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Higher order Goh conditions for singular extremals of corank 1

Francesco Boarotto, Roberto Monti, Alessandro Socionovo

Abstract

We prove Goh conditions of order n for strictly singular length minimizing curves of corank 1, under the assumption that the lower order intrinsic differentials of the end-point map vanish. This result relies upon the proof of an open mapping theorem for maps with non-singular nth differential.

Higher order Goh conditions for singular extremals of corank 1

Abstract

We prove Goh conditions of order n for strictly singular length minimizing curves of corank 1, under the assumption that the lower order intrinsic differentials of the end-point map vanish. This result relies upon the proof of an open mapping theorem for maps with non-singular nth differential.
Paper Structure (10 sections, 34 theorems, 234 equations)

This paper contains 10 sections, 34 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Intrinsic differentials
  3. Open mapping theorem of order $n$
  4. Integrals on simplexes
  5. Expansion of the end-point map
  6. Open mapping property for the extended end-point map
  7. Generalized Jacobi identities and integrals on simplexes
  8. Non-singularity via trigonometric functions
  9. Goh conditions of order $n$ in the corank 1 case
  10. An example of singular extremal

Key Result

Theorem 1.1

Let $(M,\Delta,g)$ be a sub-Riemannian manifold and $\gamma=\gamma_u \in AC ( I ; M)$ be a strictly singular length-minimizing curve of corank $1$. If $\mathrm{dom}(\mathcal{D}_u ^nF)$, $n\geq 3$, has finite codimension then any adjoint curve $\lambda \in AC( I ; T^*M)$ satisfies for all $t\in I$ and for all $j_1,\dots,j_n=1,\dots,d$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: Faà di Bruno
  • Definition 2.2: Intrinsic $n$th differential
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: Domain with finite codimension
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 65 more