Analysis and reformulation of linear delay differential-algebraic equations

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1 Electronic Journal of Linear Algebra Volume 23 Volume 23 (212) Article Analysis and reformulation of linear delay differential-algebraic equations Phi Ha Volker Mehrmann Follow this and additional works at: Recommended Citation Ha Phi and Mehrmann Volker. (212) "Analysis and reformulation of linear delay differential-algebraic equations" Electronic Journal of Linear Algebra Volume 23. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information please contact

2 ANALYSIS AND REFORMULATION OF LINEAR DY DIFFERENTIAL-ALGEBRAIC EQUATIONS PHI HA AND VOLKER MEHRMANN Abstract. General linear systems of delay differential-algebraic equations (DDAEs) of arbitrary order are studied in this paper. Under some consistency conditions it is shown that every linear highorder DAE can be reformulated as an underlying high-order ordinary differential equation (ODE) and that every linear DDAE with single delay can be reformulated as a high-order delay differential equation (DDE). Condensed forms for DDAEs based on the algebraic structure of the system coefficients are derived and these forms are used to reformulate DDAEs as strangeness-free systems where all constraints are explicitly available. The condensed forms are also used to investigate structural properties of the system like solvability regularity consistency and smoothness requirements. Key words. Delay differential-algebraic equation Differential-algebraic equation Regularization Strangeness-index Index reduction. AMS subject classifications. 34A9 34A12 65L5 65H1. 1. Introduction. In this paper we study general linear delay differentialalgebraic equations (DDAEs) of the form (1.1) A k x (k) (t)+ +A x(t)+a 1 x(t τ)+ +A κ x (κ) (t τ)=f(t) where the coefficients satisfy A i C ln i=k... κ A k /= f [ ) C l and where τ> is a single constant delay. We consider the time interval t [ ). Note that most of our analysis also carries over to multiple and nonconstant delays but here we restrict ourselves to the constant single delay case. An important special case of (1.1) is the initial value problem for a first order linear delay differential-algebraic equation with single delay (1.2) A 1 ẋ(t)+a x(t)+a 1 x(t τ)=f(t) where A 1 A A 1 C ln f [ ) C l. Received by the editors on April Accepted for publication on August Handling Editor: James G. Nagy. Institut für Mathematik MA 4-5 TU Berlin Strasse de 17. Juni 136 D-1623 Berlin Germany (ha@math.tu-berlin.de mehrmann@math.tu-berlin.de). Supported by DFG Collaborative Research Centre 91 Control of self-organizing nonlinear systems: Theoretical methods and concepts of application. 73

3 74 P. Ha and V. Mehrmann To achieve uniqueness of solutions for DDAEs one typically has to prescribe initial functions which for the special case (1.2) take the form (1.3) x [ τ] =φ [ τ] C n. Ordinary delay differential equations (DDEs) of the form (1.2) with A 1 being the identity matrix arise in various applications see [3 1] and the references therein. If the states of the physical system are constrained e.g. by conservation laws or interface conditions then algebraic equations have to be included and one has to analyze delay differential-algebraic equations (DDAEs). DDAEs may be considered from two different perspectives. On the one hand they are differential-algebraic equations (DAEs) that involve delayed terms. On the other hand DDAEs are ordinary delay differential equations (DDEs) subject to constraints that also may involve time-delayed variables. Of course DDAEs inherit all the difficulties that are associated with both DAEs and DDEs. Their interaction however leads to new effects that do not arise in either DAEs or DDEs as has been pointed out in [2 6 8]. Although DDEs are well studied analytically and numerically see e.g. [3 1] and a similar maturity has been reached for the simulation and control of DAEs see e.g. [ ] the theoretical understanding and the development of appropriate numerical methods for DDAEs however is far from complete even for the case of linear systems with constant coefficients. Only very few results are available see e.g. [ ] and these are mainly for the special case of DAEs where the delay component is nothing else than an additional part of the inhomogeneity. The main difficulty so far is the lack of a suitable regularity analysis (via the concept of an index) and a canonical form which allows to investigate structural properties like existence uniqueness of solutions consistency and smoothness requirements for the initial function. In this paper we derive such a canonical form for the linear constant coefficient case by extending the algebraic approach introduced in [13 2] and combining it with the behavior approach [18]. Surprisingly already in order to deal with (1.2) it is necessary to study linear high-order differential-algebraic equations of the form (1.4) A k x (k) (t)+ +A 1 ẋ(t)+a x(t)=f(t) with associated initial conditions of the form (1.5) x (k 1) ()=x (k 1)... ẋ()=x (1) x()=x(). We study the theoretical aspects of (1.4) (1.5) in Section 3 and then use these to study the general case of DDAEs in Section 4. The analysis is based on reformulation procedures which bring the systems into a strangeness-free form and allows also to

