在线控制多粒子系统，II -周期情况
Controlling Multiparticle System on the Line, II - Periodic case
摘要：
在arXiv中：数学0809.2365我们考虑线上的相互作用粒子$\mathcal{P}_1, ..., \mathcal{P}_n$的经典系统，其中只有相邻粒子参与相互作用。与arXiv相比：现在将{ \it周期边界条件}施加到系统上，即认为$\mathcal{P}_1$和$\mathcal{P}_n$是相邻的。周期Toda晶格就是一个典型的例子。我们研究了通过对周期性多粒子系统中的少数粒子施加力来控制它的可能性，主要是研究由单力控制的系统。多粒子系统在周期和非周期情况下的自由动力学有很大的不同。我们看到，受控周期多粒子系统也不模仿它的非周期对应物。主要结果是建立了具有广义相互作用势的多粒子系统的单控制力全局能控性。我们研究的非通用的潜力，可控性和可访问性可能缺乏。在第2，3节中给出了结果的公式化和证明。
英文摘要：
As in arXiv: math. 0809.2365 we consider classical system of interacting particles $\mathcal{P}_1, ..., \mathcal{P}_n$ on the line with only neighboring particles involved in interaction. On the contrast to arXiv: math. 0809.2365 now {\it periodic boundary conditions} are imposed onto the system, i.e. $\mathcal{P}_1$ and $\mathcal{P}_n$ 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.
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