Horizon-Uniform Sensitivity and Decay of Terminal Reward Perturbations in Discrete-Time Pontryagin Systems

Abstract

We study local stationary solutions of finite-horizon discrete-time Pontryagin systems near a steady extremal. Suppose that the stationarity equation for the control is regular, the reduced state--costate map is hyperbolic, and the endpoint conditions satisfy a scaled transversality condition with respect to the stable and unstable subspaces. Then the linearized boundary-value problem admits an inverse whose Green estimate is uniform in the horizon. The Green kernel separates interior decay from the two reflections induced by the endpoint conditions. For $x_0=x_{\mathrm{in}}$ and $p_T=r_x(x_T,y)$, a contraction argument in a weighted norm proves existence and uniqueness in a neighborhood independent of $T$, together with uniform Lipschitz estimates and a pointwise quadratic remainder. We also derive an explicit admissible data radius and an a posteriori criterion for existence and local uniqueness near an approximate trajectory. For these graph boundary conditions, a one-sided Green estimate shows that a perturbation of the terminal reward changes the initial control and the gradient with respect to the initial state of the stationary objective by $O(e^{-{\alpha }_{\mathrm{ter}}T})$ for every ${\alpha }_{\mathrm{ter}}$ below the dichotomy rate. For linear-quadratic systems with invertible $A$, stabilizable $(A,B)$, $Q\succ 0$, $R\succ 0$, and a nonpositive terminal Hessian, a symplectic graph condition verifies the assumptions, and the finite-horizon Riccati matrix and initial feedback gain converge at rate $O(e^{-2\gamma T})$. Numerical experiments verify the certificates and the predicted decay rates.

14 pages