I'm new to ODE and currently working on an optimal control problem. I'm trying to calculate the trajectories of a wheeled robot. I can steer it only continously but in theory it makes also sense to allow $L^1$ steerings, because the initial value problem has also a local solution for them (existencetheorem of caratheodory).
My dynamical system is nonholonomic with smooth and Lipschitz vectorfields $P_1,\dots,P_l: R^3 \rightarrow R^3$ and controls $u:[0,T]\rightarrow R^l$
$\dot{p}(t)=\sum^l_{i=1}P_i(p(t))u_i(t)$
I want to show that it is for practical applications not relevant if I allow also $L^1$ functions, since $C^0([0,T],R^l)$ is dense in $L^1([0,T],R^l)$.
I found a stability theorem in the book of Bartels (3x9 Numerik 20.4) which states the fact that a small variation of continous dynamical systems results in a small variation of the solution trajectory, but I can't apply it to my problem since the dynamical system with the $L^1$ steering is not continous.
Does anyone know a more general result that I can use?