GOH Conditions for Minima of Nonsmooth Problems with Unbounded Controls

¹ Department of Information Engineering, Computer Science and Mathematics (DISIM), University of L’Aquila, L’Aquila, Italy
² Department of Mathematics “T. Levi-Civita”, University of Padova, Padova, Italy

^* Corresponding author: francescaangrisani@gmail.com

Received: 30 September 2022
Accepted: 28 December 2022

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.