Optimal control problems for some ordinary differential equations with behavior of blowup or quenching
type: Item
instance of: scholarly article
title: Optimal control problems for some ordinary differential equations with behavior of blowup or quenching
zbMATH Open document ID: 1419.49005
author: Weihan Wang
author: Ping Lin
published in: Mathematical Control and Related Fields
publication date: 2019-07-03T00:00:00Z
review text: \noindent This paper is on the differential system \[ y'(t) = f(t, y(t), u(t)) \quad (t > 0) \, , \qquad y(0) = y_0,\tag{1} \] where solutions in an interval $[0, T_1)$ are defined by the integrated version \[ y(t) = y_0 + \int_0^t f(s, y(s), u(s)) ds \, , \quad (0 \le t < T_1). \] A solution of (1) has a blowup at $t = T_1$ if $\|y(t)\|$ is unbounded as $t \to T_1 - ;$ for a regular right side $f(t, y, u)$ this is the only type of singularity possible. Another type of singularity is quenching at $t = T_1,$ where $y(t)$ remains bounded but $\|y'(t)\|$ becomes unbounded as $t \to T_1-.$ Both blowup and quenching describe physical phenomena (for instance, blowup may model the catastrophic increase of temperature in a chemical reaction causing ignition). There has been a number of recent results on minimizing/maximizing blowup or quenching time. \par For the blowup case, the authors consider measurable controls that satisfy a constraint $u(t) \in U$ and a functional \[ J(y(\cdot), u(\cdot)) = \int_0^{T_1} f_0(t, y(t), u(t)) dt \] where $T_1$ is the blowup point of $y(t).$ The optimal problem is that of finding a control $\bar u(t)$ that minimizes $J(y, u).$ The authors show an existence theorem for relaxed controls and a version of Pontryagin's maximum principle based on the regular maximum principle in subintervals $[0, T], T < T_1.$ The results for the quenching case are similar but there is a complication; while it doesn't make sense to continue a solution past a blowup point, continuation past a quenching point may be possible, but there may be loss of uniqueness.
reviewed by: Hector O. Fattorini
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 49K15
zbMATH DE Number: 7076369
zbMATH Keywords: optimal control
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: blowup
zbMATH Keywords: quenching
MaRDI profile type: Publication
full work available at URL: mcrf.2018036
OpenAlex ID: W2891225392
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Wikidata QID: Q129110831
DOI: 10.3934/MCRF.2018036
description: scientific article; zbMATH DE number 7076369