Introduction In the classical theory of linear homogeneous ordinary dierential equations (ODEs), the adjoint equations are derived as those which are

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1 Linear dierential algebraic equations of index and their adjoint equations Katalin Balla Computer and Automation Research Institute, Hungarian Academy of Sciences Budapest, Hungary H-58 Budapest P.O.Box 63 Roswitha Marz Humboldt University D-0099 Berlin, Germany Unter den Linden 6 iam@mathemati.hu-berlin.de Abstract For linear dierential algebraic equations of tractability index the notion of the adjoint equation is analysed in full detail. Its solvability is shown at the lowest possible smoothness. The fundamental matrices of both equations are dened and their relationships are characterized. Keywords: linear dierential algebraic equations of index, adjoint equation, solvability, fundamental matrices. Mathematics Subject Classication: 34A09, 34A30 The wor of this author was supported by the Hungarian National Science Foundation, Grant No. T

2 Introduction In the classical theory of linear homogeneous ordinary dierential equations (ODEs), the adjoint equations are derived as those which are satised by the inverse adjoint matrices to the fundamental matrices of solutions [5]. Due to Lagrange identity, they play a central role when one characterizes the linear subspaces of solutions of the original ODEs. This fact explains their importance in the solution of boundary value problems for linear ODEs []. The monograph [6]was devoted to dierential algebraic equations (DAEs). A number of problems for which the similarity or dissimilarity to regular ODEs were essential was highlighted there. Some questions were raised later. In papers [2, 3], we have proven that for the linear homogeneous dierential algebraic equations of index with dierentiable coecients there exist differential algebraic equations { we called them adjoint DAEs { such that any pair of solutions of an original DAE and that of its adjoint equation satises an identity which may be considered as an analogue of the Lagrange identity. Moreover, we succeeded to prove that this identity appears in the characterization of linear subspaces of solutions of the original DAEs, as well. Due to dierentiability, we could also apply a theorem of reduction type and we were able to show that the adjoint equation to the regular ODE obtained by the reduction naturally leads to the adjoint equation for the DAE. The aim of this paper is twofold. First, we relax the dierentiability of the coecients of the DAE. We show that a pendant of the original DAE { we call it adjoint equation { is solvable at the smoothness appearing in the denition which assures the solvability of the original DAE of index. This condition is weaer than dierentiability of the coecients. Note that without dierentiability, the adjoint equation is not a DAE in standard form. Consequently, neither an index can be assigned directly. Therefore we give a proper denition. At this low smoothness, neither the theorems of reduction type wor. Secondly, we show that one can arrive to the homogeneous adjoint equation by construction of an equation which is satised by a properly transformed reexive generalized inverse of a fundamental matrix of the original DAE. The statement is as similar to the regular ODE case as possible. Meanwhile, we clarify the relationships between the dierent fundamental matrices of a DAE. This question was not addressed in [6]. We discuss the notion of the fundamental matrix for the adjoint equation and its properties, too. We obtain the analogue of Lagrange identity at the lowest smoothness 2

3 of coecients. In contrast to [2, 3] we do not assume that the projectors appearing in our analysis are orthogonal. In the previous wors, real DAEs were considered. The extension of the study to the complex case does not require extra eorts, so we describe the results for the complex case. 2 The DAE of index and its adjoint Let A : I! L( j C m j C m ), I IR. Let Q : I! L( j C m j C m ) be an arbitrary projector onto Ker A, i.e., for all x 2 I, A(x)Q(x) = 0 2 L( j C m j C m ). Let us denote P = I ; Q, where I is the identity operator: Iy = y for all y : I! j C m. Let P be as above. If y : I! j C m Py 2 C (I j C m ) and P 2 C (I L( j C m j C m )) () then A(Py) 0 ; AP 0 y is well dened. We can chec A(Py) 0 ; AP 0 y is independent of the special choice of projector P. Let B : I! L( j C m j C m ). Then, for all y : I! j C m and all projectors Q 2 C (I L( j C m j C m )) onto Ker A such that Py 2 C (I j C m ), the operator L given by Ly := A(Py) 0 +(B ; AP 0 )y is well dened and Ly 2 f~y : I! C m g. Moreover, Ly does not depend on j specic choice of Q. Inordertomae an accent on this independence, i.e., on the dependence of L on A and B, only, we write sometimes Ly = Ay 0 + By. Notice that the right hand side is meaningful and coincides with the left hand side provided y 2 C (I C m ). j We may also verify that P y 2 C (I C m ) and P j P 2 2 C (I L( C m C m )) j j involve P 2 y 2 C (I C m ) for all y : I! C m,aswell. Thus, the function space j j fy : I! C m Py 2 C (I C m )g where Q 2 C (I L( C m C m )) is an arbitrary j j j j projector onto Ker A does not depend on the special choice of Q. So is with the function space fy 2 C(I j C m ): Py 2 C (I j C m )g. Thus, the latter function space is dened by Ker A uniquely, we denote it by C A(I j C m ). When axedbasis is given in j C m,we spea about the elements of j C m as column vectors, and, correspondingly, the elements of L( j C m j C m ) are matrices. Our basic assumptions will be 3

