6 The Index of a DAE
One of the most important characteristics of a DAE is its index. Informally, the index measures how many times the algebraic constraints must be differentiated with respect to time in order to rewrite the DAE as an explicit ODE. Several index notions exist in the literature—such as the tractability index and perturbation index—but we focus primarily on the differential index, which is most directly connected to the process of index reduction and integration.
6.1 Differential Index Definition
Throughout this chapter, an index statement is local: it is valid only where the Jacobians required to solve the differentiated equations have full rank. This regularity condition is essential both for the definition and for the examples below.
A DAE has differential index q if q is the smallest integer such that repeated differentiation yields a system where \bm{x}'(t) can be expressed explicitly in terms of \bm{x}(t) and t:
\begin{aligned} \dfrac{\mathrm{d}}{\mathrm{d}t} \bm{f}(\bm{x}', \bm{x}, t) &= \bm{0}, \\ \dfrac{\mathrm{d}^2}{\mathrm{d}t^2} \bm{f}(\bm{x}', \bm{x}, t) &= \bm{0}, \\ &\vdots \\ \dfrac{\mathrm{d}^q}{\mathrm{d}t^q} \bm{f}(\bm{x}', \bm{x}, t) &= \bm{0}. \end{aligned}
Definition 6.1 (Differential Index) A DAE system
\bm{f}(\bm{x}'(t), \bm{x}(t), t) = \bm{0}
has differential index q if q is the minimum number of time differentiations needed to obtain an explicit expression
\bm{x}'(t) = \bm{g}(\bm{x}(t), t)
for \bm{x}'(t).
6.2 Example 1
Index-2 DAE
Consider the system:
x(t) = \sin(t), \qquad x'(t) + y(t) = 0. \tag{6.1}
The first equation fixes x(t) and the second fixes y(t), but neither equation directly provides an ODE for the complete state (x,y). First, differentiate the constraint x(t)-\sin(t)=0 once:
x'(t)=\cos(t).
Together with x'(t)+y(t)=0, this gives y(t)=-\cos(t), but it does not yet provide an equation for y'(t).
Differentiate the constraint a second time and the differential equation once:
x''(t)=-\sin(t), \qquad x''(t)+y'(t)=0 \quad\Rightarrow\quad y'(t)=-x''(t)=\sin(t).
Since two differentiations were needed to convert to an explicit ODE system, the differential index is q = 2.
6.3 Example 2
Time-Varying Coefficient DAE
Consider the time-dependent system from [1]:
\bm{E}(t) \begin{pmatrix} x'(t) \\ y'(t) \end{pmatrix} + \bm{A}(t) \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = \bm{c}(t),
where:
\bm{E}(t) = \begin{pmatrix} 1 & t \\ 0 & 0 \end{pmatrix}, \quad \bm{A}(t) = \begin{pmatrix} 0 & 0 \\ 1 & t \end{pmatrix}, \quad \bm{c}(t) = \begin{pmatrix} c_1(t) \\ c_2(t) \end{pmatrix}.
Step 1: System Equations
From the matrix form:
\left\{ \begin{aligned} x'(t) + t y'(t) &= c_1(t), \\ x(t) + t y(t) &= c_2(t). \end{aligned} \right.
Differentiate the second equation:
x'(t) + t y'(t) + y(t) = c_2'(t).
Substitute from the first equation:
c_1(t) + y(t) = c_2'(t) \Rightarrow y(t) = c_2'(t) - c_1(t).
Step 2: Derive ODEs
Differentiate y(t):
y'(t) = c_2''(t) - c_1'(t).
Substitute back into the first equation:
x'(t) = c_1(t) + t (c_1'(t) - c_2''(t)).
Thus, the system becomes:
\left\{ \begin{aligned} x'(t) &= c_1(t) + t (c_1'(t) - c_2''(t)), \\ y'(t) &= c_2''(t) - c_1'(t). \end{aligned} \right.
Two differentiations were needed: hence, the differential index is 2.
6.4 Example 3
Index-1 DAE
Consider the system:
\begin{aligned} x'(t) &= y(t), \\ 0 &= x(t)^2 + y(t)^2 - 1. \end{aligned}
Differentiate the algebraic constraint:
2x(t)x'(t) + 2y(t)y'(t) = 0.
Substitute x'(t) = y(t):
2x(t)y(t) + 2y(t)y'(t) = 0 \Rightarrow y'(t) = -x(t),
provided y(t)\neq 0. Thus the DAE is index 1 locally on the regular part of the constraint manifold. At points where y=0, the Jacobian of the algebraic constraint with respect to the algebraic variable is singular, so this example does not satisfy the regularity hypothesis there.
Now we have an explicit ODE system:
x'(t) = y(t), \qquad y'(t) = -x(t).
Only one differentiation was needed: the system has index 1.
Summary
- The differential index measures how far a DAE is from being an ODE.
- Higher-index systems are more challenging to analyze and simulate.
- Index-1 DAEs are typically well-posed and numerically tractable.
- Index-2 or higher systems often require index reduction techniques before numerical integration.