4 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 75 study theoretical aspects like existence and uniqueness of solutions as well as the consistency and smoothness requirements for the initial functions. 2. Notation and preliminaries. In the following we denote by I n C nn (or I) the identity matrix and by A T the transpose of a matrix A. For an interval I [ ) by C k (IC n ) we denote the space of k-times continuously differentiable functions from I to C n. We use the following solution concept for (1.2). Definition 2.1. A function x [ ) C n is called a (classical) solution to (1.2) if x C 1 ([ )C n ) and x satisfies (1.2) pointwise. An initial function φ is called consistent with system (1.2) if the associated initial value problem (1.2) (1.3) has at least one classical solution. System (1.2) is called solvable if for every sufficiently smooth f and every consistent initial function φ the associated initial value problem (1.2) (1.3) has a solution. It is called regular if it is solvable and the solution is unique. x (k 1) Introducing X = as initial vector of the initial value problem consisting x () of (1.4) (1.5) Definition 2.1 extends to higher order systems i.e. an initial vector X C (k+1)l iscalledconsistent forsystem(1.4) iftheinitial valueproblem(1.4) (1.5) has a solution and system (1.4) is called solvable if for every sufficiently smooth f and every consistent initial vector X the associated initial value problem (1.4) (1.5) has a solution. Systems of differential-algebraic equations arise in real-time simulation and automated physical modeling via software packages such as Dymola[9] or Matlab Simulink [16] in all areas of technology. In general they may be over- or under-determined i.e. they may contain free variable (controls) and/or redundant equations. Usually then a reformulation procedure is necessary to remove redundant equations to make the system consistent or to achieve better properties for simulation and control [5 13]. The same procedure is necessary for delay differential-algebraic equations. To achieve a good reformulation also for delay differential-algebraic equations one must add hidden constraints to the system and decouple the equations into two subsystems one which describes the dynamics of the system and one which describes all the constraints together with redundancy and consistency conditions that allow a regularization. We will derive such a reformulation and regularization procedure for systems(1.4) and (1.1) in Sections 3 4.

5 76 P. Ha and V. Mehrmann To illustrate the difficulties that may arise we present a simple example. Example 2.2. Consider the first order DDAE 1 ẋ 1 (t) 1 x 1 (t) (2.1) 1 ẋ 2 (t) + 1 x 2 (t) ẋ 3 (t) x 3 (t) 1 x 1 (t τ) f 1 (t) + 1 x 2 (t τ) = f 2 (t) 1 x 3 (t τ) f 3 (t) and the associated non-delay equation (2.2) 1 ẋ 1 (t) 1 x 1 (t) f 1 (t) 1 ẋ 2 (t) + 1 x 2 (t) = f 2 (t) ẋ 3 (t) x 3 (t) f 3 (t) in the time interval[ ). Obviously system (2.2) is under-determined and f 3 = is a consistency condition. If this holds and if x 1 and f 1 f 2 are continuous then the solution x is continuous as well. However if the consistency condition x 3 (t τ)=f 3 (t) t [τ] holds then it immediately follows that system (2.1) has the general solution 1 1 f 1 (t) f 2 (t)+ f 3 (t+τ) x(t)= 1 x(t τ)+ ẋ(t τ)+ f 2 (t) f 3 (t+τ) f 3 (t+τ) for all t [ ). To obtain a unique solution we have to provide an initial function φ=x [ τ] which is consistent if and only if φ 3 (t τ)=f 3 (t) t [τ]. Note however that the inhomogeneity f has to be at least twice continuously differentiable and since x(t) is differentiated in the delay further smoothness requirements are necessary for the initial function φ depending on the length of the considered time interval. Example 2.2 shows that further differentiability and consistency conditions may be required for DDAEs. To characterize these for general linear DDAEs with constant coefficients we will derive a condensed form. For this we need the following preliminary results. For matrices Q C qn P C pn the matrix pair(qp) is said to have no hidden redundancy if rank([ Q P ])=rank(q)+rank(p).

6 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 77 Lemma 2.3. Suppose that for Q C qn P C pn the pair(qp) has no hidden redundancy. Then for any matrix U C qq and any V C pp the pair(uqvp) has no hidden redundancy. Proof. The proof follows from the observation that a matrix pair has no hidden redundancy if and only if the intersection of the two vector spaces spanned by the rows of the two matrices contains only the vector. If[ Q P ] is of full row rank for two matrices Q Cqn P C pn then obviously the pair(qp) has no hidden redundancy. However the converse is not true as is obvious for Q=[ 1 1 ] P=[ ] since(qp) has no hidden redundancy but[q P ] does not have full row rank. Lemma 2.4. For Q C qn P C pn there exists [ S Z 1 Z 2 ] C qq+p where[ S Z 1 ] C qq is nonsingular and the rows of S C pq are the rows of a permutation matrix such that and(sqp) has no hidden redundancy. [ S Z 1 Z 2 ][ Q P ]=[SQ ] Proof. The proof follows by taking[z 1 Z 2 ] to be a full rank matrix spanning the left nullspace of[ Q ] and completing it to a full rank matrix by rows of a permutation P matrix so that[ S Z 1 ] is invertible and(sqp) has no hidden redundancy. Lemma 2.4 will be used later to recursively remove hidden redundancy in the coefficients of linear DAEs and DDAEs. Lemma 2.5. Consider k+1 full row rank matrices R C rn...r k C r kn such that none of the matrix pairs (2.3) R j R j 1 R j=k...1