4 (T) dim Im A(x) r<mfor all x 2I, (T2) the matrix pencil fa(x) B(x)g is regular and indfa(x) B(x)g =, for all x 2I and (T3) there exists a projector Q 2 C (I L( j C m j C m )) onto Ker A. With these assumptions, equation Ly = f (2) is called transferable DAE, or, equivalently, DAE of tractability index. Let A B 2 C(I L( j C m j C m )) f 2 C(I j C m ): (3) A function y 2 C A(I j C m ) is called a solution of (2) if it turns equation (2) into identity. For the above, in more general setting, see &.2. and &.2.2 in the basic boo [6] or the survey paper [8]. In the following we use notations B 0 := B;AP 0, A = A+B 0 Q, observing that by Theorem 3, Appendix A in [6], A is invertible provided (T2) holds. The state variable form is the decomposition of (2) into the inherent regular ordinary dierential equation and the algebraic equation where L s Py = PA ; f (4) Q s y = QA ; f (5) L s z := z 0 +(PA ; B 0 ; P 0 )z Q s z := Qz + QA ; BPz: (6) This splitting is a result of rearrangement of the terms in (2) followed by the application of A ; to (2). Due to Q 2 s = Q s, Q s turns to be a projector. We will characterize this projector in detail later. At the moment we recall from [6], Appendix A, Lemma 4 - where real coecients are considered - that Q s (x) projects onto Ker A(x) along S(x) = fv 2 j C m : B(x)v 2 Im A(x)g. We will recover this fact anew in a dierent context below. for denition of the index of a matrix pencil see e. g. Appendix A in [6] or [8] 4

5 L s is well-dened for z 2 C (I j C m ) and the initial value problem L s z = g z(^x) =^z ^x 2I ^z 2 j C m g 2 C(I j C m ) (7) is uniquely solvable in C (I j C m ). Moreover, z 2 Im P is valid provided that ^z 2 Im P (^x), g 2 Im P hold. Let L s be dened as the adjoint ofl s, for w 2 C (I j C m ). Then, again, the initial value problem L s w := w0 ; (B 0 A; P ; P 0 )w (8) L s w = h w(^x) = ^w ^x 2I ^w 2 j C m h 2 C(I j C m ) (9) is uniquely solvable in C (I j C m ). Meanwhile, P w 2 C (I j C m ), too, and (P w) 0 ; (P 0 + P B 0 A; )P w = P h P (^x)w(^x) =P (^x)^w: (0) Let u := A + P w, where A + u 2 C(I C m ) and j is the Moore-Penrose inverse of A. Then, A u = A A + P w =(A + A) P w = P w 2 C (I j C m ): () Noting that P A = A, A = A A ; A,Q B 0 = Q A and with one has P s := A ; P A = A ; A (2) P h =(A u) 0 ; (P 0 + P B 0 A; )A u =(A u) 0 ; (P 0 A ; Q B 0 A; A + B 0 A; A )u =(A u) 0 ; (P 0 A ; Q A A; A + B 0 A; A )u =(A u) 0 ; (P 0 A A ; A + B 0 A; A )u =(A A ; A u) 0 ; (P 0 A + B0)A ; A u =[A (A ; A u)] 0 ; B (A ; A u)=[a (P s u)] 0 ; B (P s u): We obtained that at least when s = P h, h 2 C(I j C m ), equation L = s where L := (A ) 0 ; B (3) 5

6 has at least one solution. Later we will call equation (3) adjoint equation of the DAE (2), but rst we are going to approve this notation. P s is obviously a projector and P s 2 C(I L( C m C m )). Let Q j j s := I ; P s. We have Q s = A ; Q A : (4) By the formal denition, dim Im P s = r, dim Im Q s = m ; r. Lemma The projector pair P s Q s is independent of the special choice of projector P 2 C (I L( j C m j C m )). Proof. When (T3) taes place, then, above the projector P 2 C (I L( j C m j C m )) indicated in (T3), the orthogonal projector P onto Im A is differentiable, as well. The latter is unique and P = A + A = P + P (A is not necessarily dierentiable but P is.). Therefore, it is sucient to prove that for an arbitrary projector P 2 C (I L( j C m j C m )) and the orthogonal projector P, the projector Ps dened by (2) and Ps dened in a similar way, coincide, i.e., P s = P s : (5) Denote A := A +(B ; A P 0 ) Q. Instead of showing (5), we chec that for R := A ; A A ; A, both P R = 0 and Q R = 0 hold. Then, we use R =(P + Q )R. P R = P A ; (P A ) A ; A = A ; A A ; A = A ; (A A ; )A = A ; P A = Q A =(A Q) =0: In order to verify that Q R vanishes, we recall that orthogonality means P = P, Q = Q and PQ = (A + A)Q = A + (AQ) = 0. Thus, P = P (P + Q) = PP + PQ = PP and Q = ( P + Q)Q = PQ + QQ = QQ. Further, P Q = (A ; A) Q = A ; (A Q) = 0. For P we also use the adjoint of the identities AP 0 = AP 0 Q and Q 0 = ;P 0,which holds for arbitrary P. Now the required result comes from the following chain of equalities: Q R = Q A ; Q A A ; A = ;Q (B ; P 0 A ) A ; A = ;Q B A ; A + Q P 0 A A ; A = ;( QQ) B A ; A +( QQ) (P P) 0 A A ; A = ;Q Q B A ; A +[(Q Q PP ) 0 ; (Q Q) 0 PP A A ; A 6

7 2 = ;Q Q B A ; A ; (Q Q) 0 A A ; A = ;Q Q B A ; A ; [Q 0 ( QA ) ; Q P 0 A ] A ; A = ;Q ( QB ; Q P 0 A ) A ; A = ;Q Q A A ; A = ;Q QA =0: In this context it requires a slightly more preparation to verify that Q s and its complement P s := I;Q s are also projectors independentofp. Meanwhile, we get a representation of Q s P s as symmetric as (2) and (4) are. While by the help of the formal denition we could state only Im Q s Ker A, and the rearrangement Q s := Q + QA ; BP = QA; (A + BP) = QA ; [A +(B ; AP 0 )Q + BP] =Q(A ; A)(I ; P 0 )+QA ; B = QP (I ; P 0 )+QA ; B = QA; B does not mae the value of dim Im Q s transparent, now we will be able to nd this value as well. For A B : I! L( j C m j C m ) denote A := A ; B 0 Q = A ; B Q where Q be a projector onto Ker A. The complement of Q is denoted by P, P := I ; Q. For orthogonal projectors and functions derived by their use we eep notation and one has P = AA +. When (T2) holds then ind fa (x) ;B (x)g = for all x 2Iand A is invertible due to Theorem 3, Appendix A in [6]. Remar Since both Q and Q are arbitrary, Q = Q, P = P cannot hold, in general. Neither P = P is valid. Since P = AA +, P = P = A + A, one obtains a necessary and sucient condition for P = P, namely Im A = Im A (or Im A = Ker A? ). Clearly, for the selfadjoint matrices this holds true. Neither can one claim that for a given Q there exists Q suchthatq = Q would be true, in general. For arbitrary projectors Q and Q, QA ; = ;A ; Q. Indeed, premultiplication by A and multiplication from the right by A would yield ;Q BQ 7

