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

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 175
where B n,n−1 B n−1,n−2 · · · B 21 is nonsingular
Example 9.2. Write the Hessenberg forms of size 2 and 3 in block form.
Solution. The size 2 form:
The size 3 form:
x ′ + B 11 x + B 12 y = f(t) (9.80)
B 21 x = g(t) (9.81)
x ′ + B 11 x + B 12 y + B 13 z = f(t) (9.82)
y ′ + B 21 x + B 22 y = g(t) (9.83)
B 32 y = h(t) (9.84)
Theorem 9.10. A linear differential algebraic equation in Hessenberg form of
size n is solvable and has local index n.
For the general, nonlinear DAE
F(t, y, y ′ ) = 0 (9.85)
we say the DAE is in Hessenberg form of size n if it can be written explicitly as
y ′ 1 = F 1 (t, y 1 , y 2 , . . . , y n ) (9.86)
y ′ 2 = F 2 (t, y 1 , y 2 , . . . , y n−1 ) (9.87)
. (9.88)
y ′ i = F i (t, y i−1 , y i , . . . , y n−1 , t) (9.89)
. (9.90)
0 = 0F n (t, y n−1 ) (9.91)
and the product
is nonsingular.
∂F n
∂x n−1
∂F n−1
∂x n−2
· · · ∂F 2
∂x 1
∂F 1
∂x n
(9.92)
The Hessenberg form of size 1 is defined as
y ′ = f(t, y, x) (9.93)
0 = g(t, y, x) (9.94)
where the Jacobian ∂g/∂x is nonsingular. In principle, one could solve the constraint
for x 4 and reduce the DAE to an ordinary differential equation in y. However, it
will turn out that uniquenss of the solution is not guaranteed.
4 This is a consequence of the implicit function theorem and depends on the nonsingularity of
the Jacobian.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge