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Symplectic Transversality and Endpoint Green Estimates for Finite-Horizon Pontryagin Systems
Pyuyi Chufeng Huang, Zikang Song, Xingshu Chen
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Summary
The paper investigates the existence, uniqueness, and Lipschitz dependence of local branches for finite-horizon discrete-time Pontryagin boundary value systems. The authors develop a framework for 'horizon-uniform' estimates, ensuring that constants in first-order expansions and Green estimates remain independent of the time horizon $T$. A central contribution is the 'endpoint-corrected Green estimate' derived from scaled stable-unstable boundary transversality. The research demonstrates that for a wide class of systems, including stabilizable linear-quadratic (LQ) systems with noncommuting coupled data, the inverse of the linearized two-point endpoint system can be verified using symplectic and Riccati-based criteria. This allows for robust sensitivity analysis and numerical certification near stationary Pontryagin branches.
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Relation Signals (4)
Symplectic criteria → verifies → inverse hypothesis
confidence 100% · Symplectic and Riccati criteria verify the inverse hypothesis at the level of the matrix data;
Riccati criteria → verifies → inverse hypothesis
confidence 100% · Symplectic and Riccati criteria verify the inverse hypothesis at the level of the matrix data;
Pontryagin boundary value systems → characterizedby → Endpoint-corrected Green estimate
confidence 90% · We study horizon-uniform local branches of finite-horizon discrete-time Pontryagin boundary value systems... prove the associated endpoint-corrected Green estimate,
Scaled stable-unstable boundary transversality → verifies → Endpoint-corrected Green estimate
confidence 90% · We verify this inverse from scaled stable--unstable boundary transversality, prove the associated endpoint-corrected Green estimate,
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Abstract
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.
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- Source: https://arxiv.org/abs/2606.17762v1
- Canonical: https://arxiv.org/abs/2606.17762v1
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Symplectic Transversality and Green EstimatesP. C. Huang, Z. Song, and X. Chen Symplectic Transversality and Endpoint Green Estimates for Finite-Horizon Pontryagin Systems Pyuyi Chufeng Huang School of Cyber Science and Engineering, Sichuan University, Chengdu, Sichuan, China (). Zikang Song School of Mathematics, Sichuan University, Chengdu, Sichuan, China (). Xingshu Chen School of Cyber Science and Engineering, Sichuan University, Chengdu, Sichuan, China (). Corresponding author. 