Certified differentiable IVP approximation interface for shooting

Fix an interval $[a,b]$ and an open set $U\subseteq {\mathbb{R}}^d$ on which each exact IVP flow $\varphi (t;s)$ exists for all $t\in [a,b]$. A certified differentiable IVP evaluator on $U$ declares a $C^1$ approximate endpoint map (here $C^1$ means total differentiability with continuous derivative entries) $\tilde{\mathrm{\Phi }}:U\to {\mathbb{R}}^d$ and a stated number ${\delta }_{\mathrm{I}\mathrm{V}\mathrm{P}}\ge 0$. A call at $s\in U$ either returns failure or supplies a continuous approximate trajectory $\tilde{\varphi }(\cdot ;s)$, the endpoint $\tilde{\mathrm{\Phi }}(s)$ and the exact derivative $D\tilde{\mathrm{\Phi }}(s)$ for that numerical map, satisfying

${\sup }_{t\in [a,b]}\mathrm{\parallel }\tilde{\varphi }(t;s)-\varphi (t;s){\mathrm{\parallel }}_2\le {\delta }_{\mathrm{I}\mathrm{V}\mathrm{P}},\tilde{\mathrm{\Phi }}(s)=\tilde{\varphi }(b;s)\mathrm{.}$

The same ${\delta }_{\mathrm{I}\mathrm{V}\mathrm{P}}$ bounds every successful call, uniformly in its initial state; no approximate trajectory or error bound is required at a failed call. The declared endpoint map remains $C^1$ on $U$ even when some calls fail. The returned derivative is not asserted to equal the derivative of the exact flow. This is an interface contract for a supplied method; it does not assert that any particular integrator satisfies the contract.