8 at the both sides. Next we have Q s := QA ; B = ;A; (Q B)=;A; (;Q A ) =A ; Q A : (6) The latter expression does not depend on Q, thatiswhy neither can Q s. And vice versa, by the original denition Q s did not depend on specic choice of Q, that is why neither can A ; Q A. Further, dim Im P = r. We obtain directly Lemma 2 The projectors Q s = A ; Q A P s = A ; P A (7) do not depend on the specic choice of Q or Q. Moreover, dim Im P s = r and dim Im Q s = m ; r. From the above considerations it also follows that Thus, we can state A ; Q =(QA ; ) = ;(A ; Q ) = ;Q A ; Ker A : (8) Corollary Ker A =ImQ s and Ker A =ImQ s : In the next section we return to the complementary subspaces Im P s and Im P s. When A 2 C(I L( j C m j C m )), then there exist projectors Q onto Ker A and P := I ; Q, which are continuous. Indeed, the orthogonal projectors Q = I ; P, P = AA + have this property by their denition. P projects onto Im A. Thus, under assumption (T3) and (3) there exist A A 2 C(I L( j C m j C m )) since A A 2 C(I L( j C m j C m )). When 2 C(I j C m ) is such that A 2 C (I j C m ), we can split L : L =[(A (P s )) 0 ; B P s ] ; B Q s. Now we use the following chain of identities: A Q s = A [QA ; (A + BP)] = A QA ; (A + B ; AP 0 Q) = A QP + A QA ; B ; A QP P 0 Q = A QA ; B = Q s B: Therefore, the function 2 := Q s A ; s satises B Q s 2 = B Q s A ; s = Q s s. On the other hand, since Im P s =ImA P A ; =ImA A ; =ImP, 8

9 if the considerations leading us to (3) are taen into account, we can state that equation L = P s s has a solution 2 Im P s. Summarizing the above for equation (3) with arbitrary s 2 C(I j C m ), with the decomposition s = P s s + Q s s, we are able to construct a solution := + 2. For this solution 2 C(I j C m ) and A 2 C (I j C m ) hold. Further we denote the function space C A(I j C m ):=f 2 C(I j C m ): A 2 C (I j C m )g: Remar 2 Clearly, the function space C A(I j C m )isby no means connected to any projector. We alsomust underline that the function spaces C A (I j C m ) and C A(I j C m ) are dierent as the following example shows. When, however, A 2 C (I L( j C m j C m ) holds, then C A (I j C m )=C A (I j C m ). Example: Given a positive scalar function : I! IR that is Lipschitz but not continuously dierentiable, m =2 A(x) =diag ((x) 0). We have A(x) = A(x) P (x) = diag (,0). The function space CA contains all con- tinuous functions y : I! j C 2 the rst component y of which iscontinuously dierentiable. On the other hand, y 2 C A means that the rst and second components are continuous but just the product y is continuously dierentiable. Hence constant functions belong to CA but they do not belong to CA. In the contrary, CA contains the function y =, y 2 =0butCA does not. Theorem Under the assumptions (T)-(T3), for arbitrary 0 2 j C m and s 2 C(I j C m ), there exists a solution 2 C A(I C m ) of (3) such that holds. Proof. Let A (^x)((^x) ; 0 )=0 (9) (^x) :=P s (^x) 0 + Q s (^x)a ; (^x)s(^x): (20) Then (9) trivially holds. Let w be the solution of the ODE L s w = P s s with initial value w(^x) =A 0. Then, := P s A + P w solves the problem L = P s s and (^x) = P s 0 taes place. A solution claimed in the Theorem is := + Q s A ;T s. 2 9

10 Denitions Equation of the form (3) is called adjoint equation of the DAE (2). An initial value 0 2 C m such that Q j s (^x) 0 = Q s (^x)a ; (^x)s(^x) holds is called consistent with (3). Note that another transparent form of consistent initial value can be obtained if one uses the identity Q s A ; = A ; Q s. Theorem 2 Under the assumptions (T)-(T3), for arbitrary 0 2 j C m and s 2 C(I j C m ), the solution of the initial value problem (3),(9) is unique. Proof. Let () (2) be a pair of dierent solutions. Due to linearity, 0 = L ( () ; (2) )must hold. Multiply this identity by Q. Then, 0=;Q 0 A ( () ; (2) ) ; Q B ( () ; (2) ) = ;Q B 0( () ; (2) )=;Q A ( () ; (2) ) i.e., A ( () ; (2) ) = A ( () ; (2) )+Q A ( () ; (2) ) = A ( () ; (2) ): As a consequence, A ( () ; (2) ) 2 C (I j C m ): Let := A ( () ; (2) ). We have 0 ; B A ; =0 and (^x) =A ( () (^x) ; (2) (^x)) = A ( 0 ; 0 )=0: (2) For, however, (2) is a homogeneous ODE with zero initial value, thus, the only solution is =0. Therefore, () ; (2) = A ; = 0,incontrast to the assumption.2 Accomplishing the section, we call attention to other representations of canonical projectors P s and P s. A reason for doing so is that the Moore- Penrose inverse A + is dened fully by A alone, while our problem is characterized by (T)-(T3). It turns out that other generalized inverses become useful to overcome this gap. Let P P be arbitrary but xed projectors as before and let A A be constructed by their use. It is an elementary exercise to chec that both A r (P ):=PA ; and A r (P ):=(P A ; ) (22) are reexive generalized inverses of A. Indeed, A r (P )AA r (P )= (PA ; )A(PA ; )=P(A ; A)PA ; = PPPA ; = PA ; = A r (P ), AA r (P )A = A(PA ; )A = AP (A ; A) =AP P = A 0

