
Fy0(0 0 t) = 0
jFy0(y z t)j1"
hence the matrixI;Fy0(y x t) remains nonsingular uniformly for y2B("),z 2D("), t 20 1). The relations
fz0(0 t) =;UQA;12 BPP1 ft0(0 t) = 0 t20 1) are obtained immediately by dierentiating (4.4) and considering
Fz0(0 0 t) = ;UQA;12 BPP1 Ft0(0 0 t) = 0:
2
Next we derive some further properties of the function f to be used in the proof of Theorem 3.3 below.
Corollary 4.1
With3 :=jPjjPP1+V Q+UQA;12 BPP1j, for allz 2D("),t 20 1),it holds that
jft0(z t)j "1
1;"1 (") (4.5)
jPfz0(z t)j "1
1;"1 3 (4.6)
fzt00(0 t) = 0: (4.7)
Moreover, there are constants 4, 5 such that jfzz00(z t)j4 and
jfzt00(z t)j"5 for z 2D(") t20 1): (4.8)
Proof:
First of all we havej(I;Fy0(y z t));1j 1 1;"1
for all y2B("),z 2D("),t 20 1). From ft0 = (I;Fy0);1Ft0 and (4.3) we conclude
jft0(z t)j "1
1;"1 jf(z t) + (PP1+V Q)zj "1
1;"1 ("): 15
Fromfz0=(I;Fy0);1Fz0=;(I;Fy0);1(PQ1+UQ)A;12 ~h0x(PP1+V Q);(I;Fy0);1UQA;12 BPP1 we obtain
jfz0(z t) +UQA;12 BPP1j
1;1"1j(PQ1+UQ)A;12 ~h0x(f(z t) + (PP1+V Q)z t)fPP1+V Q;UQA;12 BPP1gj
"1
1;"1jPP1+V Q;UQA;12 BPP1j hence,
jPfz0(z t)j = jP(fz0(z t) +UQA;12 BPP1)j
jPj1 1
;"1"1jPP1+V Q+UQA;12 BPP1j "1 1;"1 3:
Since fz0 = (PQ1+UQ)fz0 =fz0(PP1+V Q) and (PP1+V Q)(PQ1+UQ) = 0, we may express the second derivative fzz00 simply as
fzz00 = (I ;Fy0);1Fzz00:
Due to (B)(iv), Fzz00 is uniformly bounded, therefore fzz00 is so, too.
Finally, using the above arguments once more we nd the expression fzt00 = (I;Fy0);1fFyt00fz0 +Fzt00g
= ;(I ;Fy0);1(PQ1+UQ)A;12 ~h00xtfPP1+V Q;fz0g:
In particular, ~h00xt(0 t) = 0 (cf. (4.1)) leads now to fzt00(0 t) = 0. Moreover, we may estimate
jfzt00(z t)j 1
1;"1 "(jPP1+V Qj+jfz0(z t)j)
5 ":
2
Let us stress once more that, if a CN1-function x() in the neighbourhood of the origin solves the DAE (1.1), then it satises also the identity
(PQ1+UQ)x(t) =f((PP1+V Q)x(t) t): With the denotations u:=PP1x, v :=PQ1x, w=Qxthis reads
v(t) +Uw(t) =f(u(t) +V w(t) t): In particular, it holds that
v(t) =Pf(u(t) +V w(t) t):
Now, the structural condition (C)(ii) casts the nullspace component out from the func- tion Pf such that
v(t) =Pf(u(t) t) (4.9)
16
results. Dierentiating yields the expression
v0(t) =Pfz0(u(t) t)u0(t) +Pft0(u(t) t): (4.10) Rewrite equation (1.1) scaled by A;12 , that is equation (3.2), as
PP1x0(t) +A;12 BPP1x(t) + (I;PP1)x(t) +QQ1(PQ1x)(t)
;QQ1(PQ1x)0(t) +A;12 h(PP1x0(t) + (PQ1(x)0(t) x(t) t) = 0: (4.11) This formulation suggests to replace the terms PQ1x, (PQ1x)0 by means of (4.9) and (4.10), respectively. Then we arrive at the DAE
PP1x0(t) +A;12 BPP1x(t) + (I ;PP1)x(t) +H(x0(t) x(t) t) = 0 (4.12) where the nonlinearity H is introduced as the following map from IRmIRmIR into IRm:
H(x0 x t) := A;12 h(PP1x0+Pfz0(x t)PP1x0+Pft0(x t) x t)
;QQ1fPfz0(x t)PP1x0+Pft0(x t);Pf(x t)g:
Recall that Pf(x t) = Pf(PP1x t) due to (C)(ii). For t 2 0 1), jPP1x0j 12("), PP1x2D(") the inequalities (4.5), (4.6) yield the estimation
jPP1x0+Pfz0(x t)PP1x0 +Pft0(x t)j
12(") + "1
1;"1(3+ 2jPj) 12(")(") supposed " is chosen small enough to realize
"1
1;"1 (3+ 2jPj)1 "1 < 12: (4.13) By this, the function H is well-dened forjPP1x0j 12("),jxj ("), PP1x 2D("), t 20 1).
Lemma 4.2
Given a regular index-2 matrix pencilfA Bg, ~A:=PP1, ~B :=A;12 BPP1+ (I ;PP1). Then, fA~ B~g is a regular pencil of index 1 the nite spectrum of which coincides with that of fA Bg, i.e.,fA~ B~g=fA Bg:
Proof:
Obviously, ~Az = 0, ~Bz 2 im ~A imply z = 0, i.e., fA~ B~g is regular and indfA~ B~g= 1. Consider (A+B)z = 0, or equivalently,(A;12 A+A;12 B)z = 0 i.e.
((PP1;QQ1) +A;12 BPP1+Q1+Q)z = 0: (4.14) 17
