Controllability of second-order equations in \(L^2(\Omega)\)
Summary: We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert space \(L^2(\Omega)\): \(\ddot y+Ay=1_\omega u(t)\), \(t\in (0,\tau]\), \(y(0)=y_0\), \(\dot y(0)= y_1\), where \(\Omega\) is a domain in \(\mathbb R^N\) \((N\geq 1)\), \(y_0,y_1\in L^2(\Omega)\), \(\omega\) is an open nonempty subset of \(\Omega\), \(1_\omega\) denotes the characteristic function of the set \(\omega\), the distributed control \(u\) belongs to \(L^2(0,\tau; L^2(\Omega))\), and \(A:D(A)\subset L^2(\Omega)\to L^2(\Omega)\) is an unbounded linear operator with the following spectral decomposition: \(\Lambda z= \sum_{j-1}^\infty \lambda_j \sum_{k-1}^{\gamma_j}\langle z,\varphi_{j,k}\rangle \varphi_{j,k}\), with the eigenvalues \(\lambda_j\) given by the following formula: \(\lambda_j= j^{2m}\pi^{2m}\), \(j=1,2,3,\dots\) and \(m\geq 1\) is a fixed integer number, multiplicity \(\gamma_j\) is equal to the dimension of the corresponding eigenspace, and \(\{\varphi_{j,k}\}\) is a complete orthonormal set of eigenvectors (eigenfunctions) of \(A\). Specifically, we prove the following statement: if for an open nonempty set \(\omega\subset\Omega\) the restrictions \(\varphi_{j,k}^\omega= \varphi_{j,k}|_\omega\) of \(\varphi_{j,k}\) to \(\omega\) are linearly independent functions on \(\omega\), then for all \(\tau\geq 2/\pi^{m-1}\) the system is approximately controllable on \([0,\tau]\). As an application, we prove the controllability of the 1D wave equation.
second-order equations in Hilbert spaces
scientific article; zbMATH DE number 5849023
scientific article