Finite parameterized state-step graphs and objectives

Let $\mathrm{\Theta }\subseteq {\mathbb{R}}^m$ be open, let $N\ge 1$, and for $0\le n\le N$ let $U_n\subseteq {\mathbb{R}}^{d_n}$ be open. A finite parameterized state-step graph consists of a $C^1$ initial map $x_0:\mathrm{\Theta }\to U_0$ and $C^1$ step maps ${\mathrm{\Psi }}_n:U_n\times \mathrm{\Theta }\to U_{n+1}$ for $0\le n<N$, such that the recursion $x_{n+1}(\theta )={\mathrm{\Psi }}_n(x_n(\theta ),\theta )$ is defined for every $\theta$ in the chosen parameter neighborhood. A $C^1$ running objective consists of maps ${\mathrm{\ell }}_n:U_n\times \mathrm{\Theta }\to \mathbb{R}$ and a terminal objective $\varphi :U_N\times \mathrm{\Theta }\to \mathbb{R}$, all $C^1$. Its reduced objective is $J(\theta )=\varphi (x_N(\theta ),\theta )+{\sum }_{n=0}^{N-1}{\mathrm{\ell }}_n(x_n(\theta ),\theta )\mathrm{.}$ The maps and objective define one finite real-arithmetic graph. A gradient of this graph is the derivative of this finite composition; it is not the derivative of an exact implicit solution unless the step map itself is that solution map.