Symplectic Transversality and Endpoint Green Estimates for Finite-Horizon Pontryagin Systems

2026-06-16 · Source: Artificial Intelligence · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Robotics & Autonomous Systems · Depth: Expert, quick

Summary

This research investigates horizon-uniform local branches of finite-horizon discrete-time Pontryagin boundary value systems, specifically after smooth control elimination. A central element is a two-point endpoint inverse for the linearization, which the authors verify using scaled stable--unstable boundary transversality. They prove an associated endpoint-corrected Green estimate and combine it with weighted contractions to establish existence, uniqueness, Lipschitz dependence, and first-order expansions. These findings maintain constants independent of the horizon. The framework accommodates smooth nonlinear endpoint maps, including original Pontryagin rows that fix initial states and couple terminal costates to terminal states. Symplectic and Riccati criteria validate the inverse hypothesis at the matrix data level, covering all stabilizable linear-quadratic systems with invertible dynamics and definite weights, even with noncommuting coupled data. A numerical section demonstrates the certificates and the horizon-uniform first-order expansion.

Key takeaway

For research scientists developing or analyzing optimal control systems, particularly those involving finite-horizon discrete-time Pontryagin systems, this framework offers rigorous analytical tools. You can apply the proposed methods to ensure existence, uniqueness, and predictable expansions with constants independent of the horizon. Consider leveraging the Symplectic and Riccati criteria to verify inverse hypotheses for your specific matrix data, especially in complex linear-quadratic systems.

Key insights

A robust framework provides horizon-uniform analytical tools for finite-horizon discrete-time Pontryagin systems.

Principles

• Scaled stable--unstable boundary transversality verifies endpoint inverses.
• Symplectic and Riccati criteria validate matrix data for inverse hypotheses.

Method

The method combines endpoint inverse verification, endpoint-corrected Green estimates, and weighted contractions to derive system properties independent of the horizon.

In practice

• Covers smooth nonlinear endpoint maps.
• Applies to stabilizable linear-quadratic systems.
• Numerical section illustrates certificates.

Topics

• Pontryagin Systems
• Optimal Control
• Discrete-Time Systems
• Symplectic Transversality
• Green Estimates
• Linear-Quadratic Systems

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Editorial summary, takeaway, and curation by AIssential. Original article published by Artificial Intelligence.