On time-minimal distributed control of vibrating systems governed by an abstract wave equation
type: Item
instance of: scholarly article
title: On time-minimal distributed control of vibrating systems governed by an abstract wave equation
zbMATH Open document ID: 0569.49016
DOI: 10.1007/BF01442203
published in: Applied Mathematics and Optimization
publication date: 1985-01-01T00:00:00Z
review text: The author considers the time-minimal (or time-optimal) problem for a system described by the abstract wave equation \(y_{tt}+Ay(t)=f(t)\), where the trajectory y(t) and the control f(t) take values in a Hilbert space H and A is a positive definite self-adjoint operator in H. The control is assumed to satisfy (1) \(\int^{T}_{0}| f(t)|^ 2dt\leq C^ 2\), where C is given in advance (the trajectory initiates at time 0 and T is the arrival time to the target). The author shows that the system can be driven to equilibrium in finite time; as a consequence, a unique optimal control exists. The main result is that the time-optimal control has least \(L^ 2\) norm in the interval (0,T) (T the minimum time), the minimum norm being C, the constant in (1). This result is computationally important, since (via an initial estimation of the optimal time) it reduces a ''hard constraint'' problem such as the time- optimal problem to a ''soft constraint'' problem, much easier to deal with in actual computation.
Mathematics Subject Classification ID: 49K27
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 49K30
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93C25
zbMATH DE Number: 3909191
zbMATH Keywords: time-minimal problem
zbMATH Keywords: abstract wave equation
author: Werner Krabs
reviewed by: Hector O. Fattorini
MaRDI profile type: Publication
cites work: Q5584384
cites work: Q5601627
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description: scientific article; zbMATH DE number 3909191