The control problem for equations correct in Petrovskij's sense (Q1278915)

Using the Hilbert uniqueness method introduced by J. L. Lions, the author proves the exact controllability for the following evolution equation \(u_{tt}+L^{2m}u+L_{1}u=0\) in \(\Omega \times (0,T),\) where \(m\geq 0,\) \[ Lu=\sum_{i,j=1}^{n}{{\partial }\over{\partial x_{j}}}(a_{ij}(x) {{\partial u }\over{\partial x_{j}}}) \quad\text{and}\quad L_{1}u= \sum_{| \alpha | \leq 2m-1}(-1)^{| \alpha | }D^{\alpha }(a_{\alpha }(x)D^{\alpha }u), \] with initial and boundary conditions \(u(0)=u^{0},\) \(u'(0)=u^{1},\) on \(\Omega ,\) \({{\partial ^{k}u}\over{{\partial \nu ^{k}}}}| _{\Sigma }=0,\) \(k=0,\ldots ,2m-2,\) \({{\partial L^{m-1}u}\over {\partial \nu _{L}}}| _{\Sigma }=v\) on \(\Gamma (x^{0})\times (0,T),\) while vanishing on \( \Gamma _{*}(x^{0})\times (0,T).\) Here \(\Omega \subset {\mathbb R}^{n}\) is a bounded domain with smooth boundary \(\Gamma ,\) \(T>0,\) \(\nu \) is the unit exterior normal vector to \(\Gamma ,\) \(\nu _{L}\) a normalized vector, \(\Gamma (x^{0})\) and \(\Gamma _{*}(x^{0})\) being two parts of \( \Gamma \) and \(v\) the control function.