Viscosity solutions of HJB equations with unbounded data and characteristic points
type: Item
instance of: scholarly article
title: Viscosity solutions of HJB equations with unbounded data and characteristic points
zbMATH Open document ID: 1052.49027
author: M. Motta
published in: Applied Mathematics and Optimization
publication date: 2004-09-22T00:00:00Z
review text: The author refines and extends to the ``unbounded case'' certain previous results concerning viscosity-type properties of the value functions of optimal control problems which consist in the minimization of functionals of the form \[ J(t,x,\alpha(.)):=\int_0^t e^{-\int_0^tk(y(\sigma),\alpha(\sigma)) \,d\sigma} l((y(s),\alpha(s))ds+ e^{-\int_0^tk(y(\sigma),\alpha(\sigma)) \,d\sigma} g(y(t)) \] subject to: \[ y'(s)=f(y(s),\alpha(s)), \;\alpha(s)\in A\subseteq R^m \;a.e.(0,t_x), \;y(0)=x, \;x\in \Theta\subseteq R^n. \] As stated in the Introduction, ``the main goal is to characterize the minimal and maximal (viscosity) solutions of the'' associated HJB equation \[ H(x,u(x),Du(x))=0, \;H(x,u,p):= \sup_{a\in A} [-\langle p,f(x,u)\rangle- l(x,a)+u.k(x,a)] \] ``as value functions of the optimal control problems'' which are either of ``exit-time'' type or of ``infinite horizon'' type. As in some previous papers, to treat the case of unbounded data the author assumes the existence of a continuous ``recession function'' \(\Phi^\infty(x,w):= \lim_{\rho\to 0_+}\rho^p\Phi(x,\rho^{-1} w)\) associated to each of the unbounded data \(\Phi\in\{f,l,k\}\) to introduce a certain ``space-time'' (continuous) Hamiltonian.
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 35F30
Mathematics Subject Classification ID: 49N25
zbMATH DE Number: 2102075
zbMATH Keywords: unbounded control problem
zbMATH Keywords: Hamilton-Jacobi-Bellman equation
zbMATH Keywords: viscosity solution
zbMATH Keywords: impulsive control
reviewed by: Ştefan Mirică
MaRDI profile type: Publication
full work available at URL: s00245-003-0777-3
OpenAlex ID: W2040450991
DOI: 10.1007/S00245-003-0777-3
description: scientific article; zbMATH DE number 2102075