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. keywords: discrete-time optimal control; Pontryagin maximum principle; boundary value problems; exponential dichotomy; Riccati equations; symplectic matrices MSCcodes 49K21; 49M05; 49N10; 93C55; 37C60 1 Introduction Finite-horizon discrete-time Pontryagin systems lead, after local stationarity-based control elimination, to two-point state-costate boundary value problems. Classical optimal control gives the first-order equations [bertsekas2012dynamic, vinter2010optimal], but the finite-horizon difficulty is to invert the linearized two-endpoint system with constants independent of the horizon. A finite-dimensional implicit-function argument at a fixed horizon would hide this issue in the norm of a horizon-dependent inverse. A usual exponential dichotomy controls interior forcing, but finite endpoint rows also create homogeneous corrections whose decay starts at the opposite endpoint. These corrections are small but essential in the nonlinear weighted estimates below. The paper develops the following chain. First, Proposition 4.3 proves an endpoint-corrected Green estimate from hyperbolicity and uniform invertibility of a scaled stable–unstable boundary matrix. Second, Theorems 4.7 and 4.8 use the resulting kernels in horizon-dependent weighted spaces to obtain local existence, uniqueness, Lipschitz dependence, and first-order expansions with constants independent of the horizon. Third, the abstract estimates are applied to finite-horizon Pontryagin systems with both affine endpoint rows and the original initial-state and terminal-costate rows. Finally, symplectic and LQ criteria verify the inverse hypothesis on relative open matrix-data regions; Theorem 7.2 covers every stabilizable LQ system with invertible dynamics and definite weights, including coupled noncommuting data, and Section 8 illustrates the certificates numerically. The linear mechanism behind the inverse hypothesis is classical. For two-point boundary value problems in ordinary differential equations, well conditioning is essentially equivalent to an exponential dichotomy together with compatible boundary rows [dehoog1987dichotomy, ascher1995numerical]; finite-interval dichotomy estimates and their roughness are developed in [coppel1978dichotomies, palmer1984exponential, potzsche2010geometric]. Proposition 4.3 is a discrete two-endpoint version of this mechanism that keeps the two endpoint-image kernels in (4.2) explicit, because the nonlinear weighted estimates consume exactly those kernels. A separate line of work establishes exponential decay of parametric sensitivity for finite-horizon discrete-time optimal control and for graph-structured nonlinear programs, with constants uniform in the horizon [xu2018exponentially, na2020exponential, shin2022exponential], and for abstract evolution equations [grune2020exponential]. Those results work at optimal solutions, under second-order sufficient conditions and constraint qualifications for the associated KKT systems. The present paper works instead at stationary Pontryagin branches: no optimality, definiteness, or dissipativity is assumed, the endpoint inverse of Definition 4.1 is the only quantitative hypothesis, it is certified at the level of the matrix data by