7 78 P. Ha and V. Mehrmann R k has a hidden redundancy. Then has full row rank. R Proof. Since none of the matrix pairs in (2.3) has a hidden redundancy it follows that R k R k 1 rank =rank(r k )+rank R R =rank(r k )+rank(r k 1 )+rank = R k 2 R =rank(r k )+rank(r k 1 )+ +rank(r ) R k and since R k...r have full row rank also has full row rank. R 3. Analysis and reformulations of high-order DAEs. It will turn out that the reformulation of DDAEs leads to higher order DAEs. In this section we therefore studytheanalysisofhigh-orderdaesoftheform(1.4)andoftheinitialvalueproblem (1.4) (1.5) see also [ ] and the references therein for previous work on this topic. We will extend these results by combining it with the regularization procedure for DAEs proposed in [2] in a behavior setting [18]. Let x (k) (t) M =[A k...a ] C l(k+1)n and X(t) =. x(t) Then M (resp. X(t)) is called the behavior matrix (resp. behavior vector) of system (1.4) which can be written as (3.1) MX(t)=f(t). For notational convenience in this section we omit the argument t in X x f and their derivatives. Scaling (1.4) with a nonsingular matrix P C ll we obtain (3.2) PMX=P A i x (i) =Pf. k

8 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 79 Since the systems (3.1) and (3.2) have the same solution spaces we introduce the following definition. Definition 3.1. Two behavior matrices M=[A k...a ] and M=[Ãk...Ã] in C l(k+1)n are called (strongly) left equivalent (denoted by l ) if there exists a nonsingular matrix P C ll such that M=PM or equivalently à j =PA j j=k... Lemma 3.2. Consider the behavior matrix M of system (1.4). Then M is left equivalent to a matrix A k1 A k A 1 A k A 2 (3.3) M = A k+1... r 1 r 2 r k+1 v where all the matrices A k jj+1 j=k... on the main diagonal have full row rank. The integer numbers r 1...r k+1 v are the sizes of the block rows. Proof. We first compress the first block column of M via a QR-decomposition see [11] to M=[A k...a ] l [ A k1 A k A 1 A k A 2 ] such that A k1 has full row rank. Continuing by compressing the 2nd block column from the second block row and then inductively the other columns of M we finally arrive at (3.3). We call the number r u =(k+1)r 1 +kr r k +r k+1 the upper rank of the behavior matrix M. Note that some of the r i may vanish and obviously the upper rank is invariant under left equivalence transformations. In the following without loss of generality we assume that the behavior matrix M is already in the form M. Rewriting system (3.1) block row-wise we obtain the

9 71 P. Ha and V. Mehrmann system (3.4) A k1 x (k) +A k 11 x (k 1) + +A 11 ẋ+a 1 x=f 1 A k 12 x (k 1) + +A 12 ẋ+a 2 x=f 2 A k+1 x=f k+1 =f k+2. Recall that the diagonal blocks A k1 A k 12...A k+1 have full row rank therefore in system (3.4) for every j with k j the(k+1 j)-th block row A jk+1 j x (j) + +A k+1 j x=f k+1 j represents r k+1 j = rank(a jk+1 j ) scalar differential equations of order j. The idea now is to use differential equations of order smaller than j and their derivatives to reduce the number of scalar differential equations of order j. Let us illustrate this idea for the case j=k. If the pair A k1 there exists a matrix A k 12 A k+1 such that[ S k Z kk ] C r1r1 is nonsingular has hidden redundancy then Lemma 2.4 implies that [ S k... Z kk Z kk 1... Z k ] (3.5) Z kk A k1 +[Z kk 1...Z k ] and the matrix pair S k A k1 and A k 12 A k+1 A k 12 A k+1 = has no hidden redundancy. Scaling the first equation of (3.4) with[ S k Z kk ] from the left we get S k A k1 x (k) +S k A k 11 x (k 1) + +S k A 1 x=s k f 1 Z kk A k1 x (k) +Z kk (A k 11 x (k 1) + +A 1 x)=z kk f 1.

10 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 711 From (3.5) we deduce Z kk A k1 x (k) = [Z kk 1...Z k ] This leads to the systems and k 1 = k 1 = k 1 = Z ki A ik+1 i x (k) A k 12 A k+1 x (k) Z ki ( d dt ) k ia ik+1 i x (i) ( d i 1 k iz dt ) ki ( l= A lk+1 i x (l) +f k+1 i ). S k A k1 x (k) +S k A k 11 x (k 1) + +S k A 1 x=s k f 1 (3.6) k 1 Z ki ( d i 1 k i( dt ) l= A lk+1 i x (l) f k+1 i ) +Z kk (A k 11 x (k 1) + +A 1 x)=z kk f 1. Note that (3.6) is a set of differential equations of order at most k 1. Hence we have reduced the number of scalar differential equations of order k from r 1 = rank(a k1 ) to d 1 =rank(s k A k1 ). Applyingthesameargumenttotheblockrowsnumberedj=k weobtain the following two lemmas. For notational convenience we denote by unspecified matrices. Lemma 3.3. Consider the DAE (1.4) in its behavior form (3.1). Moreover assume that the behavior matrix M is in the form (3.3). Then there exist matrices S j Z ji j=k...1 i=j... of appropriate size such that i) the matrices[ S j Z jj ] C rjrj k j 1 are nonsingular ii) for each j with k j 1 Z jj A jk+1 j +[Z jj 1...Z j ] A j 1k+2 j A k+1 =

