A Newton method for the computation of time-optimal boundary controls of one-dimensional vibrating systems
type: Item
instance of: scholarly article
title: A Newton method for the computation of time-optimal boundary controls of one-dimensional vibrating systems
zbMATH Open document ID: 0953.93042
DOI: 10.1016/S0377-0427(99)00291-5
author: Martin Gugat
published in: Journal of Computational and Applied Mathematics
publication date: 2001-01-22T00:00:00Z
review text: The problem of time-optimal boundary control for a one-dimensional vibrating system is considered. Namely, given a system described by \[ y_{tt}(x,t) + Ly(x,t) = Bu(t), \] with initial conditions: \(y(x,0) = y_0\), \(y_t(x,0) = y_1, \;x {\i}(0,1)\), one looks for \(u\) which in the shortest time \(T\) steers the system to the rest: \(y(x,T) = 0, \;y_t(x,T) = 0, \;x \in (0,1)\), and satisfies some inequality constraints and additional moment equations. To solve the problem the author introduces a parametric auxiliary problem with fixed controlling time, where the function from the inequality constraint is taken as the objective function and the moment equations are taken as constraint functions. It appears that the smallest root of the optimal value function for this problem is the minimal controlling time and it can be approximated by zeros of the optimal value functions of some discretized auxiliary problems. Thus, after proving that the latter are differentiable the author uses the Newton method to compute the zeros of these optimal value functions getting the approximations of the minimal controlling time and the time-optimal control as well. Some numerical examples for the time-optimal control of the rotating Euler-Bernoulli beam are also presented.
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 49N05
zbMATH DE Number: 1419812
zbMATH Keywords: time-optimal control
zbMATH Keywords: parametric optimization
zbMATH Keywords: moment problems
zbMATH Keywords: optimal value function
zbMATH Keywords: time-parametric auxiliary problem
zbMATH Keywords: rotating beam
zbMATH Keywords: value function's root
reviewed by: Zdzisław Denkowski
MaRDI profile type: Publication
Wikidata QID: Q104129321
cites work: Q4856165
cites work: Q4942488
cites work: Some trigonometrical inequalities with applications to the theory of series
cites work: Q3913990
cites work: Optimal control of processes governed by partial differential equations part II: Vibrations
cites work: On moment theory and controllability of one-dimensional vibrating systems and heating processes
cites work: Q4029545
cites work: Q4057193
cites work: Q3681889
cites work: Q3772036
cites work: Nonharmonic FOurier series in the control theory of distributed parameter systems
cites work: Q3666712
cites work: Numerical Solution of Ordinary Differential Equations
cites work: Q3955832
description: scientific article; zbMATH DE number 1419812