symplectic transversality, and the endpoint rows may be genuinely nonlinear. The branch results are local stationarity results. They are useful for shooting, continuation, and sensitivity analysis near a steady Pontryagin triple, and they can serve as a preliminary step before adding second-order sufficient conditions. Thus the Riccati and symplectic material is used as a verifiable transversality certificate, not as a replacement for dissipativity or optimality hypotheses. Any middle-window decay here comes from the selected local branch and should be distinguished from turnpike mechanisms based on dissipativity, Riccati normal forms, or hyperbolicity along globally optimal trajectories [willems1972dissipative, damm2014exponential, trelat2015turnpike, grune2016strict, grune2017nmpc, faulwasser2022turnpike]. 2 Model, reference point, and control elimination For a horizon T∈ℕT , the state is in ℝdR^d, the control set U⊂ℝmU ^m is compact and convex, and (2.1) xt+1=F(xt,ut),0≤t<T.x_t+1=F(x_t,u_t), 0≤ t<T. For y in an open set Y⊂ℝqY ^q, the maximization objective and Hamiltonian are (2.2) JT(u)=r(xT,y)−λ∑t=0T−1ℓ(xt,ut),λ>0,J_T(u)=r(x_T,y)-λ _t=0^T-1 (x_t,u_t), λ>0, (2.3) H(x,u,p)=−λℓ(x,u)+⟨p,F(x,u)⟩.H(x,u,p)=-λ (x,u)+ p,F(x,u) . All local branch estimates below use fixed neighborhoods, independent of T, on which the primitive maps have the differentiability needed by the displayed remainders: F and ℓ are C3C^3 near the reference, so HuH_u is C2C^2 and the reduced map after control elimination is C2C^2; terminal rewards r(⋅,y)r(·,y) are C3C^3 when nonlinear terminal rows are used. The quantitative uniform bounds are collected in Condition 2.2. An interior stationary reference satisfies (2.4) x∞=F(x∞,u∞),p∞=Hx(x∞,u∞,p∞),Hu(x∞,u∞,p∞)=0,x_∞=F(x_∞,u_∞), p_∞=H_x(x_∞,u_∞,p_∞), H_u(x_∞,u_∞,p_∞)=0, with u∞∈intUu_∞ . Lemma 2.1 (Local stationarity reduction). Assume HuH_u is C1C^1 near (x∞,u∞,p∞)(x_∞,u_∞,p_∞), Hu(x∞,u∞,p∞)=0H_u(x_∞,u_∞,p_∞)=0, and Huu(x∞,u∞,p∞)H_u(x_∞,u_∞,p_∞) is nonsingular. Then there are neighborhoods of (x∞,p∞)(x_∞,p_∞) and u∞u_∞ and a unique C1C^1 stationarity graph u=ν(x,p+)u=ν(x,p_+) such that (2.5) Hu(x,ν(x,p+),p+)=0,ν(x∞,p∞)=u∞.H_u(x,ν(x,p_+),p_+)=0, ν(x_∞,p_∞)=u_∞. If, in addition, HuH_u is C2C^2 with bounded second derivative and Huu−1H_u^-1 is uniformly bounded after shrinking, then ν is C2C^2 with bounded second derivative. This is the case under the standing C3C^3 assumptions on F and ℓ . If −Huu(x∞,u∞,p∞)≻0-H_u(x_∞,u_∞,p_∞) 0, then after shrinking −Huu⪰ηI-H_u η I and ν(x,p+)ν(x,p_+) is the strict local maximizer of H(x,⋅,p+)H(x,·,p_+) in the selected control neighborhood; without this definiteness the graph is only a stationarity graph. Proof. Let ζ=(x,p+)ζ=(x,p_+) and Θ(ζ,u)=Hu(x,u,p+) (ζ,u)=H_u(x,u,p_+). Then DuΘ(ζ∞,u∞)=Huu(x∞,u∞,p∞)D_u ( _∞,u_∞)=H_u(x_∞,u_∞,p_∞) is invertible at ζ∞=(x∞,p∞) _∞=(x_∞,p_∞), so the implicit function theorem gives neighborhoods Ω of ζ∞ _∞ and V of u∞u_∞, with V⋐UV U after shrinking, and a unique C1C^1 map ν:Ω→Vν: → V with Θ(ζ,ν(ζ))=0 (ζ,ν(ζ))=0; the