# Microlocal normal forms for regular fully nonlinear two-dimensional control systems - Mathematics > Optimization and Control

Abstract: In the present paper we deal with fully nonlinear two-dimensional smoothcontrol systems with scalar input $\dot{q} = \bs{f}q,u$, $q \in M$, $u \inU$, where $M$ and $U$ are differentiable smooth manifolds of respectivedimensions two and one. For such systems, we provide two microlocal normalforms, i.e., local in the state-input space, using the fundamental necessarycondition of optimality for optimal control problems: the Pontryagin MaximumPrinciple. One of these normal forms will be constructed around a regularextremal and the other one will be constructed around an abnormal extremal.These normal forms, which in both cases are parametrized only by one scalarfunction of three variables, lead to a nice expression for the controlcurvature of the system. This expression shows that the control curvature, apriori defined for normal extremals, can be smoothly extended to abnormals.

Author: Ulysse Serres INRIA Lorraine - Iecn - Mmas

Source: https://arxiv.org/