Synchronization Problems for Spatially Invariant Infinite Dimensional Linear Systems

This paper presents an overview of my work with Bruce Francis on asymptotic behavior of linear systems of countably many kinematic points with “nearest neighbor” dynamics. Both first and second order systems are considered. The novelty of the results considered here is that, unlike previous work in this area where the state space was a Hilbert sequence (or function) space, the state space is the Banach sequence space of bounded doubly infinite scalar sequences with the standard supremum norm. The basic problem lying at the heart of the theory for first order systems is the “serial pursuit and rendezvous problem.” Unlike the case of finitely many points where the asymptotic behavior of the system is straightforward, for infinitely many points the asymptotic behavior of the system connects with the classical study of Borel summability of sequences. The symmetric synchronizations problems are dependent on determining the subspace of initial configurations which give convergence in the serial pursuit problem. The finite dimensional version of the infinite second order system we study arises in physics in the theory of phonons, in the simplest case of one-dimensional lattice dynamics. We compare the asymptotic behavior of the finite system case to the infinite system one, both for undamped and damped systems. The results are quite unexpected. Despite the fact that the system is unbounded there are many cases where, asymptotically, synchronization takes place both in the damped and undamped case.