Abstract
ABSTRACTWe 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 and , 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.
Highlights
We are going to investigate uniform regularity and strong regularity on compact subsets of Banach spaces of mappings which are defined as a sum of a single-valued mapping and a set-valued mapping with a closed graph
We recall basic definitions from regularity theory and derive a result guaranteeing that a perturbed problem has a solution which is similar to the classical Lyusternik-Graves and Robinson theorem
We study two path-following methods for a differential generalized equation (DGE), a model introduced in [3], which is a problem to find a pair of functions x : [0, T] → Rn and u : [0, T] → Rm such that ⎧ ⎪⎨x(t) = g(x(t), u(t))
Summary
We are going to investigate uniform (metric) regularity and strong (metric) regularity on compact subsets of Banach spaces of mappings which are defined as a sum of a single-valued (possibly non-smooth) mapping and a set-valued mapping with a (locally) closed graph. By the word ‘uniform’ we mean that the constants as well as the size of neighbourhoods, appearing in the corresponding definitions, remain the same for a certain set of mappings and/or points. These properties turn out to be the key ingredients in the proofs of the non-smooth Robinson’s inverse function theorem [1] and Lyusternik-Graves. Theorem for the sum a Lipschitz function and a set-valued mapping with closed graph [2]. By a.e. we mean almost every in terms of the Lebesgue measure
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