Necessary and sufficient condition for the principle of optimality in deterministic MSOPs

Consider sets $X_t subseteq\mathbb{R}^{n\times T}$ for $0\leq t\leq T$, a control set $U\subseteq\mathbb{R}^m$, and cost functions

$$J_t:U^{T-t}\times\prod_{s=t}^{T}X_s\to\mathbb{R}.$$

Let $\mathcal{F}$ denote the set of pairs $(f,x_0)$ such that $x_0\in X_0$ and the associated multistage optimal control problems have a unique solution when initialized at $(x_0,0)$. Principle-of-optimality conjecture. For every $(f,x_0)\in\mathcal{F}$, the family of multistage optimal control problems associated with $\{J_t,f,\{X_t\}_{t\leq s\leq T},U,T\}_{t=0}^{T}$ satisfies the Principle of Optimality at $x_0$ if and only if $J_t$ is monotonically backward separable. The conjecture proposes a necessary and sufficient characterization of the Principle of Optimality for deterministic multistage optimal control problems with unique solutions; the preceding result establishes necessity under the stated assumptions, while sufficiency is the remaining direction.