Infimal Convolution and Optimal Time Control Problem I: Fréchet and Proximal Subdifferentials

We consider a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real-valued target function defined on a Banach space. If the target function is the indicator function of a closed set, this problem is a minimal time problem for a target set, studied previously in particular by Colombo, Goncharov and Mordukhovich. We investigate several properties of the Frechet and proximal subdifferentials for the infimum time function. Also explicit expressions of the above mentioned subdifferentials as well as various directional derivatives are obtained. We provide some examples to show the essentiality of assumptions of our theorems.