# difference between non-holonomic and linear control systems

I already asked this question in the mathoverflow forum, but no one could answer. Probably it is not a question of mathematics but robotics.

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?