Abstract
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 \(l^{2}\) 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 \(l^{\infty }\). The finite-dimensional version of the infinite second-order system we study arises in physics in the theory of phonons.
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References
Bamieh B, Paganini F, Dahleh M (2002) Distributed control of spatially-invariant systems. IEEE Trans Autom Control 47(7):1091–1107
Bowman F (1958) Introduction to Bessel functions. Dover publications, New York
Brillouin L (2003) Wave propogation in periodic structures. Dover publications, New York
Curtain R, Iftime O, Zwart H (2009) System theoretic properties of a class of spatially distributed systems. Automatica 45:1619–1627
Curtain R, Iftime O, Zwart H (2010) A comparison between \(\{LQR\}\) control for a long string of \(\{SISO\}\) systems and \(\{LQR\}\) control of the infinite spatially invariant version. Automatica 46:1604–1615
D’Andrea R, Dullerud G (2003) Distributed control design for spatially interconnected systems. IEEE Trans Autom Control 48(9):1470–1495
Dove MT (1993) Introduction to lattice dynamics. Cambridge University Press, Cambridge
Feintuch A, Francis BA (2012) Infinite chains of kinematic points. Automatica 48:901–908
Feintuch A, Francis BA (2012) An infinite string of ants and Borel’s method of summability. Math Intell 34(2):15–18
Hille E, Phillips RS (1957) Functional analysis and semigroups, vol 31. A. M. S. Colloquium Publications, New York
Jovanovic M, Bamieh B (2005) On the ill-posedness of certain vehicular platoon control problems. IEEE Trans Autom Control 50(9):1307–1321
Kopell N, Ermentrout GB, Williams TL (1994) On chains of oscillators forced at one end. SIAM J Appl Mathe 51:1397–1417
Kurtze DA, Hong DC (1995) Traffic jams, granular flow, and soliton selection. Phys Rev E.52:218–221
Lin Z, Broucke M, Francis B (2004) Local control stratigies for groups of mobile automomous agents. IEEE Trans Autom Control 49(4):622–629
Luke YL (1962) Integrals of Bessel functions. McGraw Hill, New York
Melzer SM, Kuo BC (1971) Optimal regulation of systems described by a countably infinite number of objects. Automatica 7(3):359–366
Motee N, Jadbabaie A (2008) Optimal control of spatially distributed systems. IEEE Trans Autom Control 53(7):1616–1629
Swaroop D, Hedrick JK (1996) String stability of interconnected systems. IEEE Trans Autom Control 41(3):349–357
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Feintuch, A. Asymptotic behaviour of infinite chains of coupled kinematic points: second-order equations. Math. Control Signals Syst. 26, 463–480 (2014). https://doi.org/10.1007/s00498-014-0125-y
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DOI: https://doi.org/10.1007/s00498-014-0125-y