Dynamic programming for some optimal control problems with hysteresis
type: Item
instance of: scholarly article
title: Dynamic programming for some optimal control problems with hysteresis
zbMATH Open document ID: 1009.47071
author: Fabio Bagagiolo
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2002-06-27T00:00:00Z
review text: With a hysteresis operator \(\mathcal F\) and a measurable control \(\alpha\) the dynamical system \[ y'(t)=f(y(t), {\mathcal F}[y,\xi_0](t),\alpha(t))\qquad (t\geq 0) \] is studied with initial state \(\xi_0\) and \(y(0)=y_0\). For the problem of minimizing the infinite horizon discount cost \[ J=\int_0^{+\infty} e^{-\lambda t}l(y(t), {\mathcal F}[y,\xi_0](t),\alpha(t)) dt \] (with given \(\lambda>0\) and \(l\)) the corresponding (discontinuous) Hamilton-Jacobi equation is obtained for the particular cases that \(\mathcal F\) is either a Play operator or a Prandtl-Ishlinskii operator. Under natural assumptions, the function \((y_0,\xi_0)\mapsto\inf_\alpha J\) is the only viscosity solution of this equation, and the equation can also be reformulated as a viscosity Neumann type boundary value problem.
Mathematics Subject Classification ID: 47J40
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 49L25
Mathematics Subject Classification ID: 49J15
zbMATH DE Number: 1761063
zbMATH Keywords: hysteresis operator
zbMATH Keywords: optimal control
zbMATH Keywords: viscosity solution
zbMATH Keywords: dynamic programming
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: Play operator
zbMATH Keywords: Prandtl-Ishlinskij operator
zbMATH Keywords: viscosity Neumann type boundary value problem
MaRDI profile type: Publication
full work available at URL: s00030-002-8122-0
OpenAlex ID: W2058127280
DOI: 10.1007/S00030-002-8122-0
description: scientific article; zbMATH DE number 1761063