Microlocal normal forms for regular fully nonlinear two-dimensional control systems

In the present paper we deal with fully nonlinear two-dimensional smooth control systems with scalar input $\dot q=\mathbf f(q,u)$, $q\in M$, $u\in U$, where $M$ and $U$ are differentiable smooth manifolds of respective dimensions two and one. For such systems, we provide two microlocal normal forms, i.e., local in the state-input space, using the fundamental necessary condition of optimality for optimal control problems: the Pontryagin maximum principle. One of these normal forms will be constructed around a regular extremal, and the other one will be constructed around an abnormal extremal. These normal forms, which in both cases are parametrized only by one scalar function of three variables, lead to a nice expression for the control curvature of the system. This expression shows that the control curvature, a priori defined for normal extremals, can be smoothly extended to abnormals.