The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 171
Theorem 9.1. Let Ay ′ + By = f be a linear constant coefficient DAE with regular
pencil λA + B. Then there exists nonsingular matrices P, Q, such that
where u = Q −1 y,
P AQ =
P AQu ′ + P BQu = P f (9.50)
( ) I 0
; P BQ =
0 N
( ) C 0
0 I
(9.51)
and N is a matrix of nilpotency k, for some integer k ≥ 0. If N = 0 define k = 1; if
A is nonsignular, P AQ = I and k = 0. If det(λA + B) is constant then P AQ = n,
P BQ = I.
Definition 9.6. The index of the pencil λA + B is the degree of nilpotency in
equation 9.51.
Theorem 9.2. Let Ay ′ + By = f be a linear constant coefficient DAE with regular
pencil λA + B. Then the index of the pencil equals the index of the DAE.
This theorem says that there exist nonsingular matrices P , Q, such that we can
write
( ) ( ) ( ( ) ( I 0 u
′ C 0 u g
0 N v ′ +
=
(9.52)
0 I)
v h)
where
Qy =
( ( u g
, P f =
v)
h)
(9.53)
Then
u ′ + Cu = g (9.54)
Nv ′ + v = h (9.55)
and N is k-nilpotent, where k is the index of the DAE. Equation 9.54 has a unique
solution that is fully determined by its initial conditions (this is the existence theorem
for ordinary differential equations). Equation 9.55 also has a unique solution,
but does not require initial conditions, because the matrix N is nilpotent (N k =0).
To see this let D be the derivative operator; then 9.55 becomes
But since
NDv + v = h (9.56)
(ND + I)v = h (9.57)
v = (ND + I) −1 h (9.58)
(I + ND) −1 = I − ND + N 2 D 2 − N 3 D 3 + N 4 D 4 + · · · + N k−1 D k−1 (9.59)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge