Interior point control and observation for the wave equation (Q1809721)

The paper studies both exact and approximate controllability problems of the wave equation via static or spatially moving actuators. Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), and \(\partial\Omega\) its smooth boundary. The wave equation with state \(y(\cdot,t)\) is described in \(\Omega\times (0,T)\) by \[ y_{tt}=\Delta y+{\mathbf L}\bigl(\widehat x(\cdot) \bigr) \circ v\text{ in }\Omega \times(0,T), \quad y|_{\partial \Omega}=0,\;y( \cdot,0)= y_t(\cdot,0)= 0\text{ in }\Omega. \] Here, \(v\) denotes control inputs belonging to a suitable control function space \(V\); \(\widehat x(\cdot)\), \(0<t<T\) a spatial curve in \(\Omega\); and \({\mathbf L}(\widehat x(\cdot))\circ v\) one of the following: \[ \delta\bigl( x-\widehat x(t)\bigr)\circ v,\quad\nabla \biggl( \delta \bigl(x- \widehat x(t)\bigr) \circ v\biggr), \text{ or }{\partial \over \partial t}\biggl( \delta\bigl(x- \widehat x(t)\bigr) \circ v\biggr). \] In the case where \(\Omega=(0,1)\) and \(\widehat x(\cdot)\) is a constant \(\overline x\), which is a special number, the so-called ``algebraic number'' of order 2, the exact controllability of the system with \(T=2\) (minimum possible) and its version are obtained in respective function spaces. In the case of general dimension \(n\geq 1\) and moving actuators, it is shown that there exist curves \(\widehat x(\cdot)\) for which the exact or approximate controllability is ensured for a given \(T>0\). The dual problem, that is, the observability problem, is also discussed, and corresponding results are obtained such that the continuity property of the state at \(t=T\) regarding the observed data on the interval \((0,T)\) is ensured.