Simultaneous control problems for systems of elastic strings and beams

Controllability problems for N strings or beams controlled from a common en dpoint are studied. We give necessary and sufficient conditions for simulta neous spectral and approximate controllability and describe the space of si multaneously reachable states as a function of the position of the endpoint . For each type of controllability result the sharp controllability time is obtained. The proofs are based on new results in nonharmonic Fourier serie s which describe Riesz bases of exponential divided differences. (C) 2001 P ublished by Elsevier Science B.V.