Notes on the Control of the Liouville Equation

摘要

In these notes we motive the study of Liouville equations having control terms using examples from problem areas as diverse as atomic physics (NMR),biological motion control and minimum attention control. On one hand,the Liouville model is interpreted as applying to multiple trials involving a single system and on the other,as applying to the control of many identical copies of a single system; e.g.,control of a flock. We illustrate the important role the Liouville formulation has in distinguishing between open loop and feedback control. Mathematical results involving controllability and optimization are discussed along with a theorem establishing the controllability of multiple moments associated with linear models. The methods used succeed by relating the behavior of the solutions of the Liouville equation to the behavior of the underlying ordinary differential equation,the related stochastic differential equation,and the consideration of the related moment equations.