uniqueness is local in V, which is the only uniqueness used later. Differentiating the identity gives Dν=−Huu−1(Hux,Hup)Dν=-H_u^-1(H_ux,H_up) on the graph, so DνDν is bounded on a smaller neighborhood whenever the appearing derivatives and Huu−1H_u^-1 are bounded there. If HuH_u is C2C^2, then ν∈C2ν∈ C^2, and differentiating once more gives, for directions a,ba,b in the ζ variables and all derivatives evaluated at (x,ν(x,p+),p+)(x,ν(x,p_+),p_+), HuuD2ν[a,b]=−D2Θ[(a,Dν[a]),(b,Dν[b])];H_uD^2ν[a,b]=-D^2 [(a,Dν[a]),(b,Dν[b])]; on a closed tube where ‖D2Θ‖\|D^2 \|, ‖Dν‖\|Dν\|, and ‖Huu−1‖\|H_u^-1\| are bounded this bounds D2νD^2ν, and such tubes exist under the standing C3C^3 assumptions on F and ℓ . Finally suppose −Huu(x∞,u∞,p∞)≻0-H_u(x_∞,u_∞,p_∞) 0. By continuity and further shrinking, V may be chosen convex and −Huu(x,u,p+)⪰ηI-H_u(x,u,p_+) η I for all (x,p+)∈Ω(x,p_+)∈ and u∈Vu∈ V. Since Hu(x,ν,p+)=0H_u(x,ν,p_+)=0, Taylor’s formula along the segment from ν=ν(x,p+)ν=ν(x,p_+) to any u∈Vu∈ V gives H(x,u,p+)−H(x,ν,p+) H(x,u,p_+)-H(x,ν,p_+) =∫01(1−s)(u−ν)⊤Huu(x,ν+s(u−ν),p+)(u−ν)s = _0^1(1-s)(u-ν) H_u(x,ν+s(u-ν),p_+)(u-ν)\,ds ≤−η2‖u−ν‖2, ≤- η2\|u-ν\|^2, so ν(x,p+)ν(x,p_+) is the strict local maximizer of H(x,⋅,p+)H(x,·,p_+) in V. With zt=(xt,pt)z_t=(x_t,p_t), the reduced state-costate step is (2.6) ℱ(zt,zt+1)=0,F(z_t,z_t+1)=0, where (2.7) ℱ(zt,zt+1)=(xt+1−F(xt,ν(xt,pt+1))pt−Hx(xt,ν(xt,pt+1),pt+1)).F(z_t,z_t+1)= x_t+1-F(x_t,ν(x_t,p_t+1))p_t-H_x(x_t,ν(x_t,p_t+1),p_t+1). Let νx=Dxν(x∞,p∞) _x=D_xν(x_∞,p_∞) and νp=Dp+ν(x∞,p∞) _p=D_p_+ν(x_∞,p_∞). In (2.8), Fx,FuF_x,F_u are the derivatives of the original dynamics F at (x∞,u∞)(x_∞,u_∞) and all Hamiltonian derivatives are evaluated at (x∞,u∞,p∞)(x_∞,u_∞,p_∞) and Hxp+H_xp_+ denotes the derivative of HxH_x with respect to the forward costate argument. For h=(ξ,π)h=(ξ,π) and h+=(ξ+,π+)h^+=(ξ^+,π^+), the linearization is L0h+L1h+L_0h+L_1h^+ with (2.8) L0h=(−(Fx+Fuνx)ξπ−(Hxx+Hxuνx)ξ),L1h+=(ξ+−Fuνpπ+−(Hxp++Hxuνp)π+). gatheredL_0h= -(F_x+F_u _x)ξπ-(H_x+H_xu _x)ξ, L_1h^+= ξ^+-F_u _pπ^+-(H_xp_++H_xu _p)π^+. gathered Condition 2.2 (Uniform local reduction and product tube). The stationarity graph supplied by Lemma 2.1 is defined on a product state-costate tube (x∞,p∞)+Z(x_∞,p_∞)+Z, where Z=Zx×Zp⊂ℝd×ℝdZ=Z_x× Z_p ^d×R^d is a neighborhood of 0, and takes values in a control tube U0⋐intU_0 containing u∞u_∞. Thus, whenever z=(ξ,π)z=(ξ,π) and z+=(ξ+,π+)z^+=(ξ^+,π^+) lie in Z, the mixed argument (ξ,π+)(ξ,π^+) entering ν(x∞+ξ,p∞+π+)ν(x_∞+ξ,p_∞+π^+) also lies in Z. There are product balls Bx×Bp⋐ZB_x× B_p Z, a terminal parameter set Y0⋐Y_0 Y, and constants M2,M3>0M_2,M_3>0, all independent of T, such that supZ‖D2ν‖≤M2,sup(Bx×Bp)2‖D2ℱ~‖≤M2, _Z\|D^2ν\|≤ M_2, _(B_x× B_p)^2\|D^2 F\|≤ M_2, where ℱ~(z,z+)=ℱ((x∞,p∞)+z,(x∞,p∞)+z+) F(z,z^+)=F((x_∞,p_∞)+z,(x_∞,p_∞)+z^+). When terminal maps r(⋅,y)r(·,y) are used, supx∈x∞+Bx,y∈Y0, 1≤j≤3‖Dxjr(x,y)‖≤M3. _x∈ x_∞+B_x,\ y∈ Y_0,\ 1≤ j≤ 3\|D_x^jr(x,y)\|≤ M_3. The weighted radii used in the reconstruction theorems are always chosen so that the resulting pointwise tube lies in Bx×BpB_x× B_p; since wT(t)≤2w_T(t)≤ 2, this choice is uniform in T. Lemma 4.2 gives an explicit hyperbolic verification route when L1L_1 is invertible; otherwise Definition 4.1 is used directly. 