On uniform regularity and strong regularity

We investigate uniform versions of (metric) regularity and strong (metric) regularity on compact subsets of Banach spaces, in particular, along continuous paths. These two properties turn out to play a key role in analyzing path-following schemes for tracking a solution trajectory of a parametric generalized equation or, more generally, of a differential generalized equation (DGE). The latter model allows us to describe in a unified way several problems in control and optimization such as differential variational inequalities and control systems with state constraints. We study two inexact path-following methods for DGEs having the order of the grid error $O\left(h\right)$ and $O\left(h^2\right)$ , respectively. We provide numerical experiments, comparing the schemes derived, for simple problems arising in physics. Finally, we study metric regularity of mappings associated with a particular case of the DGE arising in control theory. We establish the relationship between the pointwise version of this property and its counterpart in function spaces.