11 712 P. Ha and V. Mehrmann iii) for each j with k j 1 the matrix pair has no hidden redundancy. S j A jk+1 j A j 1k+2 j A k+1 Proof. For each j with k j 1 by applying Lemma 2.4 to the matrix pair A jk+1 j A j 1k+2 j A k+1 we obtain matrices S j Z ji i=j... that satisfy conditions i)-iii). Setting P =diag([ S k Z kk ]...[ S 1 Z 11 ]I rk+1 +v) C ll and scaling system (1.4) with P from the left we obtain (3.7) S k A k1... Z kk A k1... S k 1 A k Z k 1k 1 A k A k+1... x (k) x (k 1) x S k f 1 Z kk f 1 S k 1 f 2 = Z k 1k 1 f 2 f k+1 f k+2 For each j with k j 1 we then reduce the number of differential equations of order j by eliminating the block Z jj A jk+1 j of (3.7) as in the following lemma. Lemma 3.4. Let matrices S j Z ji j= k...1 i=j... be defined as in Lemma 3.3. Then the DAE (3.7) has the same solution set as the DAE (3.8) d 1 s 1 d 2 s 2 d k+1 v S k A k S k 1 A k A k+1... M x (k) x (k 1) x S k f 1 g 2k S k 1 f 2 = g 2(k 1) f k+1 f k+2

12 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 713 where j g 2j = Z ji ( d j if dt ) k+1 i j=k...1. Proof. For each j with k j 1 by inserting into the equation we have or equivalently j 1 (3.9) j 1 Z jj A jk+1 j = Z ji A ik+1 i Z jj A jk+1 j x (j) j 1 +Z jj A ik+1 j x (i) =Z jj f k+1 j j 1 ( Z ji A ik+1 i )x (j) j 1 +Z jj A ik+1 j x (i) =Z jj f k+1 j Z ji ( d j i dt ) A ik+1 i x (i) j 1 +Z jj A ik+1 j x (i) =Z jj f k+1 j. Moreover the(k+1 i)-th equation of (3.4) implies that (3.1) A ik+1 i x (i) i 1 = A lk+1 i x (l) +f k+1 i. l= Thus substituting (3.1) into (3.9) we have j 1 or equivalently Z ji ( d i 1 j i( dt ) =Z jj f k+1 j j 1 l= Z ji ( d i 1 j i( dt ) j 1 = A lk+1 i x (l) j 1 +f k+1 i )+Z jj A ik+1 j x (i) l= A lk+1 i x (l) j 1 )+Z jj A ik+1 j x (i) Z ji ( d dt ) j if k+1 i +Z jj f k+1 j =g 2j. Continuing like this inductively we obtain (3.8).

13 714 P. Ha and V. Mehrmann From (3.8) we deduce that r j =d j +s j j=1...k+1 s k+1 = and therefore the upper rank of the behavior matrix M of system (3.8) can be bounded from above via r u (k+1)d 1 +k(s 1 +d 2 )+ +(s k +d k+1 ) =(k+1)(d 1 +s 1 )+k(d 2 +s 2 )+ +(d k+1 +s k+1 ) s i =(k+1)r 1 +kr 2 + +r k+1 s i =r u s i and thus this procedure of passing the system (3.3) to (3.8) has reduced the upper rank. This reduction of the upper rank leads to the following procedure. Procedure 3.5. Input: The DAE (1.4) and its behavior form (3.1). Begin: Set α= and let M =M f =f. Step 1. Determine a nonsingular matrix P C ll (as in Lemma 3.2) such that k k A k1 A k A 1 A PM α k A 2 = A k+1... r 1 r 2 k r k+1 v where all the matrices on the main diagonal have full row rank and let r α u =(k+1)r 1 +mr r k +r k+1 be the upper rank of the behavior matrix M α in the α-th iteration. Step 2. Determine matrices S j Z ji j=k...1 i=j... of appropriate size such that i) matrices[ S j Z jj ] C rjrj k j 1 are nonsingular ii) for each j with k j 1 Z jj A jk+1 j +[Z jj 1...Z j ] iii) for each j with k j 1 the matrix pair has no hidden redundancy. S j A jk+1 j A j 1k+2 j A k+1 A j 1k+2 j A k+1 =

14 Step 3. Setting Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 715 P =diag([ S k Z kk ]...[ S 1 Z 11 ]I rk+1 +v) C ll and scaling system (1.4) with P from the left we obtain (3.11) S k A k1... Z kk A k1... S k 1 A k Z k 1k 1 A k A k+1... x (k) x (k 1) x S k f 1 Z kk f 1 S k 1 f 2 = Z k 1k 1 f 2. f k+1 f k+2 Step 4. For each j with k j 1 we then reduce the number of differential equations of order j by eliminating the block Z jj A jk+1 j of (3.11) as in Lemma 3.3. In this way we obtain the system S k A k S k 1 A k A k+1... M x (k) x (k 1) x S k f 1 g 2k S k 1 f 2 = g 2(k 1) f k+1 f k+2 f d 1 s 1 d 2 s 2 d k+1 v with j g 2j = Z ji ( d j if dt ) k+1 i j=k...1. Let s α = k s i we then increase α by 1 set M α = M f α = f and repeat the process from Step 1. End. Since ru α+1 ru α sα Procedure 3.5 terminates after a finite number of iterations and thus we have the following theorem.