3 Pontryagin equations and sign convention This section fixes the sign convention used later and records precisely which part of the Pontryagin system is treated as a boundary value problem. We use the maximization convention for (2.2) and the forward Hamiltonian (2.3); the costate paired with the step from t to t+1t+1 is therefore pt+1p_t+1. Consequently Hx(x,u,p+) H_x(x,u,p_+) =∂xF(x,u)⊤p+−λ∂xℓ(x,u), = _xF(x,u) p_+-λ _x (x,u), Hu(x,u,p+) H_u(x,u,p_+) =∂uF(x,u)⊤p+−λ∂uℓ(x,u). = _uF(x,u) p_+-λ _u (x,u). If a trajectory is locally optimal for (2.2), then one-sided variations of a single control value inside the convex set U give the discrete-time first-order conditions (3.1) xt+1=F(xt,ut),pt=Hx(xt,ut,pt+1),0≤t<T,pT=∂xr(xT,y),⟨Hu(xt,ut,pt+1),v−ut⟩≤0,v∈U. gatheredx_t+1=F(x_t,u_t), p_t=H_x(x_t,u_t,p_t+1), 0≤ t<T,\\ p_T= _xr(x_T,y), H_u(x_t,u_t,p_t+1),v-u_t ≤ 0, v∈ U. gathered These are the standard discrete-time Pontryagin conditions [bertsekas2012dynamic, vinter2010optimal]; the signs reflect maximization of terminal reward minus accumulated cost, and all formulas below use (2.3) and (3.1). Since u∞∈intUu_∞ , all sufficiently small local branches generated below remain in the interior of U; on those branches the variational inequality in (3.1) is equivalent to Hu(xt,ut,pt+1)=0H_u(x_t,u_t,p_t+1)=0. Lemma 2.1 then eliminates utu_t as ut=ν(xt,pt+1)u_t=ν(x_t,p_t+1), and the first two equations in (3.1) become exactly the reduced implicit equation (2.6). The remaining endpoint conditions, such as x0=xinx_0=x_ in and pT=rx(xT,y)p_T=r_x(x_T,y), are handled later as affine or smooth endpoint rows. Thus a stationary Pontryagin branch in this paper means a local branch of (3.1) after interior stationarity reduction. 4 Endpoint inverse estimates and nonlinear reconstruction Throughout this section, constants denoted by C may change from line to line and they may depend on fixed norms, local C2C^2 bounds, boundary-row norms, and the Green constants in Definition 4.1. In parameterized statements, uniformity is asserted only after the parameter set is shrunk to a fixed compact neighborhood on which the local tubes, smoothness bounds, spectral gaps, transversality margins, and row-perturbation bounds are chosen once and for all. The main finite-horizon input is a two-sided inverse estimate for the linearized boundary value problem. We use a form with endpoint-correction terms, because this is what is produced by finite-horizon boundary conditions and it is exactly what the nonlinear proof needs. Definition 4.1 (Endpoint-corrected inverse property). Let L:ℝn×ℝn→ℝnL:R^n×R^n ^n be linear and let 0,T∈ℝn×n B_0, B_T ^n× n be fixed boundary-row matrices, not necessarily invertible individually. The triple (L,0,T)(L, B_0, B_T) has the endpoint-corrected inverse property if there exist constants CG≥1C_G≥ 