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

170 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS
Definition 9.2. The function φ(t) is a solution of 9.1 on an interval I if
F(t, φ(t), φ ′ (t)) = 0 (9.43)
for all t ∈ I. We call the DAE solvable on I if a solution (or family of solutions
that depend on a parameter) exists on I.
While this construction described earlier suggests that one can solve a DAE by
converting it to an ODE, this is not always the best method, since differentiation can
be expensive and knowing how to differentiate to get the desired result is a difficult
problem. However, the concepts of solvability and the definition of the Index of the
DAE are closely related.
9.2 Linear Differential Algebraic Equations with Constant
Coeffcients
Our focus will be primarily on linear equations of the form
A(t)y ′ + B(t)y = f(t) (9.44)
Definition 9.3. A linear system is said to be in standard canonical form if it
can be partitioned into the form
( ) ( ) ( ( )
I 0 x
′ C 0 x
0 N y ′ +
=
0 I)
y
( ) f(t)
g(t)
(9.45)
where N is a strictly 3 triangular matrix. For linear time-varying systems, both N
and C depend on t. Expanding,
x ′ + Cx = f(t) (9.46)
Ny ′ + y = g(t) (9.47)
We begin with the linear constant coefficient DAE (9.37).
Definition 9.4 (Matrix Pencil). Let A, B be square matrices and λ ∈ C. Then
the matrix pencil of A and B is given by
P is said to be a regular pencil if det λA + B ≠ 0.
P = λA + B (9.48)
Definition 9.5 (Nilpotent Matrix). A matrix N is said have a Nilpotency k
if k is the smallest integer such that
N k = 0 (9.49)
3 A strictly triangular matrix is a triangular matrix with all zeros on the diagonal.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007