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**Exact controllability of the wave equation with mixed boundary condition and time-dependent coefficients.**
*(English)*
Zbl 1046.35013

The author studies the wave equation with time dependent coefficients \((\alpha (t) y')' - A(t)y = 0\), where \(A(t)\) is an elliptic operator of the special form \(A(t) = \sum _{j=1}^{n}\frac {\partial }{\partial x_j} (\beta (t) a(x) \frac {\partial y}{\partial x_j})\), with initial conditions \(y(0)= y^0, \;y'(0) = y^1\). The equation is considered on a bounded domain in \(\mathbb{R}^n\) of the form \(\Omega = \Omega _0 \setminus \Omega _1\) with boundary having two disjoint components \(\Gamma _0, \;\Gamma _1\). The homogeneous Dirichlet condition is imposed on the inner boundary \(\Gamma _1\), while on \(\Gamma _0\) a Neumann condition is seeked to provide a control in the sense that for time \(T\) large enough the solution satisfies \(y(T)=y'(T) = 0\). The main result states that under natural assumptions on the coefficients and initial conditions there exists a time \(T_0 > 0\) such that for every \(T>T_0\) and initial data \(y^0, \;y^1\) there is such a control. The result is obtained by applying Hilbert Uniqueness Method, which was used previously by J. L. Lions to prove controllability in the case \(\alpha (t) = \beta (t) = a(x) = 1\).

Reviewer: Martin Kolář (Brno)

### MSC:

35B40 | Asymptotic behavior of solutions to PDEs |

35B35 | Stability in context of PDEs |

35L99 | Hyperbolic equations and hyperbolic systems |