Exact controllability of the radial solutions of the semilinear wave equation in \(\mathbb{R}^3\)
type: Item
instance of: scholarly article
title: Exact controllability of the radial solutions of the semilinear wave equation in \(\mathbb{R}^3\)
zbMATH Open document ID: 0918.93004
published in: Revista Matemática Complutense
publication date: 1999-05-05T00:00:00Z
full work available at URL: https://eudml.org/doc/44466
review text: The author considers the following semilinear equation \[ \begin{cases} y_{tt}- \Delta y+f(y)= h\chi_\omega &\text{in }\Omega\times (0,T),\\ y=0 &\text{on }\partial\Omega\times (0,T),\\ y(x,0)= y^0(x),\;y_t(x,0)= y^1(x) &\text{for }x\in\Omega, \end{cases} \tag{1} \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^n\) with boundary \(\partial\Omega\) of class \(C^2\), \(\omega\subset \Omega\) a nonempty open subset, \(f\in C^1(\mathbb{R}^n)\), \(\{y^0,y^1\}\in H_0^1(\Omega)\times L^2(\Omega)\), and \(h(x,t)\in L^2(\omega\times (0,T))\) is a control function. The aim of this paper is to solve the problem of exact controllability: to find conditions on \(T\) and \(\omega\) such that for every \(\{y^0,y^1\}, \{z^0,z^1\}\in H_0^1(\Omega)\times L^2(\Omega)\) we can find a control \(h\in L^2(\omega\times (0,T))\) such that the corresponding solution \(y\) to (1) satisfies the following: \[ y(x,T)= z^0(x), \quad y_t(x,T)= z^1(x) \text{ for }x\in\Omega. \] Making a change of variables to spherical coordinates \(r=| x|\), this problem is reduced to the problem of exact controllability of the following wave equation in \(\mathbb{R}\): \[ \begin{cases} r^2y_{tt}- (r^2y_r)_r+ r^2f(y)- r^2h\chi_\omega &\text{ in }(0,1)\times (0,T),\\ y_r(0,t)= y(1,t)=0 &\text{ for }t\in (0,T),\\ y(r,0)= y^0(r),\;y_t(r,0)- y^1(r) &\text{ in }(0,1), \end{cases} \tag{2} \] where now \(\omega= (\ell_1,\ell_2)\subset \mathbb{R}\), \(y^0,y^1\) are functions of \(| x|\) and \(h\) is a function of \((| x|,t)\). The main result states that if \(T>2\max(\ell_1, 1-\ell_2)\) and \(f\in C^1(\mathbb{R})\) satisfies \[ | f(s)|\leq C| s|\cdot \bigl|\log^{p_0}| s| \bigr|+D, \] for some constants \(C,D>0\) and for some \(1\leq p_0<2\), then the system (2) is exactly controllable in time \(T\). The proof combines HUM and a fixed point technique. The HUM was introduced by J. L. Lions and is well known, whereas the fixed point method is presented in this work by five successive steps and the author obtains a sui…
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35B37
zbMATH DE Number: 1187544
zbMATH Keywords: semilinear equation
zbMATH Keywords: exact controllability
zbMATH Keywords: spherical coordinates
zbMATH Keywords: wave equation
zbMATH Keywords: HUM
zbMATH Keywords: fixed point method
author: Luz de Teresa
reviewed by: Marian Ivanovici
MaRDI profile type: Publication
OpenAlex ID: W2151113516
DOI: 10.5209/REV_REMA.1998.V11.N1.17328
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description: scientific article; zbMATH DE number 1187544