11 A r (P )AA r (P )=(A r(p )A A r(p )) =((P A ; )A (P A ; )) =(P (A ; A )P A ; ) =(P P P A ; ) =(P A ; ), AA r (P )A = (A A r(p )A ) = (A (P A ; )A ) = (A P (A ; A )) = (A P P ) = A = A. By denition, for any reexive inverse A ; of A, the products AA ; and A ; A are projectors. In our case, when A ; = A r (P ) or A ; = A r (P ) and we have AA r (P ) = AP A ; = AA ; = (A ; A ) = Ps, A r (P )A = (PA ; )A = P (A ; A) = PP = P and AA r (P ) = A(P A ; ) = ((P A ; )A ) = (P (A ; A )) = (P P ) = P, A r (P )A = (P A ; ) A = (A P A ; ) = (A A ; ) = A ; A = P s. The adjoint of the rst identity and the latter one are worth remembering: P s = A r (P )A and P s = A r(p )A (23) meantime recalling that neither P s nor P s on the left hand side depend on P or P while the reexive inverses on the right hand side do so. Finally, we add that the expression P s u := P s A + P w occuring above can be simplied as P s u := P s A + P w = A r(p )A A + P w =(P (A + A)A r (P )) w =(P PA r (P )) w =((P P)A r (P )) =(PA r (P )) w =(PPA ; ) w 3 Fundamental solutions =(PA ; ) w = A r(p )w: If Theorems and 2 are applied to homogeneous equations (s = 0), then one can conclude that for each 0 2 Im P s (^x) there exists a unique solution of L =0 (24) such that (^x) = 0 and 2 Im P s hold. One also has that for all solutions of (24) the relation 2 Im P s holds. Thus, S := Im P s is the solution space for (24). We proved in previous section that dim S = r. Due to (2), (4) and Corollary, jc m = S (x) Ker A (x): (25)

12 Meanwhile for the homogeneous equation Ly =0 (26) Q s y =0,thus, the solution space is S := Im P s. For an arbitrary y 0 2 j C m,the consistent initial value y(^x)atx = ^x satises y(^x) =P s (^x)y 0, or, equivalently, A(^x)(y(^x) ; y 0 )=0,or P (^x)(y(^x) ; y 0 )=0. Remar 3 In [6]{ where the coecients were real { the relation IR m = S(x) Ker A(x) was obtained using geometrical arguments. Our result comes directly from the representation (7). jc m = S(x) Ker A(x) (27) Up to now, operators L and L were dened in C A(I j C m ) and C A(I j C m ), only. Solutions also were considered in these spaces. Now we want to extend the denitions to handle matrix functions also. For this purposes, we introduce some denitions. First, for each q, wexabasis in C q and denote it by j fe () q ::: e(q) q g: spanfe () q ::: e(q) q g = C q. j Denitions Let for integers q and s, the function fq s : jc s :::C s {z j } q L( C q C s ) be dened implicitly so that for all y () ::: y (q) j j 2 C s j and e (i) i = ::: q, fq s (y () ::: y (q) )e (i) q = y (i) hold. With the notationc s j I := fy : I! C s g and L qs j I := fy : I! L( C q C s )g let F s j j q : jc s I ::: j C s I {z } q and e (i) q! L qs I be dened implicitly so that for all y () ::: y (q) 2 j C s I, i = ::: q, and for all x 2I! q, F s q (y () ::: y (q) )(x)e (i) q = f s q (y () (x) ::: y (q) (x))e (i) q = y (i) (x): (28) When s = m, the upper index m will be omitted and we write shortly f q and F q, respectively. A set fy () ::: y () g of functions with y (i) 2 C A(I j C m ), i = :::, will be called fundamental set of solutions of (26) if each y (i) is a solution of (26) and spanfy () ::: y () g = S. In other words, Im F q (y () ::: y () )=S. Similarly, a set f () ::: (l) g of functions such that (i) 2 j C A (I j C m ), i = ::: l will be called fundamental set of solutions of (24) if each (i) is a 2

