Exact controllability of the radial solutions of the semilinear wave equation in \(\mathbb{R}^3\)
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 suitable observability property for the linear wave equation with a particular potential.
scientific article; zbMATH DE number 1187544