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Goh and Legendre-Clebsh conditions for nonsmooth control systems

Francesca Angrisani, Franco Rampazzo

TL;DR

This work extends classical higher-order necessary conditions for optimal control to nonsmooth, control-affine systems by leveraging set-valued Lie brackets and the Quasi Differential Quotient (QDQ) framework. The authors prove a nonsmooth maximum principle that includes Goh and Legendre–Clebsch type conditions expressed via set-valued brackets, valid for Lipschitz dynamics with interior controls. The proof blends mollification of vector fields, QDQ calculus, and a Cantor intersection argument to handle an infinite family of control variations, culminating in a rigorous nonsmooth counterpart to the smooth theory. A concrete example demonstrates the usefulness of the step-3 (LC3) condition in ruling out nonoptimal candidates that satisfy first-order conditions. Overall, the paper broadens the applicability of higher-order optimality conditions to nonsmooth dynamics with practical implications for verifying optimality in complex control systems.

Abstract

Higher order necessary conditions for a minimizer of an optimal control problem are generally obtained for systems whose dynamics is at least continuously differentiable in the state variable. Here, by making use of the notion of set-valued Lie bracket introduced in "Set-valued differentials and a nonsmooth version of Chow-Rashevski's theorem" by F. Rampazzo and H.J.Sussmann and extended in "Iterated Lie brackets for nonsmooth vector fields" by E. Feleqi and F.Rampazzo , we obtain Goh and Legendre-Clebsh type conditions for a control affine system with Lipschitz continuous dynamics. In order to manage the simultaneous lack of smoothness of the adjoint equation and of the Lie bracket-like variations, we will exploit the notion of Quasi Differential Quotient, introduced in "A geometrically based criterion to avoid infimum-gaps in Optimal Control" by M. Palladino and F.Rampazzo. We finally exhibit an example where the established higher order condition is capable to rule out the optimality of a control verifying a first order Maximum Principle.

Goh and Legendre-Clebsh conditions for nonsmooth control systems

TL;DR

This work extends classical higher-order necessary conditions for optimal control to nonsmooth, control-affine systems by leveraging set-valued Lie brackets and the Quasi Differential Quotient (QDQ) framework. The authors prove a nonsmooth maximum principle that includes Goh and Legendre–Clebsch type conditions expressed via set-valued brackets, valid for Lipschitz dynamics with interior controls. The proof blends mollification of vector fields, QDQ calculus, and a Cantor intersection argument to handle an infinite family of control variations, culminating in a rigorous nonsmooth counterpart to the smooth theory. A concrete example demonstrates the usefulness of the step-3 (LC3) condition in ruling out nonoptimal candidates that satisfy first-order conditions. Overall, the paper broadens the applicability of higher-order optimality conditions to nonsmooth dynamics with practical implications for verifying optimality in complex control systems.

Abstract

Higher order necessary conditions for a minimizer of an optimal control problem are generally obtained for systems whose dynamics is at least continuously differentiable in the state variable. Here, by making use of the notion of set-valued Lie bracket introduced in "Set-valued differentials and a nonsmooth version of Chow-Rashevski's theorem" by F. Rampazzo and H.J.Sussmann and extended in "Iterated Lie brackets for nonsmooth vector fields" by E. Feleqi and F.Rampazzo , we obtain Goh and Legendre-Clebsh type conditions for a control affine system with Lipschitz continuous dynamics. In order to manage the simultaneous lack of smoothness of the adjoint equation and of the Lie bracket-like variations, we will exploit the notion of Quasi Differential Quotient, introduced in "A geometrically based criterion to avoid infimum-gaps in Optimal Control" by M. Palladino and F.Rampazzo. We finally exhibit an example where the established higher order condition is capable to rule out the optimality of a control verifying a first order Maximum Principle.
Paper Structure (27 sections, 23 theorems, 173 equations)

This paper contains 27 sections, 23 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. The problem and the main result
  3. Organization of the paper
  4. The main result
  5. A simple example
  6. Quasi Differential Quotients
  7. $QDQ$ approximating cones and set separation
  8. Proof of Theorem 3.2
  9. Variations
  10. Composing multiple variations
  11. Set-valued Lie brackets as $QDQ$s
  12. Finitely many variations
  13. Infinitely many variations
  14. Some possible generalizations
  15. Appendix
  16. ...and 12 more sections

Key Result

Theorem 2.2

\newlabelTeoremaPrincipale0 Let us assume that $f, g_1, \ldots, g_m \in C^{0,1}_{\mathrm loc} (\mathbb{R}^n,\mathbb{R})$ and $\Psi \in C^{0,1}_{\mathrm loc}(\mathbb{R}^n,\mathbb{R})$. Let $(T,\overline{u},\overline{x})$ be a singular, local $L^1$ minimizer for problem $(P)$, and let $\mathfrak{C}$ Moreover, the following higher order conditions are verified:

Theorems & Definitions (50)

  • Definition 2.1
  • Theorem 2.2: A non-smooth Maximum Principle with Goh and Legendre–Clebsch conditions
  • Definition 3.1: Quasi Differential Quotient
  • Proposition 3.2: Chain rule
  • Proposition 3.3
  • Definition 3.4
  • Lemma 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • ...and 40 more