13 solution of (24) and spanf () ::: (l) g = S. In other words, Im F q ( () ::: (l) )=S. For brevity, a function Y : I! L( j C j C m ) is called fundamental solution of (26) if Y = F (y () ::: y () ) for some fundamental set fy () ::: y () g of solutions of (26). Similarly, : I! L( C l C m ) is called fundamental solution of (24) if j j =F q ( () ::: (l) ) for some fundamental set f () ::: (l) g of solutions of (24). When Y and are fundamental solutions, then Y 2 C A(I L( j C j C m )) and 2 C A(I L( j C l j C m )) where C A(I L( j C j C m )) := fy 2 C(I L( j C j C m )) : PY 2 C (I L( j C j C m ))g and C A(I L( j C l j C m )) := f 2 C(I L( j C l j C m )) : A 2 C (I L( j C l j C m ))g, respectively, with corresponding and l. Remar 4 Since dim S = dim S = r, for a fundamental set of solutions r and l r, respectively, must hold. In contrast to the regular explicit ODEs (A = I), where r = m, and their adjoint equations, the fundamental sets of solutions are not linearly independent when r < m, r < l m. From this point of view, it would be enough to dene fundamental solutions only for = r and l = r. Then, for example, the values of Moore{Penrose inverse Y + of a fundamental solution Y would be easily computable, namely, Y + = (Y Y ) ; Y. This restricted denition, however, would mae some other computations complicated or impossible. The less computational inconvenience occurs if one uses = m and l = m. We will see below that there is a somewhat natural normalization and group properties also can be obtained. Clearly, it is not reasonable to use fundamental solutions with > m, l>m. Denitions We call fundamental solutions Y and minimal if = r and l = r, respectively. Fundamental solutions Y and will be called maximal, if = m and l = m. We remind that for r the theoretical existence of fundamental solutions of (26) becomes trivial due to (27) and unique solvability of initial value problem. The construction using the scheme of the previous paragraph is as follows: Let p (i) 2 C (I C m ), i = ::: be the solutions of the initial j value problems for ODEs: L s z =0 3 z(^x) =^p (i) (29)

14 where P 2C (I L( C m C m ) appearing in (6) is the complement of an arbitrary j j j projector Q onto Ker A which satises (T3), ^x 2Iis arbitrary but xed and the arbitrary but xed set f^p () ::: ^p() g is such that spanf^p () ::: ^p() g = Im P (^x). Then, together with an arbitrary projector Q 2 C(I L( j C m j C m )) onto Ker A,we can construct A A 2 C(I L( j C m j C m )). Now, Y := A ; A F (p () ::: p() ) (30) is a fundamental solution. For verication, rst we recall a property mentioned after (4): if ^p (i) Im P (^x), and this is the case now, then p (i) = Pp (i). Consequently, 2 yielding F (p () ::: p() )=PF (p () ::: p() ) (3) Y = A ; A PF (p () ::: p() )=A ; AF (p () = A ; P A F (p () ::: p() )=P s F (p () On the other hand, = dim Im f (^p () ::: p() ) ::: p() ) Im P s = S: dim Im Y =dimimf (p () ::: p() ) ::: ^p() )=dimimp(^x) =r: In this chain rst we used that dim Im A ; = dim Im A = m, then we exploited that L s is an ordinary dierential operator, nally we too the fact that f^p (i) g i= spans Im P (^x) into account. This completes the chec that Im Y = S. The above scheme requires the construction of the set f^p (i) g i= which might be inconvenient when r < m. When = m then, however, the above result also reads as follows: Let U be a fundamental solution of the regular ODE L s z =0such that U(^x) =I. Then, there exists a fundamental solution Y of (26) such that Y = P s UP(^x). Indeed, we may set ^p (i) m = P (^x)e (i) m, and since UP(^x) =PUP(^x), one obtains A ; A UP(^x) =A ; A UP(^x) =A ; A PUP(^x) = P s UP(^x). It is worth to note that in a lot of cases, it is easy to construct P (^x). Even when the dimension in the denition is xed, the fundamental solution is not unique. It belongs to the basic nowledge that for the regular 4

15 ODEs they may dier only by a constantinvertible multiplier M 2 L( j C m j C m ) (from the right). First we derive here some important relationships between the fundamental solutions of the DAE (26) including the case when the dimensions dier. The same will be done for the adjoint equation (24). Then we turn to highlighting the connection between the fundamental solution pairs of (26) and (24). Theorem 3 For any pair of minimal fundamental solutions Y () Y (2) there exists a T 2 L( j C r j C r ) such that Y (2) = Y () T and T is invertible. The statement extends the result nown for regular ODEs to DAEs. Proof. It is enough to prove the statement forapairy () Y (2) where Y () is a minimal fundamental solution constructed above with a xed ^x 2Iand a xed basis f^p (i) r g r i= and Y (2) is an arbitrary minimal fundamental solution. We may assume that for any x 2 I, Y () (x) 6= Y (2) (x), otherwise, due to uniqueness, Y () = Y (2) would hold and we could set T = I, where I 2 L( j C r j C r ) is the identity operator, which is clearly invertible. On the other hand, by denition, Im Y (2) (x) = S(x) = Im Y () (x), for x 2 I, so this holds at x = ^x, too. Thus, with some ft (i) g r i=, ^t(i) 2 j C r, i = ::: r, Y (2) (^x)e (i) = A ; (^x)a (^x)f r (^p () r ::: ^p(r) r )t (i) i = ::: r (32) is valid, or simply, using the notation ^P := fr (^p () r ::: ^p(r) r ) and T := f r (t () ::: t (r) ), Y (2) (^x) = A ; (^x)a (^x) ^PT. dim Im T = r, otherwise dim Im Y (2) (^x) would be less than r. Let us construct the minimal fundamental solution Y by the previous receipt proceeding from ^PT instead of ^P. Clearly, Y (^x) =Y (2) (^x) and therefore, by uniqueness theorem, Y = Y (2). On the other hand, Y = Y () T holds by construction. 2 Lemma 3 Let Y 2 C A(I L( j C j C m )) be an arbitrary fundamental solution and Y m be an arbitrary minimal fundamental solution. Then, there exists a unique W 2 L( j C j C r ) such that Y = Y m W and dim Im W = r. Proof. With no restriction of the generality, we assume Y m = Y () from Theorem 3. Then, by the same argument, Y (^x) = A ; (^x)a (^x) ^PW, W 2 L( C C r ) and dim Im W = r must hold. The proof is accomplished if an j j auxiliary fundamental solution has been constructed proceeding from ^PW since it must coincide with Y. 2 5

