Boundary and distributed controllability of the damped wave equation: Reduction of control time
type: Item
instance of: scholarly article
title: Boundary and distributed controllability of the damped wave equation: Reduction of control time
zbMATH Open document ID: 0940.93038
author: Marianna A. Shubov
author: Clyde Martin
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-07-19T00:00:00Z
review text: The authors continue the study of the controllability problem for the damped wave equation (see the papers of the authors' references (1-7)). Also, one considers the one-dimensional wave equation which represents the vibration of a string with spatially nonhomogeneous damping modulus of elasticity and density coefficients. The equation is defined on a finite interval with linear first-order nonselfadjoint boundary conditions containing damping terms. Section 2 contains auxiliary propositions and a formulation of the main results. All new results are proven in Section 3 and 4. In this paper, the authors study two extensions. First, they consider the boundary control combined with the aforementioned distributed control, and second they examine more general distributed control. By combining in this note the boundary and the distributed controls, it is shown that it is possible to reduce the time interval. The authors consider the wave equation \[ u_{tt}+ 2d(x) u_t+ Lu= F(x,t),\quad x\in[0,a],\quad t>0,\tag{1} \] where \(L\) is the Sturm-Liouville operator defined on smooth functions \(\varphi\) such that \[ L\varphi= -{1\over\rho(x)} {d\over dx} \Biggl(p(x) {d\varphi\over dx}\Biggr)+ q(x)\varphi. \] Equation (1) describes the forced motion of a string with density \(\rho(x)\), with modulus of elasticity \(p(x)\) and subject to an external harmonic force with rigidity \(q(x)\) and in the presence of viscous damping \(2d(x)\). The function \(u(x,t)\) satisfies the initial conditions \[ u(x,0)= u^0(x),\quad u_t(x,0)= u^1(x) \] and the boundary conditions \[ (u_x+ ku_t)(0,t)= 0,\quad (u_x+ hu_t)(a,t)= \varphi(t),\quad h,k\in\mathbb{C}\cup\{\infty\}. \]
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93B05
zbMATH DE Number: 1398593
zbMATH Keywords: generalized eigenvectors
zbMATH Keywords: Riesz basis
zbMATH Keywords: boundary controls
zbMATH Keywords: controllability
zbMATH Keywords: damped wave equation
zbMATH Keywords: nonselfadjoint boundary conditions
zbMATH Keywords: distributed control
zbMATH Keywords: Sturm-Liouville operator
zbMATH Keywords: string
MaRDI profile type: Publication
OpenAlex ID: W2018435393
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DOI: 10.1006/JMAA.1999.6554
description: scientific article; zbMATH DE number 1398593