Controlling Mean and Variance in Ensembles of Linear Systems

Abstract: We investigate the possibilities of steering probability density functions of state variables in linear control systems, using a combination of open loop and time-varying output feedback control strategies. This is an intrinsically nonlinear control problem which makes contact with earlier work by Brockett (2012) on controlling the mean and variance of linear systems via time-varying state feedback transformations. We extend Brockett’s work on the control of the Liouville equation to the more difficult output feedback case, as well as to parallel connected linear systems. Our methods depend on certain controllability results for bilinear systems Brockett (1976), Dirr et al. (2016), where the controls are defined by the output feedback gain.

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