Exact controllability of a hybrid system, membrane with strings, on general polygon domains (Q1972764)

The author uses very effectively the Hilbert uniqueness method (HUM) pioneered in the late 1980s by J. E. Lagnese, and J. L. Lions and used also by R. Triggiani, I. Lasiecka and their collaborators. The proof of Hilbert concerning uniqueness of solutions to PDEs extended to certain types of inhomogeneous equations with certain types of inner product, was cleverly applied to control theory problems, and in particular to inverse problems of boundary controllability. In this paper, the author considers a hybrid system of a convex polygonal domain occupied by a membrane, which is reinforced on its boundary by strings. The membrane equation \(u_{tt}-\Delta u=0\), valid in \(\Omega\), the string equation: \(\rho u_{tt}- \beta\partial^2 u/\partial\tau^2-\partial u/\partial\nu= 0\) valid on the boundary part \(\Gamma_1\) of \(\partial\Omega\), \(\partial u/\partial\nu= 0\) on the boundary part \(\Gamma_2\), and \(u= 0\) on \(\Gamma_3\), respectively, define the mathematical model for the time period \([0,T]\). A Neumann-type boundary control is considered. For \(T\) sufficiently large (as is usually the case for boundary input) the author derives an energy inequality for the inverse boundary control problem, subject to some geometric constraints. This result establishes exact boundary controllability. He also considers a Carleman-type interior energy inequality for Neumann controls, and shows that for certain types of geometric conditions, the unique solution vanishes at a given, but sufficiently large, time \(T\).