Physics Letters A 353 (2006) 24–29 www.elsevier.com/locate/pla Dynamics and control of oscillations in a complex crystalline lattice Eron Aero, Alexander Fradkov, Boris Andrievsky ∗ , Sergey Vakulenko Institute for Problems of Mechanical Engineering of Russian Academy of Sciences, 61 Bolshoy ave., V.O., 199178, Saint Petersburg, Russia Received 7 March 2005; received in revised form 19 December 2005; accepted 20 December 2005 Available online 28 December 2005 Communicated by A.P. Fordy Abstract A highly nonlinear system of acoustic and optical oscillations in a complex crystalline lattice consisting of two sublattices is analyzed. The system is obtained as a generalization of the linear Carman–Born–Kun Huang theory. Large displacements of atoms up to structure stability loss and restructuring are admitted. It is shown that the system has nontrivial solutions describing movements of fronts, emergence of periodic structures and defects. Strong interaction of acoustic and optical modes of oscillation for media without center of symmetry is demonstrated. A possibility of energy-excitation of the optical mode by means of controlling torque applied to the ends of the lattice is examined. Control algorithm based on speed-gradient method is proposed and analyzed numerically. Simulation results demonstrate that application of control may eliminate or reduce influence of initial conditions. An easily realizable nonfeedback version of control algorithm is proposed possessing similar properties.  2005 Elsevier B.V. All rights reserved. PACS: 61.66.D; 63.70 Keywords: Energy control; Speed gradient; Complex lattice; Nonlinear oscillations; Shock waves 1. Introduction Nonlinear lattice or chain models of spatially distributed systems have been used in various areas of physics for a long time, e.g., Frenkel–Kontorova model, Fermi–Pasta–Ulam model, Toda lattice and others [1,2]. Models of such kind, al- lowing to describe complex behavior due to interaction between different modes arise in physics of plasma [3,4], macromolec- ular systems [5,6], etc. Properties of complex oscillatory sys- tems such as atomic lattices are determined by interaction of large number of degrees of freedom. Studying such properties as structure and phase transitions, formation of defects, shock waves and others requires consideration of strongly nonlinear * Corresponding author. Tel.: +7 812 321 4766; fax: +7 812 321 4771. E-mail addresses: aero@microm.ipme.ru (E. Aero), alf@control.ipme.ru (A. Fradkov), bandri@control.ipme.ru (B. Andrievsky). URLs: http://www.ipme.ru/ipme/labs/microm/ (E. Aero), http://www.ipme.ru/ipme/labs/ccs/ (A. Fradkov), http://www.ipme.ru/ipme/labs/ccs/ (B. Andrievsky). phenomena. Some nonlinear effects may arise spontaneously and can be studied based on free oscillations theory. However, purposeful changes of the crystal state demand for development of the methods of controlling its properties, particularly, control of its nonlinear oscillations. Though control of spatially distrib- uted systems, including control of spatio-temporal chaos has been studied in quite a number of works since the 1960s [7], see references in [8,9], controlled excitation of a specific mode in solid state was not considered previously. In the paper [10] a new model for interaction of acoustic and optical modes in complex crystalline lattice was proposed and analyzed. It was shown that for quadratic potential func- tion describing interaction of atoms in an elementary cell, the proposed model is reduced to the classical Carman–Born–Kun Huang model [12]. This Letter is devoted to further analysis of the model of [10] nonlinear dynamics, as well as to demonstration of possibility of purposeful change of the system properties by means of con- trolling it. In Section 2 the model of [10] is briefly recalled. Some non- linear effects in the model of [10] are studied in Sections 3, 4. 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.12.051