Directional convexity and finite optimality conditions
Given a set of directions \(\Lambda \subseteq {\mathbb{R}}^ d\), the author calls a set \(R\subseteq {\mathbb{R}}^ d\Lambda\)-convex whenever \[ r_ 0,r_ 1\in R,\quad r_ 1-r_ 0\in \Lambda,\quad \lambda \in [0,1]\quad \Rightarrow \quad (1-\lambda)r_ 0+\lambda r_ 1\in R. \] He then shows that when the reachable set R(T) of an autonomous nonlinear control system is \(\Lambda\)-convex, the transversality condition which must be satisfied by a boundary trajectory can be strengthened somewhat. The main part of the paper presents a set of sufficient conditions for the \(\Lambda\)-convexity of R(T), first in the case where the dynamics are \(\dot x=G(x)u\), and then in the fully nonlinear setting. The key hypothesis is a certain uniqueness property for solutions to the system of Pontryagin's equations; \(\Lambda\) consists of all vectors w for which every unit vector \(p(T)\in w^{\perp}\) generates a system with the uniqueness property.
scientific article; zbMATH DE number 4019848
scientific article