Controlled diffusion processes on infinite horizon with the overtaking criterion
type: Item
instance of: scholarly article
title: Controlled diffusion processes on infinite horizon with the overtaking criterion
zbMATH Open document ID: 0637.60055
DOI: 10.1007/BF01448359
published in: Applied Mathematics and Optimization
publication date: 1988-01-01T00:00:00Z
review text: Let x(t,u) be a diffusion process depending on a control u. The paper is concerned with the minimization of functionals of the form \[ C_ T(u)=E\int^{T}_{0}f(x(t),u(t),t)dt,\quad as\quad T\to \infty, \] where f is a given function. In many cases \(C_ T(u)\) diverges for any choice of u, and the introduction of discounting factors is not justified. Thus the author studies the problem of minimizing \[ J(u)=\liminf (1/T)C_ T(u)\quad as\quad T\to \infty, \] which describes the minimal growth rate of \(C_ T(u)\) (similar problems with lim sup are also considered). The overtaking criterion is discussed for this infinite horizon optimal control problem. The aim of the paper is twofold: i) to achieve a better understanding of the Bellman equation arising in the optimal control problem (in particular, to give a cost interpretation of the solution of the Bellman equation), ii) to find conditions, in terms of the solutions of the Bellman equation, for the existence of overtaking optimal controls, or minimal time average controls, and to characterize them using the Bellman equation. Control problems with a non-integrable discount factor are also studied. The author considers both cases, where the diffusion is in R d, or it takes place in a bounded domain with reflections from the boundary.
reviewed by: Franco Flandoli
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 49J55
zbMATH DE Number: 4038931
zbMATH Keywords: controlled diffusions
zbMATH Keywords: overtaking criterion
zbMATH Keywords: infinite horizon optimal control problem
zbMATH Keywords: Bellman equation
zbMATH Keywords: reflections from the boundary
author: Arie Leizarowitz
MaRDI profile type: Publication
cites work: Tracking periodic signals with the overtaking criterion
cites work: Q3324260
cites work: Optimal impulse and continuous control: Method of nonlinear quasi- variational inequalities
cites work: Existence of Overtaking Optimal Trajectories for Problems with Convex Integrands
cites work: Linear oblique derivative problems for the uniformly elliptic Hamilton- Jacobi-Bellman equation
cites work: Stochastic differential equations with reflecting boundary conditions
cites work: Long-term average cost control problems for continuous time Markov processes: A survey
cites work: Diffusion processes with boundary conditions
cites work: On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions
description: scientific article; zbMATH DE number 4038931