Titlebook: Lectures on Variational Analysis; Asen L. Dontchev Book 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to S

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Derivative Criteria for Metric Regularity,In this lecture, we will characterize metric regularity by using generalized derivatives of set-valued mappings. To make things simpler, we limit our considerations to mappings in Euclidean spaces. Some of the results can be extended to infinite dimensions but we will not do that here.

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Strong Regularity,We begin this lecture with a basic theorem in analysis: the classical inverse function theorem.

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Nonsmooth Inverse Function Theorems,The classical inverse function theorems assume continuous differentiability of the function involved.

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Strong Subregularity,“One-point” variants of the property of metric regularity can be obtained if in the definition we fix one of the points . or . at the reference values . or .. Specifically, consider a mapping . acting between metric spaces and . in the graph of ..

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Continuous Selections,The classical inverse function theorem presented in Lecture . gives conditions under which the inverse of a function has a single-valued localization, that is, locally, the inverse is a function.

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Regularity in Nonlinear Control,In this lecture we consider a control system described by a nonlinear ordinary differential equation of the form . over the interval [0, 1]. Here, as for the linear-quadratic problem in the preceding lecture, .(.) ∈ .. is the state of the system, while .(.) ∈ .. is the control, both at time ..

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Metric Regularity, follows . and .  are metric spaces with metrics that are denoted in the same way by .(⋅, ⋅) but may be different. Recall that a set . in a metric space is . at a point . ∈ . when there exists a neighborhood . of . such that the intersection . ∩ . is a closed set.