Convexity properties of the minimum time function
type: Item
instance of: scholarly article
title: Convexity properties of the minimum time function
zbMATH Open document ID: 0836.49013
DOI: 10.1007/BF01189393
author: Carlo Sinestrari
author: Piermarco Cannarsa
published in: Calculus of Variations and Partial Differential Equations
publication date: 1995-09-06T00:00:00Z
review text: The paper deals with the minimum time optimal control problem governed by the system \[ \begin{aligned} y'(t) & = f \bigl( y(t), u(t) \bigr) \tag{*} \\ y(0) & = x \in \mathbb{R}^n, \quad u : [0, + \infty) \to U, \end{aligned} \] where \(U \subset \mathbb{R}^m\) is a compact set. Given a target \(K\) and denoting by \(\Omega \subset \mathbb{R}^n\) the set of points which can be driven in a finite time on \(K\), the minimum time function \(\tau : \Omega \to [0, + \infty)\) is defined as \[ \tau (x) : = \inf \biggl\{ T : \bigl( y(t), u(t) \bigr) \text{ solution of } (*),\;y(T) \in K \biggr\}. \] Under a Petrov type controllability assumption, it is proved that the semiconcavity of the distance function from the target (which can be considered as a mild regularity assumption on \(K)\) implies the semiconcavity of \(\tau\). For linear control systems a semiconvexity property can also be established. These conditions can be used to study the structure (Hausdorff dimension estimates, propagation of singularities) of the set of nondifferentiability points of \(\tau\).
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 93C10
zbMATH DE Number: 778936
zbMATH Keywords: minimum time optimal control problem
zbMATH Keywords: Petrov type controllability
zbMATH Keywords: semiconcavity
zbMATH Keywords: semiconvexity
zbMATH Keywords: set of nondifferentiability points
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01189393
OpenAlex ID: W1974468996
description: scientific article; zbMATH DE number 778936