# Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity

Borwein, Jonathan M. and Zhu, Qiji J. (1996) Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity. SIAM Journal on Control and Optimization, 34 (5). pp. 1568-1591. ISSN 0363-0129

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## Abstract

In Gateaux or Bornologically differentiable spaces there are two natural natural generalizations of the concept of a Fr\' echet subderivative: In this paper we study the viscosity subderivative (which is the more robust of the two) and establish a fuzzy sum rule for it in a smooth Banach space. This rule is applied to obtain existence and comparison results for viscosity solutions of the Hamilton-Jacobi-Bellman equation in $\beta$-smooth spaces. A unified treatment of metric regularity in smooth spaces completes the paper. This illustriates the flexibility of viscosity subderivatives as a tool for analysis.

Item Type: Article pubdom FALSE viscosity subderivative, fuzzy sum rule, viscosity solutions, Hamilton-Jacobi equations, smooth spaces, metric regularity 49-xx Calculus of variations and optimal control; optimization > 49Jxx Existence theories58-xx Global analysis, analysis on manifolds > 58Cxx Calculus on manifolds; nonlinear operators49-xx Calculus of variations and optimal control; optimization > 49LXX Hamilton-Jacobi theories, including dynamic programming UNSPECIFIED Users 1 not found. 17 Nov 2003 07 Sep 2014 21:05 https://docserver.carma.newcastle.edu.au/id/eprint/80

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