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

TL;DR

This paper develops microlocal normal forms for two-dimensional nonlinear control systems, enabling a unified expression for control curvature that extends smoothly from normal to abnormal extremals using Pontryagin's Maximum Principle.

Contribution

It introduces two local microlocal normal forms around regular and abnormal extremals for 2D nonlinear control systems, linking control curvature to these forms.

Findings

Normal forms are parametrized by one scalar function of three variables.

Control curvature can be smoothly extended to abnormal extremals.

Normal forms facilitate analysis of control systems near extremals.

Abstract

In the present paper we deal with fully nonlinear two-dimensional smooth control systems with scalar input $\dot{q} = \bs{f}(q,u)$, $q \in M$, $u \in U$, 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.