Semi-explicit index-one DAEs and consistent initial data

Let $d,m\ge 1$, let $\mathrm{\Omega }\subseteq \mathbb{R}\times {\mathbb{R}}^d\times {\mathbb{R}}^m$ be open, and let $f:\mathrm{\Omega }\to {\mathbb{R}}^d$ and $g:\mathrm{\Omega }\to {\mathbb{R}}^m$. A semi-explicit differential-algebraic equation (DAE) on $\mathrm{\Omega }$ is

$$x^{\mathrm{\prime }}(t)=f(t,x(t),z(t)),0=g(t,x(t),z(t))\mathrm{.}$$

The $x$-coordinates are the differential variables because their derivatives occur in the first equation. The $z$-coordinates are the algebraic variables because their derivatives do not occur. A solution on an interval $I$ is a pair of functions $x:I\to {\mathbb{R}}^d$, $z:I\to {\mathbb{R}}^m$ whose graph lies in $\mathrm{\Omega }$, with $x$ differentiable, satisfying both displayed equations at every $t\in I$. An initial pair $(x_0,z_0)$ at $t_0$ is algebraically consistent when $g(t_0,x_0,z_0)=0$. On the regular index-one branch below this is the algebraic initial condition used for reduction. For higher-index systems, algebraic consistency is only necessary: differentiated constraints may impose additional initial conditions.

At a point with $g(t_{\ast },x_{\ast },z_{\ast })=0$, the presentation is regular index one when $f,g$ are $C^1$ near the point and the square Jacobian $g_z(t_{\ast },x_{\ast },z_{\ast })=D_{z}g(t_{\ast },x_{\ast },z_{\ast })$ is invertible. The A regular index-one DAE has a local reduced ODE branch ↗ then supplies a unique local branch $z=\psi (t,x)$ through that point. Only that selected branch is used in local numerical claims. Higher-index systems and points with singular $g_z$ are outside this index-one interface.