The Hamilton-Jacobi-Bellman equation with a gradient constraint
type: Item
instance of: scholarly article
title: The Hamilton-Jacobi-Bellman equation with a gradient constraint
zbMATH Open document ID: 0664.35026
DOI: 10.1016/0022-0396(88)90043-5
author: N. Yamada
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: This paper deals with the existence and uniqueness of viscosity solutions for the Hamilton-Jacobi-Bellman equation \[ \max \{L^ 1u-f^ 1,...,L^ mu-f^ m,| Du| -g\}=0\quad in\quad \Omega;\quad u|_{\partial \Omega}=0. \] Here \(L^ p\), \(p=1,...,m\) are uniformly elliptic operators with smooth coefficients. For smooth \(f^ p\geq 0\), \(p=1,...,m\), and \(g\geq 0\), it is proved that there exists a viscosity solution which belongs to \(W^{2,\infty}_{loc}(\Omega)\cap W^{1,\infty}(\Omega)\) and it is unique in the class \(C^ 1(\Omega)\cap C({\bar \Omega})\) if \(g>0\). The solution is obtained as a limit of the system of approximate equations: \[ L^ p u^ p_{\epsilon}+\beta_{\epsilon}(| Du^ p_{\epsilon}|^ 2-g^ 2)+\gamma_{\epsilon}(u^ p_{\epsilon}- u_{\epsilon}^{p+1})=f^ p\quad in\quad \Omega, \] \[ u^ p_{\epsilon}|_{\partial \Omega}=0,\quad p=1,...,m,\quad u_{\epsilon}^{m+1}=u^ 1_{\epsilon}, \] where \(\beta_{\epsilon}\) and \(\gamma_{\epsilon}\) are standard penalty functions. The uniqueness proof is carried out by using this approximation again. The uniqueness of the solution in the class \(C(\Omega)\) is also considered and the author proves the affirmative results (1) for the case \(m=1\) of the above equation, (2) for the minimax equation \[ \min \{\max \{-\Delta u+u-f,u-\psi_ 1\},u-\psi_ 2\}=0\quad in\quad \Omega;\quad u|_{\partial \Omega}=0. \]
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 49L99
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 49M30
zbMATH DE Number: 4086157
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: viscosity solutions
zbMATH Keywords: Hamilton-Jacobi-Bellman equation
zbMATH Keywords: uniformly elliptic
zbMATH Keywords: smooth coefficients
zbMATH Keywords: limit
zbMATH Keywords: approximate equations
zbMATH Keywords: standard penalty functions
zbMATH Keywords: minimax equation
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90043-5
OpenAlex ID: W1979324094
cites work: Q4199298
cites work: A second order elliptic equation with gradient constraint
cites work: Classical solutions of fully nonlinear, convex, second-order elliptic equations
cites work: Optimal Stochastic Switching and the Dirichlet Problem for the Bellman Equation
cites work: Q3911014
cites work: Q4040640
cites work: Boundary regulatity and uniqueness for an elliptic equations with gradient constraint
cites work: Résolution analytique des problèmes de Bellman-Dirichlet
cites work: Optimal control of diffusion processes and hamilton–jacobi–bellman equations part 2 : viscosity solutions and uniqueness
Recommended article: On Hamilton-Jacobi-Bellman equations with convex gradient constraints
Recommended article: Hamilton-Jacobi-Bellman equations with fast gradient-dependence
Recommended article: Hamilton-Jacobi-Bellman equation under states constraints
Recommended article: Hamilton–Jacobi–Bellman Equations
Recommended article: Solving a Hamilton–Jacobi–Bellman equation with constraints
Recommended article: Q4280514
Recommended article: Non-convex Hamilton-Jacobi equations with gradient constraints
Recommended article: On the Hamilton-Jacobi-Bellman equations
Recommended article: Hamilton-Jacobi-Bellman equations for optimal control processes with convex state constraints
Recommended article: Q4249807
description: scientific article; zbMATH DE number 4086157