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Hessians in Birkhoff-Theoretic Trajectory Optimization

I. M. Ross

TL;DR

This work analyzes Hessians arising from universal Birkhoff-theoretic trajectory discretizations, revealing a two-block structure that separates data-dependent and data-independent components. By formulating a weighted Lagrangian linked to the Pontryagin Hamiltonian, it derives a Hamiltonian-form Birkhoff Hessian that is asymmetric yet tightly connected to optimal-control conditions. A Gershgorin-based eigenvalue theorem shows mesh-independent spectral properties, with about 80% of eigenvalues data-independent and confined to $[-2,4]$, guiding both stability and complexity considerations. Computationally, the data-dependent block scales linearly in grid size, while the data-independent block permits efficient, even matrix-free, solutions; Chebyshev-based approaches especially promise $\mathcal{O}(N_n \log N_n)$ time and $\mathcal{O}(N_n)$ space for million-point problems, suggesting a practical path toward real-time large-scale trajectory optimization.

Abstract

This paper derives various Hessians associated with Birkhoff-theoretic methods for trajectory optimization. According to a theorem proved in this paper, approximately 80% of the eigenvalues are contained in the narrow interval [-2, 4] for all Birkhoff-discretized optimal control problems. A preliminary analysis of computational complexity is also presented with further discussions on the grand challenge of solving a million point trajectory optimization problem.

Hessians in Birkhoff-Theoretic Trajectory Optimization

TL;DR

This work analyzes Hessians arising from universal Birkhoff-theoretic trajectory discretizations, revealing a two-block structure that separates data-dependent and data-independent components. By formulating a weighted Lagrangian linked to the Pontryagin Hamiltonian, it derives a Hamiltonian-form Birkhoff Hessian that is asymmetric yet tightly connected to optimal-control conditions. A Gershgorin-based eigenvalue theorem shows mesh-independent spectral properties, with about 80% of eigenvalues data-independent and confined to , guiding both stability and complexity considerations. Computationally, the data-dependent block scales linearly in grid size, while the data-independent block permits efficient, even matrix-free, solutions; Chebyshev-based approaches especially promise time and space for million-point problems, suggesting a practical path toward real-time large-scale trajectory optimization.

Abstract

This paper derives various Hessians associated with Birkhoff-theoretic methods for trajectory optimization. According to a theorem proved in this paper, approximately 80% of the eigenvalues are contained in the narrow interval [-2, 4] for all Birkhoff-discretized optimal control problems. A preliminary analysis of computational complexity is also presented with further discussions on the grand challenge of solving a million point trajectory optimization problem.

Paper Structure

This paper contains 12 sections, 2 theorems, 42 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. An Overview of Birkhoff-Theoretic Discretizations of Optimal Control Problems
  3. Connecting a Special Lagrangian to the Pontryagin Hamiltonian
  4. Hamiltonian Form of the Birkhoff Hessian
  5. A Birkhoff Hessian Eigenvalue Theorem
  6. A Preliminary Analysis of Computational Complexity
  7. Computational Cost Associated with the Data-Dependent Matrix, ${\boldsymbol A}_{data}$
  8. Computational Cost Associated with the Data-Independent Matrix, ${\boldsymbol A}_{0}$
  9. Thousands of Grid Points
  10. Addressing the Million Grid-Point Challenge
  11. A Brief Discussion on Alternative Birkhoff Hessians
  12. Conclusions

Key Result

Lemma 1

Given any $\epsilon > 0$, there exists an $N_\epsilon \in \mathbb{N}$ such that for all $N \ge N_\epsilon$ the following holds:

Figures (3)

  • Figure 1: Schematic for illustrating the plethora of possible Birkhoff-theoretic methods for trajectory optimization; figure adapted from newBirk-part-II andDIDO:arXiv.
  • Figure 2: Data-dependent sparsity pattern of ${\boldsymbol A}_{data}$
  • Figure 3: Data-independent sparsity pattern of ${\boldsymbol A}_{0}$

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1: newBirk-part-I
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5