15 716 P. Ha and V. Mehrmann Theorem 3.6. The DAE (1.4) has the same solution set as the DAE (3.12)  k1  k  1  k  2  k+1... x (k) x (k 1) x = ˆf 1 ˆf 2 ˆf k+1 ˆf k+2 where  k1  k+1 has full row rank. Proof. Clearly after carrying out Procedure 3.5 we obtain a system of the form (3.12) where Âk1...Âk+1 have full row rank and none of the matrix pairs  jk+1 j j=k...1 has a hidden redundancy.  j 1k+2 j  k+1 Applying Lemma2.5to the matricesâjk+1 j j=...k it followsthat has full row rank.  k1  k+1 Following the notation in [13] we call (3.12) the strangeness-free reformulation of the DAE (1.4). Obviously if at t = the consistency assumptions ( d dt ) i(âk 12x (k 1) (t)+ +Â12x(1) (t)+â2x(t) ˆf 2(t))= 1 (3.13)... ( d dt ) i(â k+1 x(t) ˆf k+1 (t))=...k ˆf k+2 (t)= hold then we can differentiate all but the first equation of system (3.12) to obtain an underlying ODE as in the following theorem. Theorem 3.7. Consider the DAE (1.4) and assume that the consistency condition (3.13) is satisfied. Then (1.4) has the same solution set as the underlying

16 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 717 ODE  k1  k  11  1  k 12  k  2  1k  k  k+1 where the first column  k1  k+1 has full row rank. x (k) x (k 1) x (1) x = The following corollary is a direct consequence of Theorem 3.7. ˆf 1 ˆf (1) 2 ˆf (k 1) kˆf (k) k+1 Corollary 3.8. Consider the initial value-problem (1.4) (1.5) and assume that the function f is sufficiently smooth. Then we have: x (k 1) i) Consistency conditions for f and the initial vector X = are given x () by system (3.13). ii) System (1.4) is uniquely solvable if and only if in addition the matrix is square.  k1  k+1 Motivated by the strangeness-free reformulation (3.12) of the DAE (1.4) we introduce the following definition. Definition 3.9. Consider the behavior matrix M=[A k...a ] C l(k+1)n associated with the DAE A k x (k) (t)+ +A x(t)+h(t)= where A i C ln i=k... and h [ ) C l. The matrix M is called strangeness-free if there exists a nonsingular matrix P

17 718 P. Ha and V. Mehrmann C ll such that where A k1 A k A 1 A k A 2 PM= A k+1... i) each block column contains exactly n columns and A k1 ii) the matrix A k 12 has full row rank. A k+1 Remark 3.1. One of the characteristics of the behavior approach [18] is that all the variables are treated equal. This is important for example if one wants to know which variables to choose as controls. The behavior approach however is not appropriate if for example the choice of controls is predetermined. In this case a uniform treatment for a general high-order system (with or without delay) is still not available see [14] and [15] for special cases of first and second order systems without delay. 4. Analysis and reformulation of DDAEs. This section discusses DDAEs with single delay of the form(1.2) and the initial value problem(1.2) (1.3). Analogous to Section 3 the behavior approach and the algebraic approach will be combined. Consider a behavior formulation of (1.2) as (4.1) N X =f with ẋ(t) N =[A 1 A A 1 ] and X (t)= x(t) f (t) =f(t). x(t τ) A first remarkable difference between DAEs and DDAEs is that for the DAE (1.4) of order k after applying the strangeness-free reformulation (Procedure 3.5) the resulting system is still a DAE of order at most k. However when applying a similar procedure for the DDAE (1.2) then the order of the system may increase as is illustrated in the following example. Example 4.1. Consider the system [ 1 ][ẋ(t) ẏ(t) ]=[1 1 ][x(t) y(t) ]+[ 1 ][x(t τ) y(t τ) ]+[f 1 ] t. f 2

18 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 719 In behavior form we have ẋ(t) N =[ 1 1 ẏ(t) 1 1 ] x(t) X =. y(t) x(t τ) y(t τ) Differentiating the second equation and inserting it into the first we get [ ][ẋ(t) ẏ(t) ]=[1 1 ][x(t) y(t) ]+[ 1 ][x(t τ) y(t τ) ]+[ 1 ][ẋ(t τ) ẏ(t τ) ]+[f 1+ f 2 ]. f 2 In behavior form we have N 1 X 1 =[ f 1+ f 2 f 2 ] with N 1 =[ ẋ(t) ẏ(t) x(t) ] X1 = y(t) x(t τ). y(t τ) ẋ(t τ) ẏ(t τ) Thus the size of the behavior matrix is increased. The second important difference between DAEs and DDAEs is the strangenessfree reformulation procedure. Let us illustrate this by considering the following example. Example 4.2. Consider the system (4.2) [ 1 ]ẋ(t)+[1 ]x(t)+[1 ]x(t τ)=[f(t) ] t. g(t) The associated non-delayed system is [ 1 ]ẋ(t)+[1 ]x(t)=[f(t) ] t. g(t) Using the strangeness-free reformulation in [13] for the non-delayed system we differentiate the first equation and insert it into the second equation to obtain [ ]ẋ(t)+[1 ]x(t)=[ f(t) g(t) f(t) ].

