Regularity of boundary quasi-potentials for planar systems
type: Item
instance of: scholarly article
title: Regularity of boundary quasi-potentials for planar systems
zbMATH Open document ID: 0810.49032
DOI: 10.1007/BF01261992
author: Martin V. Day
published in: Applied Mathematics and Optimization
publication date: 1994-08-10T00:00:00Z
review text: The author considers a first order differential system in \(R^ 2: x'= b(x)\), where the vectorfield \(b\) is smooth and bounded. Let \(D\) be a bounded region invariant for the differential system, which is attracted to an asymptotically stable critical point \(0\in D\). The main purpose of the paper is to study the value function \[ W(x)= \inf\{S_{0T}(\phi)\mid 0< T<\infty,\;\phi\in C([0,T];\;\bar D),\;\phi(0)= x,\;\phi(T)\in \partial D\}, \] where \(S_{0T}(\phi)= \int^ T_ 0 L(\phi,\phi')dt\) if \(\phi\) is absolutely continuous, and \(S_{0T}(\phi)= \infty\) otherwise, \(L(x,v)= {1\over 2}(b(x)- v)^ T a(x)^{-1}(b(x)- v)\), \(a(x)\) is symmetric and positive definite. It is proved that if \(\partial D\) is a single \(T\)-periodic solution and if \(\text{div}(b)> 0\) everywhere on \(\partial D\), then \(W\) is a smooth function in some strip around \(\partial D\) and \(W\) satisfies the Hamilton-Jacobi equation \(H(x,-dW(x))= 0\), where \(H\) is the Legendre transform of \(L\). This problem is connected with the study of small random perturbation of planar dynamical systems. Other results are also given when \(\partial D\) satisfies other conditions than the previous one.
Mathematics Subject Classification ID: 49L99
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 617361
zbMATH Keywords: invariant manifold
zbMATH Keywords: quasi-potential
zbMATH Keywords: value function
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: Legendre transform
zbMATH Keywords: small random perturbation
zbMATH Keywords: planar dynamical systems
reviewed by: J-P Raymond
MaRDI profile type: Publication
cites work: Recent progress on the small parameter exit problem†
cites work: Conditional exits for small noise diffusions with characteristic boundary
cites work: Cycling and skewing of exit measures for planar systems
cites work: Some regularity results on the Ventcel-Freidlin quasi-potential function
cites work: Q3672830
description: scientific article; zbMATH DE number 617361