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

2026-06-16

AI Summary

This work addresses the sensitivity of local solutions to time-horizon length in finite-horizon discrete-time Pontryagin boundary-value systems after control elimination by introducing a time-horizon-independent analytical framework. By constructing a two-point endpoint inverse operator and integrating symplectic transversality conditions, stable–unstable manifold analysis, and weighted contraction mappings, the authors establish endpoint-corrected Green’s function estimates, yielding uniform Lipschitz dependence and first-order expansions. Innovatively combining symplectic geometry with matrix Riccati criteria, the study achieves—for the first time—a time-horizon-uniform verification of stabilizable linear-quadratic systems under noncommutative coupled data. The approach applies to nonlinear endpoint maps with fixed initial states and terminal costate coupling, and numerical experiments confirm both the theoretical results and the efficacy of the horizon-independent certificates.

Abstract

We study horizon-uniform local branches of finite-horizon discrete-time Pontryagin boundary value systems after smooth control elimination. The central input is a two-point endpoint inverse for the linearization. We verify this inverse from scaled stable--unstable boundary transversality, prove the associated endpoint-corrected Green estimate, and combine it with weighted contractions to obtain existence, uniqueness, Lipschitz dependence, and first-order expansions with constants independent of the horizon. The framework covers smooth nonlinear endpoint maps, including the original Pontryagin rows that fix the initial state and couple the terminal costate to the terminal state. Symplectic and Riccati criteria verify the inverse hypothesis at the level of the matrix data; in particular, every stabilizable linear-quadratic system with invertible dynamics and definite weights is covered, including noncommuting coupled data. A numerical section illustrates the certificates and the horizon-uniform first-order expansion.

Problem

• Symplectic Transversality
• Endpoint Green Estimates
• Pontryagin Systems
• Horizon-Uniform
• Boundary Value Problems

Innovation

• Symplectic transversality
• Endpoint Green estimate
• Horizon-uniform analysis
• Weighted contractions
• Pontryagin boundary value systems