Optimization and Controllability without Differentiability Assumptions

Let $\mathcal{X}$ and $\mathcal{Y}$ be real normed vector spaces, $K \subset \mathcal{X}$ convex and compact, $C \subset \mathcal{Y}$ a convex body, $\mathcal{U} \subset K$, and $(\phi ,\Phi ):K \to \mathbb{R}^m \times \mathcal{Y}$ a function that can be appropriately approximated by functions $(\phi _i ,\Phi _i )$ whose compositions with linear maps of small finite-dimensional simplices are $C^1 $. We derive sufficient conditions for $\phi $ to be (locally) controllable on $\mathcal{U}$ subject to the restriction $\Phi (u) \in C$, and obtain, as a corollary, corresponding necessary conditions for a related restricted minimum. These conditions are formulated in terms of directional derivate containers which are a type of set-valued “derivatives” of $(\phi ,\Phi )$, and they improve on and extend previously obtained results. They are used elsewhere to obtain new conditions for controllability and restricted minimum in nonsmooth optimal control problems defined by differential or functional-integral equatio...