On exact controllability of networks of nonlinear elastic strings in 3-dimensional space

Abstract

This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions, and, by using the methods of quasilinear hyperbolic systems, prove that for tree networks the natural initial, boundary value problem has classical solutions existing in neighborhoods of the “stretched” equilibrium solutions. Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved. Finally, it is proved that, given two different equilibrium states satisfying certain conditions, it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a sufficiently large time interval.