The control problem for a high-order evolutionary equation (Q1896848)

In the paper the author considers the hyperbolic problem \[ \begin{aligned} & {\partial^2u \over \partial t^2} + \Delta^{2m} u + \sum_{|\alpha |\leq 2m - 1} (- 1)^{|\alpha |} D^\alpha \bigl( a_\alpha (x) D^\alpha u \bigr) = 0 \quad \text{on } \Omega \times (0,T) \\ & u(x,0) = u_0(x) \quad \text{on } \Omega \\ & {\partial u \over \partial t} (x,0) = u_1(x) \quad \text{on } \Omega \\ & {\partial^ku \over \partial \nu^k} = 0 \quad \text{on } \partial \Omega \times (0,T)\;(k = 0, \dots, 2m - 2) \\ & {\partial \Delta^{m - 1} u \over \partial \nu} = \begin{cases} v \quad \text{on } \Gamma (x_0) \times (0,T) \\ 0 \quad \text{on } \bigl( \partial \Omega \backslash \Gamma (x_0) \bigr) \times (0,T) \end{cases} \end{aligned} \] where \(\Gamma (x_0)\) is the subset of \(\partial \Omega\) defined by \(\Gamma (x_0) = \{x \in \partial \Omega : (x - x_0) \cdot \nu \geq 0\}\), \(x_0 \in \mathbb{R}^n\) being given. The main result is that under suitable conditions on the coefficients \(a_\alpha (x)\) if \(T > 0\) is large enough, then for any initial data \(u_0 \), \(u_1\) there exists a control function \(v\) which drives the solution \(u\) to \[ u(x,T) = {\partial u \over \partial t} (x,T) = 0. \]