Local Newton convergence for a simple nonlinear shooting root

Here $C^2$ means twice continuously differentiable. Let the exact shooting residual $R(s)=B(s,\varphi (b;s))$ be $C^2$ on an open neighborhood of a root $s^{\ast }$, with $R(s^{\ast })=0$, $DR(s^{\ast })$ invertible, and $DR$ Lipschitz on a closed ball about $s^{\ast }$. Assume there is an open neighborhood $U$ of $s^{\ast }$ on which the exact IVP flow is an evaluator satisfying Certified differentiable IVP approximation interface for shooting with $\tilde{\varphi }(\cdot ;s)=\varphi (\cdot ;s)$, $\tilde{\mathrm{\Phi }}(s)=\varphi (b;s)$, and ${\delta }_{\mathrm{I}\mathrm{V}\mathrm{P}}=0$. Also assume $B$ is $C^1$ on an open neighborhood of $\{(s,\varphi (b;s)):s\in U\}$, as required by Finite-budget single shooting for a nonlinear two-point BVP. Then there are $\rho >0$ and $C>0$ such that, for every initial trial $s_0\in {\bar{B}}_2(s^{\ast },\rho )$, every attempted exact-Jacobian Newton update is defined, remains in this ball, and satisfies

$\mathrm{\parallel }{s}_{k+1}-s^{\ast }{\mathrm{\parallel }}_2\le C\mathrm{\parallel }{s}_k-s^{\ast }{\mathrm{\parallel }}_2^2\mathrm{.}$

A finite budget may stop before the residual tolerance is met. No arbitrary-start or global-convergence guarantee follows.