16 Lemma 4 Let Y 2 CA(I L( C C m )) be an arbitrary fundamental solution j j and Y m be an arbitrary minimal fundamental solution. Then, there exists V 2 L( C r C ) such that Y j j m = YV. Proof. Let Y = F (y () ::: y () ). Since dim Im Y = r, there exists a permutation P = fi ::: i g such that fy (i ) ::: y (ir) g is a fundamental set of solutions of (26) and therefore, F r (y (i ) ::: y (ir) ) is a minimal fundamental solution. By Theorem 3, there exists a unique T 2 L( j C r j C r ) such that F r (y (i) ::: y (ir) )T = Y m. Let D 2 L( C r C ) dened by De (j) j j r = e (i j ), j = ::: r, or more compactly, Dfr r (e () r ::: e(r) r ) = fr (e (i ) ::: e (ir) ). Then, for all x 2I, Y (x)df r r (e () r ::: e(r) r )T = f r (y (i ) (x) ::: y (ir) (x))t = Y m (x): (33) Setting V = Dfr r (e () r ::: e(r) r )T, one arrives to the existence of a V 2 L( C C m ) claimed in the statement. Again, Y j j m cannot be fundamental, unless dim Im V = r holds. 2 Remar 5 In this Lemma we could not claim uniqueness of V as the following trivial example shows. Let Y m = F r (y () ::: y (r) ) Y = F (y () ::: y (r) 0 ::: 0) >r. For any matrixv the rst r rows of which form an identity bloc but the last ; r rows are arbitrarely chosen, we have the relation Y m = YV. Combining Lemmas 3 and 4 we arrive to the general Theorem 4 For any pair of fundamental solutions Y () 2 C A(I L( j C j C m )) and Y (2) 2 C A(I L( j C 2 j C m )), there exists R 2 L( j C 2 j C ) such that Y (2) = Y () R and dim Im R = r. Proof. Indeed, Y (2) = Y m W = Y () VW, where Y m is an arbitrary minimal fundamental solution and V and W are chosen correspondingly by Lemmas. Let R = VW. dim Im R = r holds otherwise Y (2) cannot be fundamental.2 It is worth noting the special case = 2 = m: Corollary 2 For any pair of maximal fundamental solutions Y () Y (2) there exists R 2 L( j C m j C m ) such that Y (2) = Y () R and dim Im R = r. 6

17 Based on decomposition (25) and Theorems and 2, the existence of fundamental solutions 2 CA(I L( C l C m ), l = r ::: m can be obtained. j j Moreover, by analogous arguments as for the original homogeneous DAE, the analogues of Lemmas 3 and 4 and Theorems 3 and 4 hold for adjoint equations, just Y must be changed for. An obvious change arises in the construction of fundamental solutions. Let p (j) l 2 C (I C m ), i = ::: l be the solutions of the initial value j problems for ODEs: L s w =0 w(^x) =^p(j) l (34) where P 2C (I L( C m C m ) appearing in (6) is the complement of an arbitrary j j j projector Q onto Ker A which satises (T3), ^x 2Iis arbitrary but xed and the arbitrary but xed set f^p () l ::: ^p(l) l g is such that spanf^p () l ::: ^p(l) l g = Im P (^x). Then, we can construct A 2 C(I L( j C m j C m )). Now, let :=A ; P F l (p () l ::: p(l) l ) 2 C(I L( j C l j C m )): (35) We obtain A =P F l (p () l ::: p(l) l ) 2 C (I L( j C l j C m )): Consequently, 2 CA(I L( C l C m )) and Im Im P j j s = S. Further, is a fundamental solution of (24). Indeed, A ; P = P s A + P and due to (8){(3), it involves that A ; P p (j) l = P s A + P p (j) l, j = ::: l satises (24). Noting that P p (j) l are solutions of the ODE v 0 ; (P 0 + P B 0 A; )v = 0, the above choice of the initial values ensures dim Im =dimima ; P F l (p () l ::: p(l) l ) = dim Im P F l (p () l ::: p(l) l )=dimimf l(p p () l ::: P p (l) ) l () (l) = dim Im f l (P ^p l ::: P^p ) = dim Im l P = r This completes the chec that Im =S. Now, the verication of the analogues of the statements about relationships between dierent fundamental solutions of the adjoint equations becomes trivial if one follows the scheme of the corresponding proofs for the original equation. Now we state a connection between fundamental solutions of (26) and (24). 2 C A(I L( j C j C m )) and 2 C A(I L( j C l j C m )) be an Theorem 5 Let Y arbitrary pair of fundamental solutions for (26) and (24). Then, ( AY ) 0 =0 (36) 7

18 or, in other words, there exists 2 L( j C l j C ) such that for all x 2 I, ((x)) A(x)Y (x) =holds. Remar 6 When A(x) = I, this result is the well-nown Lagrange identity for regular ODE-s. Proof. 2 ( AY ) 0 =( AP Y ) 0 =( A) 0 PY + A(PY) 0 = BPY ; A(PA ; B 0 ; P 0 )PY = (B ; AP A ; B 0)PY = (I ; AA ; )BPY = (I ; P s)bpy =( Q s)bpy =0: In contrast to the regular ODEs, when minimal and maximal fundamental solutions are of the same dimensions, and thus, they are invertible, for DAEs this is not the case. One may as about the relation of a generalized inverse of a fundamental solution for the DAE to the adjoint equation. Clearly, these results are closely related to the derivation of the adjoint equation described in the previous paragraph. First we concern maximal fundamental solutions with a basic value at a xed point. Denition A maximal fundamental solution Y of (26) will be called normalized at ^x if A(^x)(Y (^x) ; I) = 0. A maximal fundamental solution of (24) will be called normalized at ^x if A (^x)((^x) ; I) =0. By virtue of (7) and (2), we immediately obtain P s (^x)y (^x) = P s (^x) and P s (^x)(^x) = P s (^x) for maximal fundamental solutions normalized at ^x. Thus, solutions of the initial value problems posed with initial values Y (^x) =P s (^x) and(^x) =P s (^x) provide to us a pair of such Y and. The simplest consequence is that for each ^x, there exist both maximal fundamental solutions normalized at this ^x. Further, we can state Corollary 3 Provided Y and are maximal fundamental solutions of (26) and (24) normalized atthe same point ^x, relation (^x)a(^x) =A(^x)Y (^x) is valid. For them, Theorem 5 yields 2 L( j C m j C m ), =A(^x). Lemma 5 For each ^x, both maximal fundamental solutions Y and normalized at ^x are unique. 8