19 72 P. Ha and V. Mehrmann Clearly if g(t)+ f(t)= holds then we obtain a unique solution x(t)=f(t). Performing the same steps for system (4.2) we obtain that (4.3) [ ]ẋ(t)+[1 ]x(t)+[1 ]x(t τ)+[ 1 ]ẋ(t τ)=[ f(t) g(t) f(t) ]. The second equation of (4.3) not only gives the consistency condition but also the constraint and one obtains the system ẋ(t τ)+g(t) f(t)= t [τ] ẋ(t)+g(t+τ) f(t+τ)= t (4.4) [ 1 ]ẋ(t)+[1 ]x(t)+[1 ]x(t τ)=[ f(t) g(t+τ)+ f(t+τ) ]. Thus the step that passes from system (4.3) to (4.4) changes nothing but the inhomogeneity and we can proceed like this without ever terminating. This shows that DDAEs require a different reformulation procedure which terminates after a finite number of steps. Considering system (4.2) again we can proceed as follows. Replacing the first equation of (4.2) by its derivative gives [ 1 1 ]ẋ(t)+[ ]x(t)+[ ]x(t τ)+[1 ]ẋ(t τ)=[ f(t) g(t) ]. Subtracting the first equation from the second we get [ 1 ]ẋ(t)+[ ]x(t)+[ ]x(t τ)+[1 ]ẋ(t τ)=[ f(t) g(t) ]. g(t) Shifting the time in the first equation by τ we obtain [ 1 1 ]ẋ(t)+[ ]x(t)+[ ]x(t τ)=[ f(t+τ) g(t+τ) ] g(t) and subtracting the second equation from the first yields [ 1 ]ẋ(t)+[ ]x(t)+[ ]x(t τ)=[ f(t+τ) g(t+τ) g(t) ]. g(t) If the consistency condition in the first equation is satisfied then we have a unique solution x(t).

20 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 721 Motivated by Example 4.2 we propose a new procedure to treat the system (1.2) in the behavior form (4.1). The idea is to replace nontrivial scalar DDEs in system (1.2) by (appropriately chosen) derivatives. Since in this way the order of the system may be increased we study directly general DDAEs of the form (1.1). Setting N =[A k...a A κ... A 1 ]= [N + N ] x (k) (t) x (κ) (t τ) X + (t) = X (t τ) = x(t) x(t τ) we have the behavior form of (1.1) given by (4.5) [N + N ][ X +(t) X (t τ) ]=f(t). Set r = rank(n ) and d = l r and perform a column compression of N as in the following lemma. Lemma 4.3. Consider the DDAE (1.1) in its behavior form (4.5). Then there exists a nonsingular matrix P C ll such that by scaling system (4.5) with P from the left we obtain the system (4.6) [ F G H ][ X +(t) X (t τ) ]=[f 1(t) f 2 (t) ] r d where G has full row rank. Proof. First we determine a matrix P 2 C dl whose rowsspan the left nullspace of N i.e. P 2 N = and then we complement P 2 as P =[ P 1 P 2 ] to a nonsingular matrix. Then and G=P 1 N has full row rank. P[N + N ]=[ P 1N + P 1 N P 2 N + ]= [F G H ] Since in (4.6) G has full row rank we see that the behavior system (4.5) has r nontrivial scalar delay differential equations and d scalar differential equations. Since typically the matrix[ F ] is not strangeness-free then a first idea would be H carry out the strangeness-free formulation (Procedure 3.5) for the DAE (4.7) [ F H ]X +(t)=[ g 1(t) g 2 (t) ]

21 722 P. Ha and V. Mehrmann where g(t) =[ g 1(t) g 2 (t) ] = [G ]X (t τ)+[ f 1(t) f 2 (t) ]. However as pointed out in Example 4.2 this may not lead to a procedure that terminates in a finite number of steps. In order to overcome this difficulty we propose the following approach. Since the order of the DAE HX + (t)=f 2 (t) is at most k we replace the first equation of system (4.6) by its(k+1)-st derivative and obtain the system (4.8) [ F H ] ( d dt )k+1 X+ (t) +[ G X + (t) ]( d k+1x dt ) ( (t τ)= d dt )k+1 f 1 (t). f 2 (t) To guarantee that system (4.8) has the same solution set as (4.7) we must require that the following consistency condition holds at t = (4.9) ( d dt ) j(fx + (t)+gx (t τ) f 1 (t))= j=...k+1. Thus we have shown the following lemma. Lemma 4.4. Consider system (4.7) and assume that the consistency condition (4.9) is satisfied at t=. Then system (4.7) has the same solution set as the DDAE (4.8). Setting [ f 1 (t) f 2 (t) ] = [G ]( d dt ) k+1 we can apply Procedure 3.5 to the DAE [ F H ] ( d dt )k+1 X+ (t) X + (t) and thus immediately obtain the following lemma. ( X (t τ)+ d dt )k+1 f 1 (t) f 2 (t) =[ f 1 (t) f 2 (t) ] Lemma 4.5. Consider the DDAE (1.1) in its behavior form (4.5) and assume that the consistency condition (4.9) is satisfied at t=. Then system (1.1) has the same solution set as the DDAE (4.1) r s s d v v F 1 F2 G 1 f 1 (t) ( H d dt )k+1 X + (t) + G 2 X + (t) X (t τ)= f 2 (t) f 3 (t) f 4 (t)

