Inverse problems of dynamics of controllable systems with distributed parameters
type: Item
instance of: scholarly article
title: Inverse problems of dynamics of controllable systems with distributed parameters
zbMATH Open document ID: 0856.93004
published in: Russian Mathematics
publication date: 1997-02-25T00:00:00Z
review text: Inverse problems are perhaps more important in engineering and applied physics than the so called direct problems. In a typical classroom problem one assumes that the behavior of a dynamical system is known and predicts the output of the system for a given input. In many real-life problems, the properties of the system are not known and experiments may be carried out to find some characteristics of the system by relating its output to the input. Thus, whole classes of problems may be defined. For example, let the state of the system be given by \(y=y(t)\) on some time interval \(t\in T\). An admissible control \(u(t)\) acts on the system which evolves according to some law \(y(\cdot,t)=A(u(\cdot,t))\). Here, \(A\) is a Volterra operator (or one with the Volterra property). This means that only past and present actions of the control \(u\) affect the behavior of the system. In other words, there is no ``peaking into the future''. Since most control and design problems may be classified as inverse problems, a huge amount has been written on this broad subject. The author proceeds to characterize certain classes of such problems. Let the initial state be known \(y(0)= y_0\). A class of feasible controls is known. Controls could be either inputs into the system deliberately applied, or some perturbations such as noise, wind, which some authors may call uncontrollable inputs. The system evolves according to some law \(y(\cdot)= Au(\cdot)\). Observation, or measurement of the system, is described by the observation operator \(C\): \(z=Cy\), or \(z=CAu\), and the control or input may be determined by the measurement \(u=Bz\). If the operator \(CA\) is only approximately known, or guessed, several techniques exist for minimizing the discrepancy functional: \(J(u)=|z(\cdot)-CAu(\cdot)|\). Most popular is the least squares technique. However, some serious problems arise at this point. Stability of this procedure should be questioned. But even if the problem is well posed, to …
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 93B30
Mathematics Subject Classification ID: 93C20
zbMATH DE Number: 923055
zbMATH Keywords: inverse problems
zbMATH Keywords: real time
zbMATH Keywords: ill-posed problems
zbMATH Keywords: Volterra's property
author: A. I. Korotkii
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 923055