Finite-budget indirect shooting for an interior-control ODE optimality system

This procedure is restricted to a smooth, unconstrained-interior control branch. Let the smooth data be those of Finite-horizon ODE optimal-control transcription data, with no active path or endpoint constraints apart from the prescribed initial state. Any optional inequality constraints must remain strictly inactive throughout the branch; additional endpoint equalities or active endpoint inequalities require an augmented boundary residual and are outside this procedure. Let $H(t,x,p,u)=L(t,x,u)+p^{T}f(t,x,u)$. Supply a pointwise-defined twice continuously differentiable feedback $\psi (t,x,p)$ on an open tube such that $H_u(t,x,p,\psi (t,x,p))=0$ and $\psi (t,x,p)\in \mathrm{int}{U}_c$ throughout that tube; differentiability uses the convention of $C^k$ Euclidean maps and diffeomorphisms. Every successful IVP evaluation must keep its trajectory in this tube. If the evaluator cannot certify that condition for a queried trajectory, return inadmissible-interior-branch with no shooting result. Define $H_x^{\mathrm{f}\mathrm{i}\mathrm{x}}(t,x,p,u):=D_x[L(t,x,u)+p^{T}f(t,x,u)]$, with the derivative taken holding $u$ fixed, and only then substitute $u=\psi (t,x,p)$. Define $z=(x,p)\in {\mathbb{R}}^{2n}$ and $G(t,z)=\left(\begin{array}{c}f(t,x,\psi (t,x,p))\\ -H_x^{\mathrm{f}\mathrm{i}\mathrm{x}}(t,x,p,\psi (t,x,p))\end{array}\right),B(z_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}},z_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}})=\left(\begin{array}{c}{x}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}-x_0\\ p_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}-{\varphi }_x(x_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}})\end{array}\right),$ where $z_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}=(x_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}},p_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}})$ and $z_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}}=(x_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}},p_{\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}})$. The indirect shooting procedure takes a certified differentiable IVP evaluator for $z^{\mathrm{\prime }}=G(t,z)$, an initial trial $z_0$, residual tolerance $\tau \ge 0$, and correction cap $M\in {\mathbb{N}}_0$, and calls Finite-budget single shooting for a nonlinear two-point BVP on the square BVP $(G,B)$. It returns that algorithm's trajectory and status without changing the root-tolerance meaning. In particular, root-tolerance-met certifies only the computed boundary residual for the numerical IVP map; it is not an exact BVP, optimality, global-optimality, or discretization-error certificate. If the control is on a boundary or no such smooth feedback is supplied, this procedure is inapplicable.