Controlling Multiparticle System on the Line, II - Periodic Case

As in [7] we consider classical system of interacting particles P_1, ... , P_n on the line with only neighboring particles involved in interaction. On the contrast to [7] now periodic boundary conditions are imposed onto the system, i.e. P_1 and P_n 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 non-periodic counterpart. Main result established is global controllability by means of single controlling force of the multiparticle system with ageneric 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.