22 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 723 where X x ( κ) (t τ) 1 F2 (t τ)= for some κ N and where the matrix[ F is x(t τ) H] strangeness-free and of full row rank. Since the second equation in system (4.1) is a DAE of the variable x(t τ) we can shift it to obtain a DAE for x(t). We summarize this and Lemmas in the following theorem. Theorem 4.6. Consider the DDAE (1.1) in its behavior form (4.5). Moreover assume that the consistency condition (4.9) is satisfied at t= and that (4.11) ( G2 X (t τ) f 2 (t)) t [τ] = holds. Then system (1.1) has the same solution set as the DDAE (4.12) where F 1 F2 G 1 H ( d dt )k+1 X + (t) + X + (t) G X + (t)+ 2 X (t τ)+ f 1 (t) r s = f 3 (t) d v f 2 (t+τ) s f 4 (t) v x ( κ) (t) X + (t) =. x(t) 1 F2 In (4.12) the matrix[ F is strangeness-free and of full row rank. H] Bypassingfromsystem(4.6)to(4.12)wehavereducedthenumberofscalardelay differential equations from r (in system (4.6)) to r s (in system (4.12)). However the number of system equations (number of rows) is still l. Definition 4.7. The step that passes system (4.6) to (4.12) is called a reformulation step. The natural numbers r d s v are called characteristic invariants of system (1.1) and of its behavior form (4.6). Setting k new =max{2k+1 κ} κ new = κ wecanbringsystem(4.1)intobehavior form and perform a new reformulation step. Since the number of nontrivial DDEs decreases every time that we perform a reformulation step this process terminates after finitely many steps.

23 724 P. Ha and V. Mehrmann We summarize the discussion above in the following procedure. Procedure 4.8. Input: A DDAE of the form (1.1). Begin Set and N =[A k...a A κ... A 1 ]= [N + N ] x (k) (t) x (κ) (t τ) X+(t) = X (t τ) = x(t) x(t τ) X (t)=[ X +(t) X (t τ) ] f (t)=f(t) and k =k κ =κ r =rank(n ) d =l r. Step 1. Determine a nonsingular matrix P C ll such that by scaling P from the left of the behavior system we obtain N i X i (t)=f i (t) (4.13) [ F G H ][ X +(t) X (t τ) ]=[f 1(t) f 2 (t) ] r i d i where G has full row rank. If[ F ] is strangeness-free and has full row rank then STOP H else proceed to Step 2. Step 2. Check the consistency conditions d j dt j(fx +(t)+gx (t τ) f 1 (t))= j=...k+1 at t=. If they are satisfied then transform the behavior system (4.13) into (4.14) [ F H ] ( d dt )k+1 X+ (t) +[ G X + (t) ]( d k+1x dt ) ( (t τ)= d dt )k+1 f 1 (t). f 2 (t) Step 3. Apply Procedure 3.5 to system (4.14) to obtain (4.15) F 1 F2 H G 1 ( d dt )k+1 X + (t) + G 2 X + (t) f 1 (t) X (t τ)= f 2 (t) f 3 (t) f 4 (t) r i s i s i d i v i v i

24 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 725 where x ( κ) (t τ) X (t τ)= x(t τ) 1 F2 for some κ N and the matrix[ F is strangeness-free and of full row rank. H] Step 4. Check the consistency condition ( G2 X (t τ) f 2 (t)) t [τ] =. If it is satisfied then shift the second equation of system (4.15) and permute the second and the third block rows to get (4.16) where F 1 F2 H ( d dt )k+1 X + (t) + X + (t) f 1 (t) r i s i = f 3 (t) d i v i f 2 (t+τ) s i f 4 (t) v i G 2 X + (t)+ G 1 X (t τ) x ( κ) (t) x ( κ) (t τ) X + (t) = X (t τ)=. x(t) x(t τ) Step 5. Reorganize system (4.16) in the form à k1 x (k1) (t)+ +Ãx(t)+à 1x(t τ)+ +à κ 1 x (κ1) (t τ)= f 1 (t) where Ãj C ln j= κ 1...k 1 à k1 /= à κ1 /= k 1 max{2k+1 κ} κ 1 κ. Set N i+1 =[à k1...ã à κ1... à 1 ] =[N+ i+1 N i+1 ] x + (k1) (t) x (t) = X i+1 (t τ) = x(t) (κ1) (t τ) x(t τ) X i+1 X i+1 (t) =[ Xi+1 + (t) X i+1 (t τ) ] fi+1 (t) = f 1 (t)

