Global Directional Controllability
Jack Warga · SIAM Journal on Control and Optimization · 1989
This paper derives sufficient conditions for the image of a function $\varphi _1 :Q \to \mathbb{R}'''$ to cover an open segment originating at $\varphi _1 (\bar q)$ and with direction w (or even a neighborhood of such an interval) subject to restrictions of the form $\varphi _2 (q) \in C$ or $\varphi _2 (q) + G \subset C$. Here $\varphi _2 :Q \to \mathcal{Z}$, Q is an arbitrary set, C a convex subset of the topological vector space $\mathcal{Z}$, and G a neighborhood of 0 in $\mathcal{Z}$. Also derived are similar directional controllability conditions subject to the additional restriction $q \in \mathcal{U} \subset Q$ for an “abundant” subset $\mathcal{U}$ of Q. These conditions are applicable to unilateral problems of control theory and of (infinite-dimensional) mathematical programming. These results are global and apply to problems defined by functions whose restrictions to certain finite-dimensional sets are differentiable (but not necessarily $C^1 $) or are locally uniform limits of ifferentiable functions.