Editing Controllability (section)

== Collective controllability -- Control of the state transition via feedback ==

Collective controllability represents the ability to steer <math>n</math> linear dynamical systems that obey identical dynamics

: <math>\dot{\mathbf{x}}^{(i)}(t) = A \mathbf{x}^{(i)}(t) + B \mathbf{u}^{(i)}(t)</math>

where <math>n</math> equals the dimension of <math>\mathbf{x}</math>, 
between specified starting and ending configurations by way of a common state feedback gain matrix <math>K(t)</math>, and thereby, each instantiating a control input

: <math>\mathbf{u}^{(i)}(t)=K(t){\mathbf{x}}^{(i)}(t) </math>

for <math>i\in\{1,\ldots,n\}</math>, respectively.

The accessibility distribution <math>R</math> having full column rank is trivially a necessary condition. It is also sufficient, and in fact, the collective is strongly controllable, in that it can be steered from an initial
configuration

: <math>\Phi(0)= \begin{bmatrix} \mathbf{x}^{(1)}(0)\ldots \mathbf{x}^{(n)}(0) \end{bmatrix} </math>

to any specified terminal configuration

: <math>\Phi(T)= \begin{bmatrix} \mathbf{x}^{(1)}(T)\ldots \mathbf{x}^{(n)}(T) \end{bmatrix}, </math>
provided <math>\det(\Phi(0)\Phi(T))>0 </math>, over any specified time interval <math>[0,T] </math>  through a choice of a common time-varying feedback gain matrix <math>K(t)</math> provided <math>R</math> has full column rank
<ref>{{cite journal  |author1=Mahmoud Abdelgalil |author2=Tryphon T. Georgiou | title = Collective Steering in Finite Time: Controllability on <math>GL^+(n,\mathbb{R})</math>  | journal = IEEE Transactions on Automatic Control | doi = 10.1109/TAC.2025.3574186 | year = 2025  | url = https://ieeexplore.ieee.org/document/11015961  | access-date = 2025-27-05 }}</ref>