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I'm struggling with the definitions and differences between a non-holonomic and a linear control systems.

I'm working with a phase space $P\subset R^{m}$ and a control space $U\subset R^{l}$ and my control system

$\dot{p}=f(p,u)$

is of the form

$\dot{p}=\sum^{l}_{i=1}u_{i}P_{i}(p)$

where $P_{1},\dots,P_{l}$ are smooth vector fields on $P$.

In Frederic Jeans book Definition 1.1 a control system of this form is called non-holonomic. I already read that there is the adjective "linear" which would fit better in my opinion since whenever I use the form of the control system I use the fact that it depends linearly on the control.

Can someone advice me literature where it is stated like that?

When I read more about non-holonomicity I found that it is a question whether the vectorfields $P_{i}$ span the whole $TP$.

Can someone explain how linearity and non-holonomicity are connected?

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