Minimal time function and viscosity solutions
type: Item
instance of: scholarly article
title: Minimal time function and viscosity solutions
zbMATH Open document ID: 0634.49012
DOI: 10.1007/BF00938801
published in: Journal of Optimization Theory and Applications
publication date: 1989-01-01T00:00:00Z
review text: Two theorems of \textit{O. Hàjek} [Funkc. Ekvacioj, Ser. Int. 20, 97-114 (1978; Zbl 0371.49003)] are generalized. It is proved that, if \(V(A,\Gamma)\) is the set of points which can be steered to the origin along a solution of the control system \(x'=Ax-c\), if c(t)\(\in \Gamma\), \(\Gamma\) is a compact subset of R n, \(0\in intrelco \Gamma\), and if a rank condition holds, then the minimal time function T(.) is a viscosity solution of the Bellman equation \[ \max \{<DT(x),\gamma -Ax>: \gamma \in co \Gamma \}-1=0,\quad x\in V(A,\Gamma)\setminus \{0\}, \] and of the Hàjek equation \[ 1-\max \{<DT(x),\exp [-AT(x)]>: \gamma \in co \Gamma \}=0,\quad x\in V(A,\Gamma). \]
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 4032476
zbMATH Keywords: minimal time function
zbMATH Keywords: viscosity solution
zbMATH Keywords: Bellman equation
zbMATH Keywords: Hàjek equation
author: Vasile Staicu
MaRDI profile type: Publication
cites work: Viscosity Solutions of Hamilton-Jacobi Equations
cites work: Q3701821
cites work: Q3961832
cites work: On existence and uniqueness of solutions of Hamilton-Jacobi equations
cites work: Q4722436
cites work: On a generalized Bellman equation for the optimal-time problem
cites work: On the sufficiency of the Hamilton-Jacobi-Bellman equation for optimality of the controls in a linear optimal-time problem
cites work: Q5593460
full work available at URL: https://doi.org/10.1007/bf00938801
OpenAlex ID: W2084193039
description: scientific article; zbMATH DE number 4032476