1, γ>0γ>0, and T0≥1T_0≥ 1 such that, for every T≥T0T≥ T_0, every interior forcing f=(f0,…,fT−1)f=(f_0,…,f_T-1), and every boundary datum b∈ℝnb ^n, the linear problem (4.1) L[ht,ht+1]=ft,0h0+ThT=b,0≤t<T,L[h_t,h_t+1]=f_t, B_0h_0+ B_Th_T=b, 0≤ t<T, has a unique solution. With fixed finite-dimensional norms and (4.2) κT(t,s)=e−γ|t−s|+e−γ(t+T−s)+e−γ(T−t+s),0≤s<T, _T(t,s)=e^-γ|t-s|+e^-γ(t+T-s)+e^-γ(T-t+s), 0≤ s<T, this solution satisfies (4.3) ∥ht∥≤CG[(e−γt+e−γ(T−t))∥b∥+∑s=0T−1κT(t,s)∥fs∥] h_t ≤ C_G [(e^-γ t+e^-γ(T-t)) b + _s=0^T-1 _T(t,s) f_s ] for all 0≤t≤T0≤ t≤ T, with constants independent of T. The two extra terms in (4.2) are endpoint corrections: forcing near one endpoint can create a small mode decaying from the other endpoint. We denote by ℒTbdbL_T^bdb the boundary-data solution, obtained by setting f=0f=0, and by ℒTforfL_T^forf the forcing solution, obtained by setting b=0b=0. Lemma 4.2 (Implicit linearization in first-order form). Let L[h,h+]=L0h+L1h+L[h,h^+]=L_0h+L_1h^+ be the linearization of an implicit equation ℱ(zt,zt+1)=0F(z_t,z_t+1)=0. If L1L_1 is invertible, then L[ht,ht+1]=ft⟺ht+1=Mht+gt,M=−L1−1L0,gt=L1−1ft.L[h_t,h_t+1]=f_t h_t+1=Mh_t+g_t, M=-L_1^-1L_0, g_t=L_1^-1f_t. Thus Proposition 4.3 verifies Definition 4.1 for the implicit linearization whenever the boundary rows and the matrix M satisfy scaled transversality. If L1L_1 is singular, Definition 4.1 is an independent two-point inverse assumption and is not implied by this explicit hyperbolic test. Proof. Because L1L_1 is invertible, the interior equations L0ht+L1ht+1=ftL_0h_t+L_1h_t+1=f_t are equivalent to ht+1=Mht+gth_t+1=Mh_t+g_t, while the endpoint condition involves only h0h_0 and hTh_T and is unchanged; the implicit and the explicit boundary value problems therefore have identical solution sets. If the explicit problem is covered by Proposition 4.3, substituting gs=L1−1fsg_s=L_1^-1f_s into its estimate bounds the forcing sum by ∥L1−1∥∑sκT(t,s)∥fs∥ L_1^-1 _s _T(t,s) f_s , and absorbing the fixed factor ∥L1−1∥ L_1^-1 into the Green constant gives Definition 4.1 for the implicit linearization. Existence and uniqueness transfer because the solution sets coincide. If L1L_1 is singular, this reduction is unavailable. For the reduced linearization (2.8), L1L_1 is block triangular in (ξ+,π+)(ξ^+,π^+), so it is invertible exactly when Hxp++HxuνpH_xp_++H_xu _p is invertible; at the reference Hxp+=Fx⊤H_xp_+=F_x , and for the LQ data of Section 6 the condition is invertibility of A⊤A . A concrete verification comes from a hyperbolic first-order form and a uniformly invertible scaled boundary matrix, the finite-horizon version of the exponential-dichotomy viewpoint [coppel1978dichotomies, palmer1984exponential, potzsche2010geometric]. Proposition 4.3 (Endpoint inverse from scaled transversality). Consider, for a horizon N, (4.4) ht+1=Mht+gt,0≤t<N,0h0+ThN=b,h_t+1=Mh_t+g_t, 0≤ t<N, B_0h_0+ B_Th_N=b, on ℝnR^n. Suppose M is hyperbolic and let the columns of VsV_s and VuV_u be bases of its stable and unstable subspaces, with dimEs+dimEu=n