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

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 x0=xrm in and pT=rx(xT,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^-αrm ter T) for every αrm ter below the dichotomy rate. For linear-quadratic systems with invertible A, stabilizable (A,B), Qsucc0, Rsucc0, 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-2γ T). Numerical experiments verify the certificates and the predicted decay rates.