On the Bellman equation for infinite horizon problems with unbounded cost functional
type: Item
instance of: scholarly article
title: On the Bellman equation for infinite horizon problems with unbounded cost functional
zbMATH Open document ID: 0952.49023
author: Francesca Da Lio
published in: Applied Mathematics and Optimization
publication date: 2001-01-07T00:00:00Z
review text: The author investigates a class of nonlinear infinite horizon optimal control problems minimizing a cost functional of the type \[ J(x,\alpha)=\int_0^\infty l(y_x(t),\alpha(t))e^{-\lambda t}\, dt. \] Since the discount rate \(\lambda\) is only required to be nonnegative and the set of control values \(A\) is allowed to be unbounded this class in particular includes the linear quadratic (LQ) control problem. Under standard regularity assumptions on the problem data it is shown that the optimal value function \(V\) of this problem is the unique viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation \[ \lambda V(x) + \sup_{a\in A}\{-f(x,a)DV(x)-l(x,a) \}=0. \] In addition, the regularity of \(V\), the dependence of \(V\) on the problem data (in terms of stability of the solution to the HJB equation) and the behaviour of optimal controls for the LQ problem under perturbations are addressed.
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49N10
zbMATH DE Number: 1406319
zbMATH Keywords: optimal control
zbMATH Keywords: viscosity solutions
zbMATH Keywords: Hamilton-Jacobi-Bellman equation
MaRDI profile type: Publication
full work available at URL: s002459911010
OpenAlex ID: W2119937635
DOI: 10.1007/S002459911010
description: scientific article; zbMATH DE number 1406319