Trajectory and final controllability in hyperbolic and pseudohyperbolic systems with generalized actions
type: Item
instance of: scholarly article
title: Trajectory and final controllability in hyperbolic and pseudohyperbolic systems with generalized actions
zbMATH Open document ID: 1037.93047
DOI: 10.1023/A:1013871026026
author: S. I. Lyashko
author: D. A. Nomirovskij
author: T. I. Sergienko
published in: Cybernetics and Systems Analysis
publication date: 2003-09-16T00:00:00Z
review text: The authors consider a hyperbolic system: \[ \partial^2u/ \partial t^2+A(u)= f(h) \] in a cylindrical domain \(Q=(0,T)\times \Omega\), where \(h\) is a control parameter, and \(A\) is an elliptic operator of the form: \[ A(u)=\Sigma_{ij} \partial/ \partial x_i(a_{ij} (x)\partial u/\partial x_j)+\Sigma_ia_i(x)\partial u/\partial x_i+a(x)u. \] All coefficients are continuously differentiable, except \(a\), which is allowed to be only continuous. In their previous work the authors utilized the concept of negative norms, which otherwise satisfy all requirements for a norm, such as for example the Cauchy-Schwarz inequality. They derive a sequence of inequalities valid in these negative norms. A major project of this paper consists in finding a dense imbedding of the space of functions which satisfy their requirements and are of the class \(C_0^\infty (Q)\), into the space \(L_2(Q)\). Using these techniques the authors prove controllability (subject to a rather general choice of the quality functional), which is valid for a class of hyperbolic and pseudohyperbolic systems of equations similar to the one displayed above.
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 35L82
zbMATH DE Number: 1981958
zbMATH Keywords: negative norms
zbMATH Keywords: dense imbedding
zbMATH Keywords: systems with generalized action
zbMATH Keywords: hyperbolic systems
zbMATH Keywords: pseudo-hyperbolic systems
zbMATH Keywords: final controllability
zbMATH Keywords: optimal control
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: a:1013871026026
OpenAlex ID: W175652419
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description: scientific article; zbMATH DE number 1981958