On the Minimum Time Function Around the Origin

TL;DR

This paper investigates the geometry of the minimum time function and reachable sets in control systems, extending convexity and regularity results from linear to certain nonlinear cases using linearization and bang-bang principles.

Contribution

It extends convexity and regularity properties of reachable sets and the minimum time function from linear to nonlinear control systems under normality assumptions.

Findings

Polynomial strict convexity of reachable sets for linear systems.

Extension of convexity results to nonlinear systems with small nonlinearity.

Proof of positive reach of the minimum time function's epigraph in nonlinear settings.

Abstract

We deal with finite dimensional linear and nonlinear control systems. If the system is linear and autonomous and satisfies the classical normality assumption, we improve the well known result on the strict convexity of the reachable set from the origin by giving a polynomial estimate. The result is based on a careful analysis of the switching function. We extend this result to nonautonomous linear systems, provided the time dependent system is not too far from the autonomous system obtained by taking the time to be 0 in the dynamics. Using a linearization approach, we prove a bang-bang principle, valid in dimensions 2 and 3 for a class of nonlinear systems, affine and symmetric with respect to the control. Moreover we show that, for two dimensional systems, the reachable set from the origin satisfies the same polynomial strict convexity property as for the linearized dynamics, provided the nonlinearity is small enough. Finally, under the same assumptions we show that the epigraph of the minimum time function has positive reach, hence proving the first result of this type in a nonlinear setting. In all the above results, we require that the linearization at the origin be normal. We provide examples showing the sharpness of our assumptions.