Exact controllability for the semilinear string equation in non cylindrical domains. (English) Zbl 1166.93310
Summary: We investigate the exact controllability for a mixed problem for the equation
\[
u^{\prime \prime} - [\frac{\tau_0}{m} + \frac{k}{m} \frac{\gamma(t)-\gamma_0}{\gamma_0}]u_{xx}+f(u)=0
\]
in a non cylindrical domain. This model, without the resistance represented for \(f(u)\), is a linearization of Kirchhoff’s equation for small vibrations of a stretched elastic string when the ends are variables, see Medeiros, Limaco, Menezes (2002). We employ a variant, due to Zuazua (1990b), of the Hilbert Uniqueness Method (HUM), idealized by Lions (1988a, b).