Exponential decay for distributed bilinear control systems with damping
type: Item
instance of: scholarly article
title: Exponential decay for distributed bilinear control systems with damping
zbMATH Open document ID: 0916.93063
DOI: 10.1007/BF02844383
publication date: 1999-07-12T00:00:00Z
review text: The system is \[ \begin{aligned}{ &y''(t) + Ay(t) + Dy'(t) + u(t)By(t) = 0, \\ &y(0) = y_0, \quad y'(0) = y_1 }\end{aligned} \] in a Hilbert space \(H.\) All operators are linear, \(D\) and \(B\) are bounded and \(A : V \to E,\) where \(V \hookrightarrow E\) with dense, continuous and compact imbedding. Under suitable assumptions, the author shows uniform exponential decay of the energy \[ E(t) = \tfrac 12\big( \| y(t)\| _V^2 + \| y'(t)\| _E^2 \big) \] by means of a scalar nonlinear feedback \(u(t) = f(\langle By(t), y(t) \rangle).\) This is related to results of Ball and Slemrod, where there is no damping \((D = 0)\) and only weak stabilization is obtained.
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1279155
zbMATH Keywords: distributed systems
zbMATH Keywords: bilinear systems
zbMATH Keywords: stabilization by damping
zbMATH Keywords: feedback
author: Larbi Berrahmoune
reviewed by: Hector O. Fattorini
MaRDI profile type: Publication
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1279155