Asymptotic behaviour of infinite chains of coupled kinematic points: second-order equations

The behaviour of infinite chains of coupled kinematic points is studied. The points are second order, that is, they have mass. The chains could have been designed in any number of ways, including linear-quadratic optimal control. Behaviour means what happens as time goes to infinity. It is not assumed that the initial state is in the Hilbert space l2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^{2}$$\end{document} because it has been seen in our previous work that in some situations this assumption has anomalous results. Instead, the initial state taken to be l∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^{\infty }$$\end{document}. The finite-dimensional version of the infinite second-order system we study arises in physics in the theory of phonons.