19 Proof. Let Y be a maximal fundamental solution of (26) and U be the fundamental solution of ODE L s z = 0 such that U(^x) = I. By denition, L s (PYe i ) = 0, i = ::: m and we showed that PUP(^x) = UP(^x). Since Q s Y = 0, we have Y = P s Y = P s PY = P s UP(^x)N with some N 2 L( j C m j C m ). Since Y is assumed to be normalized at ^x, we obtain P s (^x)n = P s (^x). Now, taing into account that P s (^x) =A ; (^x)a (^x)p (^x) with arbitrary P and P,we arrive atp (^x)n = P (^x), i.e., Y = P s UP(^x). Let W be the fundamental solution of ODE L sw = 0 such that W (^x) =I. We checed that P W = P WP (^x) and L (P s A + P (We i )) = 0. Noting that = A ; P s A + P W = A ; A A + P W PP W = A ; P W = A ; P WP (^x) (37) we can state that any maximal fundamental solution of (24) is of the form = A ; P WP (^x)m with some M 2 L( C m C m ). Since P j j s (^x) = A ; A (^x), the condition of the normalization yields now P (^x)m = A (^x), i.e. = A ; P WA (^x) =A r(p )WA (^x). We underline again that in spite of their form obtained above, both Y and are independent of P and P since the initial values P s (^x) and P s (^x) do not depend on them.2 We recall now that W = U ;. In order to get the analogue for the pair of maximal fundamental solutions Y and normalized at the same ^x, we introduce the proper generalized inverses. Let Y ; be the reexive inverse of Y such that YY ; = P s and Y ; Y = P s (^x), while ; be the reexive inverse of such that ; = P s and ; = P s (^x). One checs that they can be given explicitly, namely, Y ; = P s (^x)u ; P and ; = A r(p )(^x)w ; A. Since the reexive generalized inverses with the prescibed products are unique and neither Y nor depend on P or P, Y ; and ; are independent of them, as well. Clearly, Y ; and ; are reexive inverses of Y and, respectively. Note, however, that for these reexive inverses Y ; := Y ; and ; := ; one has Y Y ; = P s (^x) Y ; Y = P s and ; = P s(^x) ; = P s : In the next Theorem we use Y ; Y ; ; ; dened in this way. 9

20 Theorem 6 Given the maximal fundamental solution Y of (26) normalized at ^x, then the maximal fundamental solution of (24) normalized at ^x is :=A r(p )Y ; A (^x): (38) Given the maximal fundamental solution of (24) normalized at ^x, then the maximal fundamental solution of (26) normalized at ^x is Y := A r (P ) ; A(^x): (39) Proof. By Lemma 5, = A r(p )WA (^x). We also use A r(p ) = A r(p )P and A = AP s and the explicit form of the inverse Y ; and get =A r(p )WA (^x) =(A r(p )P )U ; (A(^x)P s (^x)) = A r (P )(P U ; P s (^x) )A (^x) =A r (P )Y ; A (^x): For the second part, above Y = P s UP(^x) obtained in Lemma 5, we use A r (P )A = P s and A r (P )A = P. Then, using the explicit form of the inverse ;, 2 Y = P s UP(^x) =(A r (P )A)W ; (A r (P )(^x)a(^x)) = A r (P )(AW ; (A r (P )(^x))a(^x)) = A r (P ) ; A(^x): Among the fundamental solutions the maximal ones stand out by a computational comfort that is quite similar to the regular ODE case. This is due to the natural normalization used above. What concerns minimal fundamental solutions, so there seems to be no canonical way to normalize since there is no distinguished basis of Im P s (^x) to start with. Let us nish this section by realizing the wanted group properties of maximal fundamental solutions Y (: ^x) (: ^x) of (26) and (24) normalized at ^x. Theorem 7 For normalized maximal fundamental solutions of (26) and (24), it holds that for all x ^x z 2I. Y (x ^x) ; = Y (^x x) Y (x ^x) =Y (x z)y (z ^x) (x ^x) ; =(^x x) (x ^x) =(x z)(z ^x) 20

21 Proof. From above we have the representations Y (x ^x) = P s (x)u(x ^x)p (^x) Y (x ^x) ; = P s (^x)u(^x x)p (x) (x ^x) = (x ^x) ; = A r(p )(x)w (x ^x)a (^x) A r(p )(^x)w (^x x)a (x): Hence, the rst pair of relations is obvious. Further, compute Y (x z)y (z ^x) = P s (x)u(x z)p (z)p s (z)u(z ^x)p (^x) = P s (x)u(x z)p (z)u(z ^x)p (^x) = P s (x)u(x z)u(z ^x)p (^x) = P s (x)u(x ^x)p (^x) =Y (x ^x): (x z)(z ^x) = A r(p )(x)w (x z)a (z)a r(p )(z)w (z ^x)a (^x) = A r(p )(x)w (x z)p (z)w (z ^x)a (^x) = (A r(p )(x)p (x))u (z x)p (z)u (^x z)a (^x) = A r(p )(x)(p (z)u(z x)p (x)) U (^x z)a (^x) = A r(p )(x)p (x)u (z x)u (^x z)a (^x) = A r(p )(x)p (x)u (^x x)a (^x) = A r(p )(x)w (x ^x)a (^x) =(x ^x): 2 For the normalized maximal fundamental solution Y (x ^x)andeach arbitrary fundamental solution Y ~ (x) we have the indentity ~Y (x) =Y (x ^x) ~ Y (^x): (40) Supposed ~ Y (x) is minimal and the columns of ~ Y (^x) form an orthogonal basis of the subspace S(^x), the relation ~Y (^x) ~ Y (^x) =I (4) becomes true while ~ Y (^x) ~ Y (^x) represents the orthoprojector of j C m onto S(^x). Then, due to ~ Y (^x) ~ Y (^x) P s (^x) =P s (^x) Y (x ^x)p s (^x) =Y (x ^x) multiplying the indentity (40) by ~ Y (^x) P s (^x) leadsto ~Y (x) ~ Y (^x) P s (^x) =Y (x ^x): (42) If we start with two dierent orthogonal basic systems of S(^x) we obtain two dierent minimal fundamental solutions but both of them satisfy (4) and (42). Therefore, thining of normalizing minimal fundamental solutions via (4) maes no sense. A similar argument is true for the minimal fundamental solutions of the adjoint equation (24). 2

