Hamilton-Jacobi-Bellman equation under states constraints
type: Item
instance of: scholarly article
title: Hamilton-Jacobi-Bellman equation under states constraints
zbMATH Open document ID: 0954.35045
author: Kyung-Eung Kim
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-02-05T00:00:00Z
review text: This paper is concerned with the uniqueness of discontinuous solutions of the Hamilton-Jacobi-Bellman equation \[ \begin{cases} -{\partial V\over\partial t} (t,x)+ H(t,x,{\partial V\over\partial x}(t, x))= 0,\\ V(t,x)= \psi(x)\text{ when }g(T,x)\leq 0,\end{cases} \] where \(H(t,x,p)= \sup_{v\in F(t,x)}\langle p,v\rangle\), arising in Mayer's problem under state constraints \[ \min\left\{\psi(x(T))\left|\begin{aligned} & x'(t)\in F(t,x(t))\text{ a.e. in }[0,T],\\ & g(t,x(t))\leq 0,\quad t\in[0,T],\\ & x(0)= \xi_0,\\ & x(\cdot)\in AC(0,T),\end{aligned}\right. \right\}, \] where \(T> 0\), \(F:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n\), \(g: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}\), \(\xi_0\in \mathbb{R}^n\), \(\psi: \mathbb{R}^n\to \mathbb{R}\cup\{+ \infty\}\). The author used two types of discontinuous solutions, bilateral solutions and contingent solutions, and showed that they coincide with the value function \(V\) for state constraints, that are backward invariant and forward viable.
Mathematics Subject Classification ID: 35F20
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 1441387
zbMATH Keywords: uniqueness of discontinuous solutions
zbMATH Keywords: bilateral solutions
zbMATH Keywords: contingent solutions
reviewed by: Vasile Iftode
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6684
OpenAlex ID: W1996904445
cites work: Q3324260
cites work: Viability theory
cites work: Semicontinuous Viscosity Solutions For Hamilton–Jacobi Equations With Convex Hamiltonians
cites work: Optimal Control and Semicontinuous Viscosity Solutions
cites work: Hamilton-Jacobi Equations with State Constraints
cites work: Viscosity Solutions of Hamilton-Jacobi Equations
cites work: Measurable viability theorems and the Hamilton-Jacobi-Bellman equation
cites work: Q3753007
cites work: Hamilton-Jacobi equations: Viscosity solutions and generalized gradients
cites work: Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation
cites work: Lower Semicontinuous Solutions of Hamilton–Jacobi–Bellman Equations
cites work: Asymptotic behavior of some nonlinear heat equations
DOI: 10.1006/JMAA.1999.6684
description: scientific article; zbMATH DE number 1441387