Boundary control and a matrix inverse problem for the equation $u_{tt}-u_{xx}+V(x)u=0$

The authors solve the problem of recovering the matrix-valued potential $V(x)$, $x>0$, from the given reaction operator $R\colon u(0,t)\mapsto u_x(0,t)$, $t>0$. They show the connections between this problem and the theory of boundary control, which allows them to obtain analogues of the classical Gel'fand–Levitan–Krein equations. They establish the basis property for a family of vector-valued exponentials; this property is connected with the spectral characteristics of the boundary value problem. They prove the controllability of the corresponding system under a boundary control $u(0,t)=f(t)$.