E_s+ E_u=n; write MVs=VsDsMV_s=V_sD_s and MVu=VuDuMV_u=V_uD_u, and assume that, for some K≥1K≥ 1 and γ>0γ>0, ∥VsDsk∥≤Ke−γk V_sD_s^k ≤ Ke^-γ k and ∥VuDu−k∥≤Ke−γk V_uD_u^-k ≤ Ke^-γ k for all k≥0k≥ 0. For a fixed hyperbolic matrix these bounds hold, after increasing K, for every rate below the spectral gap, including possible Jordan blocks. For each horizon define the scaled boundary matrix (4.5) Γ^N=[0Vs+TVsDsN0VuDu−N+TVu], _N= [ B_0V_s+ B_TV_sD_s^\,N B_0V_uD_u^-N+ B_TV_u ], where the superscripts are powers, not transposes. If Γ^N _N is invertible for every N≥T0N≥ T_0 and supN≥T0∥Γ^N−1∥<∞ _N≥ T_0 _N^-1 <∞, then the endpoint inverse property of Definition 4.1 holds for (4.4), with constants depending only on the dichotomy bounds, the spectral-projector norms, the fixed boundary matrices, and the displayed uniform bound for Γ^N−1 _N^-1. Moreover, the uniform inverse bound follows if (4.6) Γ^∞:=[0VsTVu] _∞:= [ B_0V_s B_TV_u ] is invertible and, for some cutoff N∗N_* large enough for the Neumann argument, each matrix Γ^N _N with T0≤N<N∗T_0≤ N<N_* is invertible. We say that scaled boundary transversality holds if Γ^N _N in (4.5) is invertible for every N≥T0N≥ T_0 with supN≥T0‖Γ^N−1‖<∞ _N≥ T_0\| _N^-1\|<∞. The bases are fixed once and for all: replacing them by a fixed pair VsAs,VuAuV_sA_s,V_uA_u right-multiplies Γ^N _N by diag(As,Au)diag(A_s,A_u), so the property is basis independent up to condition numbers, while no N-dependent rescaling is allowed. Proof. Let V=[VsVu]V=[V_s\ V_u] and write V−1=(WsWu)V^-1= W_sW_u. Set Πs=VsWs _s=V_sW_s and Πu=VuWu _u=V_uW_u. After absorbing the fixed basis and projector norms, ‖MkΠs‖≤Ce−γk\|M^k _s\|≤ Ce^-γ k and ∥(M|Eu)−kΠu∥≤Ce−γk\|(M|_E_u)^-k _u\|≤ Ce^-γ k for k≥0k≥ 0, where the inverse powers are taken only on EuE_u. Every homogeneous solution has a unique representation qt=VsDsta+VuDut−Nηq_t=V_sD_s^ta+V_uD_u^t-Nη: indeed, write q0=Vsa+Vucq_0=V_sa+V_uc and put η=DuNcη=D_u^Nc. For such a solution, 0q0+TqN=Γ^N(aη) B_0q_0+ B_Tq_N= _N aη. Hence, for homogeneous boundary datum b, scaled transversality gives ‖(a,η)‖≤C‖b‖\|(a,η)\|≤ C\|b\|. The dichotomy estimates then yield ‖qt‖≤C(e−γt+e−γ(N−t))‖b‖.\|q_t\|≤ C(e^-γ t+e^-γ(N-t))\|b\|. This bound is uniform in N because the bases and the inverse bound for Γ^N _N are uniform. For forcing, use the empty-sum convention and define pt=∑j=0t−1Mt−1−jΠsgj−∑j=tN−1(M|Eu)t−1−jΠugj.p_t= _j=0^t-1M^t-1-j _sg_j- _j=t^N-1(M|_E_u)^t-1-j _ug_j. In the second sum the exponent is negative; it means the corresponding inverse power of the automorphism M|EuM|_E_u. Set StS_t equal to the first sum and UtU_t equal to the whole signed second term in this display. Then St+1−MSt=ΠsgtS_t+1-MS_t= _sg_t and Ut+1−MUt=ΠugtU_t+1-MU_t= _ug_t, so pt+1−Mpt=gtp_t+1-Mp_t=g_t. The dichotomy bounds give ‖pt‖ \|p_t\| ≤C∑j=0N−1e−γ|t−j|‖gj‖, ≤ C _j=0^N-1e^-γ|t-j|\|g_j\|, ‖p0‖+‖pN‖ \|p_0\|+\|p_N\| ≤C∑j=0N−1(e−γj+e−γ(N−j))‖gj‖. ≤ C _j=0^N-1(e^-γ j+e^-γ(N-j))\|g_j\|. The formulas also cover the endpoints: p0=−∑j=0N−1(M|Eu)−1−jΠugjp_0=- _j=0^N-1(M|_E_u)^-1-j _ug_j and pN=∑j=0N−1MN−1−jΠsgjp_N= _j=0^N-1M^N-1-j _sg_j. The boundary residual for arbitrary b is r=b−0p0−TpNr=b- B_0p_0- B_Tp_N, so ‖r‖≤C‖b‖+C∑j=0N−1(e−γj+e−γ(N−j))‖gj‖.