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Local controllability and non-controllability for a 1D wave equation with bilinear control. (English) Zbl 1221.35221

The author considers the following wave equation
\[ \left\{\begin{aligned} &\frac{\partial^2 w}{\partial t^2}(t,x)=\frac{\partial^2 w}{\partial x^2}(t,x)+u(t)\mu(x)w(t,x), \quad x\in(0,1),\;t\in(0,T), \\ &\frac{\partial w}{\partial x}(t,0)=\frac{\partial w}{\partial x}(t,1)=0,\end{aligned}\right. \tag{1} \]
where \(\mu\in W^{2,\infty}((0,1),\mathbb R)\). It is proved that under generic assumptions on \(\mu\), the system (1) is locally controllable around the reference trajectory (\(w(t,x)=1,u(t)=0\)) if and only if \(T>2\). The cases \(T<2\) and \(T=2\) are also analyzed, proving negative results.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35L05 Wave equation
93B05 Controllability
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