On Separating Points for Ensemble Controllability

On Numerical Examination of Uniform Ensemble Controllability for Linear Ensemble Systems

“…Remark 2.2. Although the matrix D(s) in (2.3) was called the "ensemble controllability matrix" for RR-ensembles in [13], we will introduce a new concept of ensemble controllability matrix for general R-ensembles in Section 3, which also involves information of the drift vector field A(•). Therefore, to avoid abuse of notations, we will refer the matrix D(s) in (2.3) as the s-separation matrix associated with the ensemble in (2.2).…”

“…Then, it is proved (see Theorem 4 in [13]) that the ensemble in (2.1), with its state variable X(t, •) ∈ C(K, R n ) defined in an infinite-dimensional space, is uniformly ensemble controllable if and only if the following system in (2.4), with its state variable (Z 1 , . .…”

“…Although much of the work has been done, the discussion on ensemble controllability of linear ensemble systems is not finished yet. In [13], a transparent view on UEC of real-valued linear ensemble systems under real-valued control inputs (R-ensembles) is provided by connecting UEC of an ensemble system to controllability of each subsystem in the ensemble. A sufficient and necessary condition on UEC of R-ensembles is given based on the separating properties of the control vector field.…”

“…Furthermore, since realistic systems are all steered by real-valued control inputs, the conditions in [4] appear to be restrictive and demanding for checking UEC of practical RC-ensembles. Aside from [13] and [4], a thorough investigation placed directly on UEC of RC-ensembles is missing in the literature.…”

“…The paper is organized as follows. In Section 2, we briefly summarize our previous work in [13] and provide preliminary knowledge of the ensemble control problem. In Section 3, leveraging Stone-Weierstrass theorem for modules, we propose a new algebraic framework to examine UEC of linear ensembles through checking controllability of subsystems in the ensemble.…”