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Higher order necessary conditions for optimal controls not ranging in the interior

Francesca Angrisani, Franco Rampazzo

TL;DR

The paper extends classical higher-order necessary conditions for optimal controls to affine systems when the minimizing control may lie on the boundary of the control set. By introducing the concept of $i$-balanced controls and leveraging a Lie-bracket–driven variation framework, it derives a higher-order maximum principle that includes $(i,j)$-Goh conditions, $(i)$-Legendre–Clebsch conditions, and, in the single-input case, a third-order condition involving $[g,[f,g]]$. The results are established via a set-separation argument using Boltyanski approximating cones and a detailed construction of finite and infinite families of variations, culminating in multiplier conditions that subsume the classical interior control case. The work broadens the applicability of singular-control theory to problems with target sets that may exclude interior controls, enabling more robust optimality analysis in noninnerable control landscapes.

Abstract

Goh's and Legendre-Clebsch necessary conditions for optimal controls of affine-control systems are usually established under the hypothesis that the minimizing control lies in the interior of the control set $U$. In this paper we investigate the possibility of establishing Goh's and Legendre-Clebsch necessary conditions without this assumption, so that even control sets with empty interiors or optimal controls touching the boundary of $U$ can be taken into consideration.

Higher order necessary conditions for optimal controls not ranging in the interior

TL;DR

The paper extends classical higher-order necessary conditions for optimal controls to affine systems when the minimizing control may lie on the boundary of the control set. By introducing the concept of -balanced controls and leveraging a Lie-bracket–driven variation framework, it derives a higher-order maximum principle that includes -Goh conditions, -Legendre–Clebsch conditions, and, in the single-input case, a third-order condition involving . The results are established via a set-separation argument using Boltyanski approximating cones and a detailed construction of finite and infinite families of variations, culminating in multiplier conditions that subsume the classical interior control case. The work broadens the applicability of singular-control theory to problems with target sets that may exclude interior controls, enabling more robust optimality analysis in noninnerable control landscapes.

Abstract

Goh's and Legendre-Clebsch necessary conditions for optimal controls of affine-control systems are usually established under the hypothesis that the minimizing control lies in the interior of the control set . In this paper we investigate the possibility of establishing Goh's and Legendre-Clebsch necessary conditions without this assumption, so that even control sets with empty interiors or optimal controls touching the boundary of can be taken into consideration.
Paper Structure (19 sections, 14 theorems, 135 equations)

This paper contains 19 sections, 14 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Remarks on the notation
  3. The main result
  4. An worked out example of not optimal control
  5. Variations
  6. Variation builders
  7. Primitives of the variation generators $\gamma_{i,j},\gamma_{0,i}$ $\gamma_{1,1,0}$
  8. Control variations
  9. Approximation of solutions
  10. The case with sub-linear controls
  11. Proof of Theorem 3.7
  12. Proof of 2) in Theorem \ref{['eccolestime']}
  13. Proof of 3) in Theorem \ref{['eccolestime']}
  14. Proof of 4) in Theorem \ref{['eccolestime']}
  15. Proof of Theorem 2.3
  16. ...and 4 more sections

Key Result

Theorem 2.3

\newlabelTeoremaPrincipale0 Let $(\overline{u},\overline{x})$ be a local weak minimizer of problem (P), and let us set Furthermore, let $C$ be the Boltyanski approximating cone to the target ${\mathfrak{T}}$ at $\overline{x}(T)$. Then there exist multipliers $(p,\lambda) \in AC([0,T],(\mathbb{R}^n)^*) \times \mathbb{R^*}$, with $\lambda\geq 0$ such that the following properties are satisfied:

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: A higher order maximum principle
  • Remark 2.4
  • Example 2.5
  • Definition 3.1: Variation signals
  • Definition 3.2: Variation builders
  • Lemma 3.3
  • Proof 1
  • Lemma 3.4
  • ...and 22 more