On Uniform Metric Regularity

Radek Cibulka

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 covers a large territory in control and optimization, such as differential variational inequalities, control systems with constraints, as well as necessary optimality conditions in optimal control. We propose two inexact path-following methods for DGEs having the order of the grid error $O(h)$ and $O(h^2)$, respectively. We provide numerical experiments, comparing the schemes derived, for simple problems arising in physics.
Further, we study metric regularity of mappings associated with a particular case of the DGE arising in control theory by focusing on the interplay between the pointwise versions of these properties and their infinite-dimensional counterparts.