f i Section 14-23 S 989 /

RENE LAMOUR

Floquet-Theory for Differential-Algebraic Equations (DAE)

f i e Theorem of Lyapunov says that, in case of a periodic ODE, there exists a transformation to an ODE with

fie generalization of ODE-Floque't theory to index 1 and index 2 DAEs is given using an adapted transformation concept.

matrices. This fact is very useful in stability investigations of nonlinear ODES, too.

1. Equivalent Transformations of Linear DAEs The presented concept works for index-1 and index-2 DAEs, too. But we restrict the representation here to the index 1 case only. Consider linear homogeneous DAEs

A ( t ) z ' ( t ) + B ( t ) x ( t ) = 0 ,where A , B E C(R, L(IR")). (1)

Suppose the nullspace N ( t ) := kerA(t) to be smooth, i.e. to be spanned by continuously differentiable basis functions. Obviously, all solutions of (1) belong to the subspace s ( t ) := { z E IRm : B( t ) z E im A ( t ) } c Rm. Assume that (1) is index-1-tractable, i.e., S ( t ) n N ( t ) = (0). Then, exactly one solution passes through each point of S ( t ) at time t .

Using any C' projector function Q ( t ) onto N ( t ) and P ( t ) := I - Q ( t ) , the solutions of the DAE (1) belongs to the function space Ch := {x E C : Px E C ' } . It is possible to show that starting with index 1 systems in Kronecker normal form and using C' transformations F we arrive at DAEs with continuously differentiable canonical projectors, which is not true in general. As a consequence, looking for a Kronecker normal form for continuous coefficient DAEs, we should apply a larger class of transformations. The class Ch seems to be the proper one also for the transformations F .

In particular, A ( t ) then has constant rank.

L e m m a The transformation of the unknown function x ( t ) = F(t) i t ( t ) with F E Ch, F nonsingular, trans- B = B F + A ( ( P F ) ' - P ' F ) =: B F + AF' are forms the DAE (1) into A(t)i t ' ( t) + B( t ) i t ( t ) = 0 , where A = AF,

continuous and A has a smooth nullspace again. A

transformation of variables x = F(t)i t with nonsingular F E Ch results further in S ( t ) := { z E JR" : B ( t ) z E im A(t)} = F - ' ( t ) S ( t ) . Because of N ( t ) n S ( t ) = F - ' ( t ) ( N ( t ) n S ( t ) ) the transformed DAE is index 1 tractable if and only if (1) is so.

D e f i n i t i o n 1. Two DAEs with the structure of (1 ) are said to be kinematically equzvalent if there are B = E ( B F + A F ' ) , and if sup IF(t)l < m, nonsingular matrix functions F E Ch, E E C with A = EAF,

tEIR sup IF(t)-l \ < m. t€R

2. Linear DAEs with periodic coefficients Consider (1) with periodic coefficients A ( t ) = A( t + T ) , B ( t ) = B ( t + T ) for all t E R. We make use of the natural splitting Rm = N ( t ) @ S ( t ) for index-1-tractable DAEs. Note that N ( t ) and S ( t ) are T-periodic since the coefficients A ( t ) and B ( t ) are so. N ( t ) is supposed to be smooth. We span it by T-periodic C'-functions N ( t ) = span{n,+l(t), . . . , n m ( t ) } , r = rank A ( t ) . S ( t ) may be only continuous. Let S ( t ) be spanned by T-periodic C-functions S ( t ) = span{sl(t), . . . , s,.(t)}. In what follows we choose a projector P ( t ) along N ( t ) so that P is not only smooth but also periodic.

Since Pea,, acts onto S along N, we have the representation PCaI,(t) = V ( t ) ( I ) V - ' ( t ) , where V ( t ) :=

[ s l ( t ) , . . . , s,.(t),n,.+l ( t ) , . . . , nm( t ) ] E L(IRm). Aiming to construct a special transformation we choose the projector p such that P(0) = Pcall(0). The fundamental matrix is given by

X ( t ) = pcan(t)U(t)P(O) = pcaii(t)U(t)Pcan(O)

s 990 ZAMM . Z. Angew. Math. Mech. 78 (1998) S3 - = V ( t ) ( I ) v - ' ( t ) U ( t ) V ( O ) ( I ) v-'(o) =: V ( t ) ( z(t) ) v-'(o), (2)

where Z E C(R, L(RP)), Z(0) = I , and the monodromy matrix

Since rank X ( t ) = r is constant, Z ( t ) E L(R') is nonsingular for all t E R. From linear algebra it is known that every nonsingular matrix C E L(R') can be represented in the form C = ew with W E L(C') and C2 = ew with W E L(IR7). Now, let Z ( T ) = eTWo, WO E L(C') and Z ( 2 T ) = Z ( T ) 2 = e Z T W o t wo E L(IRT), respectively. Here we concluded Z ( 2 T ) = Z ( T ) 2 from the corresponding property of X and the relation V ( 2 T ) = V ( T ) = V ( 0 ) . Now we introduce the special transformation

From (4) we see that this transformation is nonsingular. We remark that the transformation (4) may be not smooth since the same is true for S ( t ) and, hence, V ( t ) and X ( t ) , too. This generalizes the Theorem of FloquBt.

T h e o r e m 1. T h e fundamental matrix X ( t ) of the D A E can be wri t ten in the f o r m

X ( t ) = F K ( t ) ( etWo ) FK(O)-'

where FK E C&(IR., L(Cm)) is nonsingular and T-periodic.

Def in i t i on 2. T w o linear, homogeneous, T-periodic D A E s are said to be (periodically) equivalent iff the and B = E ( B F + AF' ) , where F E C&, E E C are T-periodic and nonsingular matrkc relation A = EAF

functions, is true for their coeficients.

Periodic equivalence means kinematic equivalence by periodic transformations. The following Theorem generalizes the result of Lyapunov.

T h e o r e m 2.

(i) If two linear homogeneous T-periodic D A E s are (periodically) equivalent t hen their monodromy matrices are similar and, hence, their characteristic multipliers coincide. (ii) If the monodromy matrices of two linear T-periodic D A E s are similar then the D A E s are (periodically) equivalent. (iii) T h e D A E (1) wi th periodic coeficients is (periodically) equivalent t o a T-periodic complex (2T-periodic real) linear sy s t em an Kronecker normal f o r m with constant coeficients.

These two Theorems are one essential part for stability investigations for nonlinear periodic DAEs ( see [3],[4])

Acknowledgements

The results presented an this paper are based on ajoant work with R. Marz and R. Wankler

References

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2 R.. LAMOUR, R. MARz, R.M.M. MATTHEIJ, On the stability behaviour of systems obtained by index-reduction,

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4 R.. LAMOUR, R. MARz, R. WINKLER, Floquet-theory for index-2 differential-algebraic equations, in preparation 5 L.S. PONTRYAGIN, Gewohnliche Differentialgleichungen,Dt. Verl. d. Wiss., Berlin, 1965 6 c. TISCHENDORF, On the stability of solutions of autonomous index-1 tractable and quasilinear index-2 tractable DAES,

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Circuits Systems Signal Process., 13, 2-3,pp. 139-164, 1994

Address: Dr. R.en6 Lamour, Institut for Mathematics, Humboldt-University of Berlin, D-10099 BERLIN