# Implicit function theorem and consistency of a semi-explicit DAE

This may be a trivial question, but here goes:

Suppose a semi-explicit differential-algebraic equation (DAE) system is defined as follows: \begin{align} &\dot x = f(x,z,\theta),\qquad x(0) = x_{0}\\ &g(x,z,\theta) = 0 \end{align} where $x \in \mathbb{R}^{n}$, $z \in \mathbb{R}^{m}$ and $\theta \in \mathbb{R}^{p}$ (a constant time-invariant vector). Suppose also that $g(\cdot)$ is at least once differentiable, and the Jacobian of $g(\cdot)$ with respect to $z$, which we call $g_{z}$ is nonsingular everywhere.

The vector $z$ is dependent, in that it is uniquely determined by $x$ and $\theta$. Its value is computed by solving the second implicit equation $g(\cdot)=0$.

Now, I want to perturb the value of $z$ by some constant quantity $d_{z} \in \mathbb{R}^{m}$, as follows: \begin{align} &\dot x = f(x,z+d_{z},\theta)\\ &g(x,z+d_{z},\theta) = 0 \end{align}

Question: Can I do so for any arbitrary value of $d_{z}$ without making the system of equations inconsistent?

My partial reasoning: I'm guessing I can, because the implicit function theorem tells me given a nonsingular Jacobian $g_{z}$, there is a neighborhood around some point ($\bar x$, $\bar z$, $\bar \theta$) which is the open-set denoted $X$, there exists a function $h$ such that: $$g(x,h(x,\theta),\theta) = g(\bar x,\bar z, \bar \theta), \qquad (x,\theta) \subset X$$ Therefore if we take ($\bar x$, $\bar z$, $\bar \theta$) to be the solution point, we can write the explicit function: $$z = h(x,\theta), \qquad (x,\theta) \subset X$$ Therefore, in the neighborhood defined by $X$, the system can be written: \begin{align} &\dot x = f(x,h(x,\theta),\theta)\\ &z = h(x,\theta) \end{align} And I have a feeling that the perturbed system should be consistent, but I don't know how show this rigorously: \begin{align} &\dot x = f(x,h(x,\theta) + d_{z},\theta)\\ &z = h(x,\theta) + d_{z} \end{align}

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I have this special case in mind: if I want to find the intersection of two or three spheres, it should be easy to make a perturbation that ensures the spheres are "far apart"... – J. M. Nov 30 '11 at 2:46
@J.M. You're right; even for the simple case of the intersection of two circles, one can conceive a perturbation large enough as to give rise to two circles that do not intersect. I'm just wondering, is there a way to specify some conditions on $d_{z}$ or some such to prevent this from happening? – Gilead Nov 30 '11 at 3:17
I'm doubtful that anything useful can be done, unless there's something special about both $f$ and $g$... – J. M. Nov 30 '11 at 3:22
Suppose I change the question: instead of two systems of equations, I'd like to consider an index-1 semi-explicit DAE system (which is my real system of interest). – Gilead Nov 30 '11 at 3:53