Semiconcavity of the value function for a class of differential inclusions

We provide intrinsic sufficient conditions on a multifunction F and endpoint data phi so that the value function associated to the Mayer problem is semiconcave

Proximal Analysis and the Minimal Time Function of a Class of Semilinear Control Systems

The minimal time function of a class of semilinear control systems is considered in Banach spaces, with the target set being a closed ball. It is shown that the minimal time functions of the Yosida approximation equations converge to the minimal time function of the semilinear control system. Complete characterization is established for the subdifferential of the minimal time function satisfying the Hamilton–Jacobi–Bellman equation. These results extend the theory of finite dimensional linear control systems to infinite dimensional semilinear control systems

The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space

The minimal time function with constant dynamics is studied in the context of a Hilbert space. A general formula for the subgradient is proven, and assumptions are identified in which the minimal time function is lower $C^2$

Variational analysis for a class of minimal time functions in Hilbert spaces

This paper considers the parameterized infinite dimensional optimization problem $$\text{minimize}\quad\bigl\{t\geq 0:\;S\cap\{x+tF\}\not= \emptyset\bigr\},$$ where $S$ is a nonempty closed subset of a Hilbert space $H$ and $F\subseteq H$ is closed convex satisfying $0\in\iint F$. The optimal value \tsf(x) depends on the parameter $x\in H$, and the (possibly empty) set $S\cap (x+T(x)F)$ of optimal solutions is the ``$F$-projection'' of $x$ into $S$. We first compute proximal and Fr\'echet subgradients of \tsf(\cdot) in terms of normal vectors to level sets, and secondly, in terms of the $F$-projection. Sufficient conditions are also obtained for the differentiability and semiconvexity of \tsf(\cdot), results which extend the known case when $F$ is the unit ball

Some new regularity properties for the minimal time function

A minimal time problem with linear dynamics and convex target is considered. It is shown, essentially, that the epigraph of the minimal time function $T(\cdot)$ is $\varphi$-convex (i.e., it satisfies a kind of exterior sphere condition with locally uniform radius), provided $T(\cdot)$ is continuous. Several regularity properties are derived from results in [G. Colombo and A. Marigonda, Calc. Var. Partial Differential Equations, 25 (2005), pp. 1-31], including twice a.e. differentiability of $T(\cdot)$ and local estimates on the total variation of $DT$