Pontryagin's principle for time-optimal problems
type: Item
instance of: scholarly article
title: Pontryagin's principle for time-optimal problems
zbMATH Open document ID: 0952.49020
DOI: 10.1023/A:1021793611520
published in: Journal of Optimization Theory and Applications
publication date: 2000-09-07T00:00:00Z
review text: The paper considers time-optimal control problems for the semilinear parabolic equation \[ y_t- \sum^n_{i,j= 1}{\partial\over\partial x_j} a_{ij}(x) y_{x_i}+ f(x, t,y,u)= 0\quad\text{in }\Omega\times (0,T), \] \[ {\partial y\over\partial v}= 0\quad\text{on }\partial\Omega\times(0, T),\quad y(x,0)= y_0(x)\quad\text{in }\Omega, \] with constraints on the state \(y\) at the terminal time \(T\); the terminal time \(T\) and \[ u\in U= \{u\in L^s_{\text{loc}}((0, \infty); L^s(\Omega))|u(\cdot, t)\in K(x)\text{ a.e. }x\in\Omega\} \] being the controls. Here \(\Omega\subset \mathbb{R}^n\) is a bounded domain with a smooth boundary \(\partial\Omega\) and \(K\) is a measurable multimapping with nonempty closed values. The target functional involves integrals over \(\Omega\times(0, T)\) and over \(\Omega\) at the terminal time \(T\). It is proved that the optimal triple \((y_*, u_*, T_*)\) satisfies Pontryagin's maximum principle. The proof uses Ekeland's variational principle and a change of variables (to pass to a problem with a fixed terminal time).
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1300020
zbMATH Keywords: state constraints
zbMATH Keywords: time-optimal control
zbMATH Keywords: semilinear parabolic equation
zbMATH Keywords: Pontryagin's maximum principle
author: J-P Raymond
author: Housnaa Zidani
reviewed by: Uldis Raitums
MaRDI profile type: Publication
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full work available at URL: a:1021793611520
OpenAlex ID: W121328656
description: scientific article; zbMATH DE number 1300020