## Interior point control and observation for the wave equation.(English)Zbl 0989.93045

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.
Reviewer: T.Nambu (Kobe)

### MSC:

 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability 35L05 Wave equation 93B07 Observability
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