25 726 P. Ha and V. Mehrmann then from (4.16) we conclude that the number of nontrivial scalar delay differential equations of system is Set [N i+1 + N i+1 ][ Xi+1 X i+1 + (t) (t) (t τ)]=fi+1 rank(n i+1 )=rank( G 1 ) r i s i. r i+1 =rank(n i+1 ) r i s i d i+1 =l r i+1 d i +s i and repeat the process from Step 1. End Definition 4.9. Consider the DDAE (1.1) in its behavior form (4.6) and the sequence(r i d i s i v i ) i N of characteristic invariants generated by Procedure 4.8. Then we call the delay index of (1.1). ω=min{i N s i =} Theorem 4.1. Consider the DDAE (1.2) and let ω be the delay-index of (1.1). Moreover suppose that the consistency conditions (4.9) at t = and (4.11) of all reformulation steps 1...ω are satisfied. Then (1.1) has the same solution set as the following DDAE (4.17) N ω +X ω +(t)+n ω X ω (t)=f ω (t) with delay-index where (4.18) F N+= ω 1 ω F ω 2 H ω N = ω f ω = f1 ω f2 ω f3 ω r ω d ω v ω X ω =[x (kω) (t) T...x(t) T x (κω) (t τ) T...x(t τ) T ] T =[ Xω + X ω ]. In (4.18) the matrix[ Fω 1 F2 ω Hω] is strangeness-free and of full row rank. Since (1.2) is a special case of system (1.1) we can apply Theorem 4.1 to study (1.2). Due to Definition 3.9 the fact that matrix[ Fω 1 F2 ω Hω] is strangeness-free and

26 Analysis and Reformulation of Linear Delay Differential-Algebraic Equations 727 of full row rank implies that there exist a nonsingular matrix P C ll such that by scaling system (4.17) with P from the left we obtain (4.19)  kω1  kω  1 x (kω) (t)  kω  2 x (kω 1) (t)  kω+1... x(t)... ˆf 1 (t)... x (κω) (t τ) ˆf + 2 (t) =... x(t τ) ˆf kω+1(t) ˆf kω+2(t) and à kω =  kω1  kω 12  kω+1 has full row rank. Rewriting (4.19) block row-wise in behavior form as (4.2) N ω 1 N ω 2 N ω k ω+1 ˆf 1 (t) ˆf X ω 2 (t) = ˆf kω+1(t) ˆf kω+2(t) we get the following consistency conditions at t= (4.21) N ω 1 Xω f 1 (t)= d i dt i(nω 2 X ω f 2 (t))= 1 d i dt i(nω k ω+1x ω f kω+1(t))=...k ω +1 f kω+2(t)=. Therefore similar to Theorem 3.7 we obtain the following theorem which stresses that every DDAE of the form (1.2) contains an underlying high-order DDE. Theorem Consider the DDAE (1.2). Let ω be its delay-index and assume that (4.19) is the delay-index formulation of (1.2). Moreover assume that the

27 728 P. Ha and V. Mehrmann consistency conditions (4.9) at t= and (4.11) of all reformulation steps 1...ω are satisfied. Furthermore suppose that the consistency condition (4.21) is also satisfied. Then (1.2) has the same solution set as the DDE Ã kω x (kω) (t)+ +Ãx(t)+Ã 1x(t τ)+ +Ã κ ω k ω x (κω+kω) (t τ)+ f ω (t)= where Ãk ω has full row rank. Proof. The proof follows by differentiating the j-th equation of system (4.2) j 1 times for each 2 j k ω +1. Based on Theorem 4.11 we have the following solvability result. Corollary Consider the DDAE (1.2). Moreover assume that the function f is sufficiently smooth. i) The DDAE (1.2) is solvable if and only if the consistency conditions (4.9) at t= and (4.11) of all reformulation steps 1...ω are satisfied and also the consistency condition (4.21) is satisfied. ii) The initial value problem (1.2) (1.3) is uniquely solvable if and only if in addition the matrix Ãk ω is square. We illustrate our result by the following example. Example Consider the first order DDAE 1 1 ẋ(t)+ 1 x(t)+ 1 x(t τ)= 1 A 1 A A 1 f 1 f 2 f 3 f t. Clearly the matrix pencil λa 1 A is singular and hence it if we insert x [ τ] = φ to determine x [τ] from the DAE A 1 ẋ(t)+a x(t)= A 1 φ(t)+f(t) t (τ) we obtain an under-determined system and therefore x [τ] is not uniquely determined. However using Procedure 4.8 we finally arrive at where 1... g 1 (t) x(t)+ 1 ẍ(t)+ ẋ(t)= g 2 (t) 1 g 3 (t)... g 1 (t) f 1 (t+τ)... f 2 (t+2τ)+ f 3 (t+2τ) g 2 (t) = f 2 (t+τ) f 3 (t+τ). g 3 (t) f 3 (t)

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