- The paper demonstrates that a uniform inverse property is achieved through an endpoint-corrected Green estimate under scaled symplectic transversality conditions.
- It constructs an explicit discrete exponential dichotomy to provide numerical certificates for invertibility in both linear and nonlinear Pontryagin systems.
- The results extend to coupled linear-quadratic systems and support robust approaches in sensitivity analysis, shooting methods, and turnpike theory.
Symplectic Transversality and Endpoint Green Estimates for Finite-Horizon Pontryagin Systems
Problem Formulation and Context
The paper addresses finite-horizon discrete-time Pontryagin systems—classical boundary value problems arising in discrete-time optimal control after elimination of controls via local stationarity. Once the controls are expressed as functions of states and costates (via the stationarity conditions of the Hamiltonian and the implicit function theorem), one obtains two-point boundary value systems coupling states and costates across the time horizon. A major technical obstacle in the analysis and numerical solution of such systems is the inversion of the associated linear two-endpoint map, especially with constants that remain uniform in the horizon length. Horizon-uniform estimates are crucial for stability analysis, sensitivity analysis, and for asymptotic results relevant in the study of long-horizon behavior (including turnpike theory).
The main contributions include: formulating an endpoint-corrected Green’s function estimate for the linearized problem, proving that such estimates follow from scaled symplectic transversality conditions, and demonstrating their robustness under perturbations. The approach is constructive, providing data-level certificates for the invertibility required, especially in the context of (coupled) linear-quadratic (LQ) systems with invertible dynamics and positive definite weights. The framework is general, extending to nonlinear endpoint maps—beyond mere affine boundary conditions. It is shown that these hypotheses can be checked purely at the matrix-data level for substantial system classes.
Main Theoretical Results
The central technical novelty is the uniform (in the time horizon) inverse estimate—an endpoint-corrected Green estimate—for the linearized Pontryagin system. The problem is recast in weighted function spaces over the time horizon, such that norm bounds and error estimates do not deteriorate as the horizon grows.
A key analytical device is the introduction of scaled stable–unstable transversality, formalized through the construction of horizon-dependent boundary matrices (invariant subspaces evolved under the discrete system). The endpoint-corrected Green estimate is then established via a discrete variant of the exponential dichotomy formalism: the solution to the inhomogeneous linear system is represented using an explicit kernel which decays exponentially from both endpoints and is controlled by uniform constants, with corrections for endpoint-forcing contributions. This two-sided structure is intrinsic to boundary-value problems and distinguishes the approach from standard initial value analyses.
The endpoint inverse property is proved robust under continuous data perturbations—a critical property for applicability to nonlinear and parametric models. The main nonlinear results—existence, uniqueness, Lipschitz dependence, and first-order expansions—are established in horizon-uniform weighted spaces using contraction mappings. The estimates rely fundamentally on the endpoint-corrected Green estimates: the weights are matched to the dichotomy parameters and convolution kernels, ensuring that nonlinearity-induced quadratic remainders are uniformly controlled.
Further, the data–oriented verification of the endpoint inverse property is established via symplectic and Riccati-theoretic conditions. The analysis leverages the symplectic structure inherent to linearized Hamiltonian systems, with the key data-level criterion reducing to invertibility of limiting scaled graph-transversality matrices constructed from the stable and unstable Lagrangian subspaces. For LQ systems, the analysis covers fully coupled, noncommuting (A, Q) and (B, R) data, under stabilizability and definiteness assumptions.
Implications and Significance
The study places the analysis of finite-horizon Pontryagin systems on a rigorous foundation that is robust to the horizon, supporting extensions to numerical methods, sensitivity analysis, shooting and continuation methods, and local control design around stationary points. A particularly strong aspect is that all key quantitative assumptions are checkable at the matrix data level, unlike approaches relying on structural dissipativity or global Riccati turnpike properties.
This has several implications:
- Numerical Algorithms: The results justify the development of robust numerics for long-horizon shooting methods, with a priori guarantees on the conditioning of the associated linear systems. The uniform error and sensitivity bounds can be exploited in Newton-Kantorovich iterations and in uncertainty quantification.
- Extension to Polyhedral/Nonclassical Endpoints: The explicit dependence on endpoint rows enables extension to constrained control sets (via active-set reduction under complementarity hypotheses) and to general nonlinear endpoint maps.
- Theoretical Control Theory: The explicit separation of the invertibility/Green estimate from dissipativity (or definiteness) allows the development of local branch theory even in the absence of optimality or (global/static) turnpike properties.
- Practical LQ Design: The results encompass coupled, noncommuting LQ systems, supporting their use in high-dimensional control and estimation problems where turnpike/invariant properties are not directly available.
Numerical Verification
The theory is supported by finite-horizon, data-level numerical verifications. Stabilizable coupled LQ systems with noncommuting A and Q are analyzed. The uniformity and explicit constants of the endpoint inverse property are certified computationally, even on high (up to T=160) horizons. The empirical Green constants saturate, indicating the sharpness of the theoretical bounds and the efficacy of the proposed criteria.
A further experiment with nonlinear perturbations confirms the predicted quadratic order and horizon-uniformity of the first-order expansion remainder.
Future Research Directions
The methodology admits generalizations:
- Non-stationary and Polyhedral References: Extension to nonautonomous (time-varying) or polyhedral-restricted references via nonautonomous dichotomies and active-set partitioning.
- Singular Reductions: Treatment of singular L1​ blocks in the equations, where the present construction may not apply and new data-level invertibility criteria are needed.
- Infinite-Horizon and Turnpike Analysis: With uniform-in-horizon estimates in hand, the asymptotic regime can be studied more rigorously, potentially clarifying the interplay of local branch properties with global turnpike phenomena.
Conclusion
The paper establishes horizon-uniform existence and regularity results for finite-horizon discrete-time Pontryagin systems, via explicit data-level verification criteria grounded in symplectic and Riccati theory. The analysis clarifies the role of endpoint-corrected Green estimates and scaled stable–unstable transversality, and provides numerically verifiable certificates for a broad class of nonlinear and LQ models, including cases with strongly coupled, noncommuting data. The work forms a rigorous baseline for the local analysis and numerical treatment of two-point boundary value problems in discrete-time control and optimization.