Boundary controllability of systems of connected strings and vector exponential families

Boundary controllability problems are considered for a class of systems described by hyperbolic equations for vector functions of one spatial variable.. Using the Fourier method, the study of controllability reduces to the investigation of families of vector exponentials {η_n exp (iω_nt )}, with ω_n being the eigenfrequencies of the system and η_n being the traces of eigenfunction derivatives at the boundary points where control is applied. It is known that, in the case of vector exponentials, the asymptotics is not by itself sufficient for the investigation of the minimality and the basis property. There is essentially only one way to examine vector exponential families, viz. to construct and study the generating function (GF). The problem under consideration naturally gives rise to the GF, since it is expressed via the solution of ordinary differential equations of the Helmholtz type. A system of strings connected elastically at one point and the control of a multichannel acoustic system are considered as examples