Ensemble control of linear systems

In this article, we study ensemble control problems which involve controlling a continuum of dynamical systems with different dynamics by using the same control signal. In particular, we look into ensemble control of linear systems. From the standpoint of mathematical control theory, the challenge is to simultaneously steer a continuum of systems between points of interest with the same control signal. This raises some new and unexplored questions about controllability and optimal control of such systems. We analyze ensemble controllability and optimal control of linear systems and study in detail the problem of controlling an uncountable family of harmonic oscillators. We show how the ideas of polynomial approximation are in connection with the ensemble controllability. We also find the optimal ensemble control for this linear ensemble control system.