Feedback Stabilization of Solitons and Phonons Using the Controlled Lax Form

We consider the problem of asymptotically stabilizing a desired family of soliton solutions of a completely integrable system. We proceed from the Lax form of the system, to which we add a suitable control term. We present a framework for making a Liouville torus, on which a particular multi-soliton lies, a global attractor. The method applies tools from nonlinear control theory to a controlled Lax form, augmented by a set of output functions. In principle there may be a large number of such outputs, but in practice we observe that it is often sufficient to use only a small subset. For the periodic Toda lattice the number of outputs should be equal to the number of lattice elements, but for lattices of up to 50 elements we show reasonable performance for as few as four output functions. We also apply the method to a discretization of the Korteweg-de Vries equation for which a complete set of independent invariant functions may be arbitrarily large, and observe reasonable performance using only three outputs. Finally we apply our results for the Toda lattice to an important potential application-the control of thermal transport at the nanoscale.