22 4 Index denition for adjoint equation Under the only assumptions (T)-(T3), equation (3) is not a DAE in standard form. We willshownow that we can extend the denition of DAE with index to include (3). Consider equations of the form (A ') 0 ; B ' = p (43) where A B : I! L( j C m j C m ) are continuous matrix functions that satisfy assumptions (T)-(T3). Note that in terms of the adjoint matrices these assumptions are the same as (T) dim Im A(x) r<mfor all x 2I (T2) the matrix pencil fa(x) B(x) g is regular and has index for x 2I, (T3) there exists a projector Q 2 C (I L( j C m j C m ) such that Im A(x) = Ker Q(x) for x 2I. With P = I ; Q P A = A we derive (A ') 0 =(P A ') 0 = P (A ') 0 + P 0 A' for all ' 2 C A (I j C m ). Recall that this function space contains all continuous functions ' which have a continuously dierentiable part A '. By means of this expression we rewrite (43) as P (A ') 0 + P 0 A' ; B ' = p: (44) Trivially, if' 2 C A solves (44), then the pair ' w, w := A ' represents a solution of the enlarged system P w 0 + P 0 w ; B ' = p (45) w ; A ' =0: (46) Written in compact form the system (45), (46) is simply P 0 0 0! w '! 0 + P 0 ;B I ;A! w '! = p 0! : (47) 22

23 Theorem 8 The coecients of (47) are continuous and satisfy (T)- (T3) correspondingly. Proof. (T) and (T3) are given trivially. It remains to chec whether the matrix G := P! 0 + P!! 0 ;B Q 0 = P! + P 0Q ;B 0 0 I ;A 0 I Q ;A remains nonsingular. In fact, G(x)z = 0, means (we drop the argument x) Q z ; A z 2 =0 P z + P 0 Q z ; B z 2 =0 what implies Q z =0 A z 2 =0 P z ; B z 2 =0 i. e. Q z =0 z 2 2 Ker A B z 2 = P z 2 Im A. Due to the index--property offa B g we nd z 2 =0,thus z =0. 2 It comes out that the enlarged system (45), (46) resp. (47) is an index- DAE in standard form. Due to the respective theory, for arbitrary continuous right hand sides, there are continuous solutions w ' that have continuously dierentiable components P w. In particular, for continuous p, equation (47) has those solutions, but due to (46) we have A ' = P A ' = P w what means that ' belongs to C A. In this sense, (43) and the standard form DAE (47) are equivalent. This gives rise to dene equation (43) to be index- tractable if (T)-(T3) are satised. By this, the original DAE (2) is index- tractable if and only if (43) is so. It should be mentioned that the enlarged system w 0 ; B ' = p (48) w ; A ' =0 (49) instead of (47) will not do, since this DAE that has also standard form is of higher index. Namely, the respective matrix becomes nonsingular. I B 0 ;A 23!

24 5 Acnowledgement The manuscript of this wor was prepared basically during the visits of the rst author at the Department of Numerical Methods, Institute of Applied Mathematics at Humboldt University. She is grateful for this possibility to all persons there arranging these visits. References [] A. A. Abramov: On the transfer of boundary conditions for systems of linear ordinary dierential equations. Zh. Vychisl. Mat. Mat. Fiz. Vol., No. 3, pp. 542{545 (in Russian English transl. in: USSR J. Comp. Math. Math. Phys.) (96) [2] K. Balla: Linear subspaces for linear DAEs of index. Computers Math. Applic. Vol. 32, No. 4/5, pp. 8{86 (996) [3] K. Balla: Boundary conditions and their transfer for dierential{ algebraic equations of index Computers Math. Applic. Vol. 3, No. 0, pp. {5 (996) [4] K. Balla, R. Marz: Transfer of boundary conditions for DAEs of index. SIAM J. Numer. Anal. Vol. 33, No. 6, pp. 238{2332 (996) [5] E. A. Coddington, N. Levinson: Theory of ordinary dierential equations. Mc Graw Hill, New Yor, 955. [6] E. Griepentrog, R. Marz: Dierential-Algebraic Equations and Their Numerical Treatment Leipzig, Teubner Verlag, 986. [7] R. Marz: Extra-ordinary dierential equations. Attempts to an analysis of dierential{algebraic systems.in: European Congress of Mathematics, Budapest, July 22-26, 996, Vol.. (eds.: A. Balog, G. O. H. Katona, A. Recsi, D. Szasz). Series \Progress in Mathematics" Vol. 68. Birhauser Verlag, pp , 998. [8] R. Marz: Numerical methods for dierential-algebraic equations Acta Numerica, pp. 4{98 (992) 24