Goh conditions for minima of nonsmooth problems with unbounded controls

TL;DR

This paper extends higher order necessary optimality conditions to control systems with Lipschitz dynamics and unbounded controls using set-valued Lie brackets and Quasi Differential Quotients, surpassing classical smoothness limitations.

Contribution

It introduces a Goh-type condition for nonsmooth control systems with unbounded controls, utilizing set-valued Lie brackets and Quasi Differential Quotients.

Findings

The new condition can exclude suboptimal controls that satisfy the maximum principle.

It applies to systems with Lipschitz continuous dynamics and unbounded controls.

An example demonstrates the condition's effectiveness in ruling out certain controls.

Abstract

Higher order necessary conditions for a minimizer of an optimal control problem are generally obtained for systems whose dynamics is continuously differentiable in the state variable. Here, by making use of the notion of set-valued Lie bracket we obtain a Goh-type condition for a control affine system with Lipschitz continuous dynamics and unbounded controls. In order to manage the simultaneous lack of smoothness of the adjoint equation and of the Lie bracket-like variations we make use of the notion of Quasi Differential Quotient. We conclude the paper with a worked out example where the established higher order condition is capable to rule out the optimality of a control verifying the standard maximum principle.