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This paper is concerned with networks of interacting linear systems at sitesof a multidimensional lattice. The systems are governed by linear ODEs withconstant coefficients driven by external inputs, and their internal dynamicsand coupling with the other component systems are translation invariant. Suchsystems occur, for example, in finite-difference models of large-scale flexiblestructures manufactured from homogeneous materials. Using the spatio-temporaltransfer function of this translation invariant network, we establishconditions for its positive realness in the sense of energy dissipation. Thelatter is formulated in terms of block Toeplitz bilinear forms of the input andoutput variables of the composite system. We also discuss quadratic stabilityof the network in isolation from the environment and phonon theoreticdispersion relations.

Spatio-temporal transfer function conditions of positive realness for translation invariant lattice networks of interacting linear systems