Singular extremals of optimal control problems with $L^1$ cost

TL;DR

This paper analyzes singular extremals in optimal control problems with an $L^1$ cost, establishing conditions for their optimality and applying the theory to geometric examples.

Contribution

It derives a generalized Legendre-Clebsch condition for singular extremals and proves its necessity and sufficiency for local optimality in $L^1$ control problems.

Findings

Legendre-Clebsch condition is sufficient for local optimality.

Absence of conjugate points ensures strong optimality.

The condition is also necessary for optimality.

Abstract

We study the optimal control problem for a control-affine system, where we want to minimize the $L^1$ norm of the control. First, we show how Pontryagin Maximum Principle (PMP) applies to this problem and we divide the extremal trajectories into two categories: regular and singular extremals. Then, we obtain a strong generalized Legendre-Clebsch condition for singular extremals and we show that this condition together with the absence of conjugate points is sufficient to ensure local strong optimality. We provide also some geometric examples where we apply our results. Finally, we prove that generalized Legendre-Clebsch condition is necessary for optimality.