Controlling Multiparticle System on the Line. II - Periodic case

As in [A. Sarychev, Controlling multiparticle system on the line. I, J. Differential Equations 246 (12) (2009) 4772–4790] we consider classical system of interacting particles P1, . . . ,Pn on the line with only neighboring particles involved in interaction. On the contrast to [A. Sarychev, Controlling multiparticle system on the line. I, J. Differential Equations 246 (12) (2009) 4772–4790] now periodic boundary conditions are imposed onto the system, i.e. P1 and Pn are considered neighboring. Periodic Toda lattice would be a typical example. We study possibility to control periodic multiparticle systems by means of forces applied to just few of its particles; mainly we study system controlled by single force. The free dynamics of multiparticle systems in periodic and nonperiodic case differ substantially. We see that also the controlled periodic multiparticle system does not mimic its nonperiodic counterpart. Main result established is global controllability by means of single controlling force of the multiparticle system with a generic potential of interaction. We study the nongeneric potentials for which controllability and accessibility properties may lack. Results are formulated and proven in Sections 2, 3.