Square nonlinear two-point boundary-value problems and endpoint residuals

Fix $a<b$, an integer $d\ge 1$, an open set $\mathrm{\Omega }\subseteq \mathbb{R}\times {\mathbb{R}}^d$ containing $[a,b]\times$ the states under consideration, and a vector field $f:\mathrm{\Omega }\to {\mathbb{R}}^d$. Let $D_B\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^d$ be open and let $B:D_B\to {\mathbb{R}}^d$. A square nonlinear two-point boundary-value problem (BVP) is

$y^{\mathrm{\prime }}(t)=f(t,y(t))(a\le t\le b),B(y(a),y(b))=0,$

where $y$ is a continuously differentiable curve whose graph lies in $\mathrm{\Omega }$, whose endpoint pair lies in $D_B$, and where the $d$ scalar components of $B$ give exactly $d$ boundary equations. A BVP solution is such a curve satisfying both the differential equation and endpoint equation. No existence or uniqueness is part of this definition.

If the IVP from initial state $s\in {\mathbb{R}}^d$ has a unique solution $\varphi (\cdot ;s)$ on $[a,b]$, the exact shooting residual is $R(s):=B(s,\varphi (b;s))$ whenever $(s,\varphi (b;s))\in D_B$. For a mesh $a=t_0<\cdots <t_N=b$ and segment initial states $s_0,\dots ,s_N$, the exact multiple-shooting residual has rows ${\varphi }_i(t_{i+1};s_i)-s_{i+1}$ for $0\le i<N$ and final row $B(s_0,s_N)$, where ${\varphi }_i$ denotes the unique segment IVP solution when it exists. These residuals are defined only on their admissible IVP and boundary domains.