  This paper is concerned with networks of interacting linear systems at sites
of a multidimensional lattice. The systems are governed by linear ODEs with
constant coefficients driven by external inputs, and their internal dynamics
and coupling with the other component systems are translation invariant. Such
systems occur, for example, in finite-difference models of large-scale flexible
structures manufactured from homogeneous materials. Using the spatio-temporal
transfer function of this translation invariant network, we establish
conditions for its positive realness in the sense of energy dissipation. The
latter is formulated in terms of block Toeplitz bilinear forms of the input and
output variables of the composite system. We also discuss quadratic stability
of the network in isolation from the environment and phonon theoretic
dispersion relations.

Systems and Control;Dynamical Systems;Optimization and Control