\|r\|≤ C\|b\|+C _j=0^N-1(e^-γ j+e^-γ(N-j))\|g_j\|. Adding the homogeneous solution q with boundary datum r yields h=p+qh=p+q. Its homogeneous correction obeys ‖qt‖≤C(e−γt+e−γ(N−t))‖b‖+C∑j=0N−1Kt,j‖gj‖,\|q_t\|≤ C(e^-γ t+e^-γ(N-t))\|b\|+C _j=0^N-1K_t,j\|g_j\|, where Kt,jK_t,j is the sum of e−γ(t+j)e^-γ(t+j), e−γ(t+N−j)e^-γ(t+N-j), e−γ(N−t+j)e^-γ(N-t+j), and e−γ(2N−t−j)e^-γ(2N-t-j). The first and fourth terms are bounded by e−γ|t−j|e^-γ|t-j|, since t+j≥|t−j|t+j≥|t-j| and 2N−t−j≥|t−j|2N-t-j≥|t-j|. The two middle terms are exactly the endpoint-image kernels e−γ(t+N−j)e^-γ(t+N-j) and e−γ(N−t+j)e^-γ(N-t+j) in (4.2). Thus h=p+qh=p+q exists for arbitrary (g,b)(g,b) and satisfies (4.3), after the harmless replacement of |t−1−j||t-1-j| by |t−j||t-j| in the particular-solution constants. If g=b=0g=b=0, the homogeneous boundary equation gives Γ^N(aη)=0 _N aη=0, hence a=η=0a=η=0 and the solution is zero; uniqueness follows by applying this to the difference of two solutions. Finally, ‖Γ^N−Γ^∞‖≤Ce−γN\| _N- _∞\|≤ Ce^-γ N. If (4.6) is invertible, Neumann’s lemma gives uniform inverses for all sufficiently large N, and the finitely many remaining invertible matrices have a positive minimum least singular value. This proves the last assertion. The scaled boundary test is useful only if it survives small modeling errors. The next proposition shows this finite-horizon robustness under continuous perturbations of the dynamics and boundary rows. Proposition 4.4 (Robustness of scaled transversality). Let M(θ)M(θ), 0(θ) B_0(θ), and T(θ) B_T(θ) depend continuously on θ near 0. Suppose M(0)M(0) is hyperbolic and the scaled matrices Γ^N(0) _N(0) in (4.5) are uniformly invertible for all N≥T0N≥ T_0. Then, using any locally continuous choice of stable and unstable bases, the matrices Γ^N(θ) _N(θ) are uniformly invertible for all N≥T0N≥ T_0 after shrinking θ to a small neighborhood. Hence Definition 4.1 holds with constants uniform in small θ. Proof. Let Ps(θ)P_s(θ) and Pu(θ)P_u(θ) be the Riesz spectral projectors around the parts of the spectrum inside and outside the unit circle. They depend continuously on θ. With fixed bases Vs(0),Vu(0)V_s(0),V_u(0), the matrices Vs(θ):=Ps(θ)Vs(0)V_s(θ):=P_s(θ)V_s(0) and Vu(θ):=Pu(θ)Vu(0)V_u(θ):=P_u(θ)V_u(0) remain bases after shrinking. Choose continuous left inverses Ls(θ)L_s(θ) and Lu(θ)L_u(θ) for these bases and set Ds(θ)=Ls(θ)M(θ)Vs(θ),Du(θ)=Lu(θ)M(θ)Vu(θ).D_s(θ)=L_s(θ)M(θ)V_s(θ), D_u(θ)=L_u(θ)M(θ)V_u(θ). Then M(θ)Vs(θ)=Vs(θ)Ds(θ)M(θ)V_s(θ)=V_s(θ)D_s(θ) and similarly on Eu(θ)E_u(θ), and the blocks are continuous. On a compact smaller neighborhood the spectra stay a positive distance from the unit circle. Hence for each parameter there are k0k_0 and β0>0 _0>0 with ‖Ds(θ0)k0‖<e−2β0k0\|D_s( _0)^k_0\|<e^-2 _0k_0 and ‖Du(θ0)−k0‖<e−2β0k0\|D_u( _0)^-k_0\|<e^-2 _0k_0; by continuity these bounds persist on a neighborhood of θ0 _0, submultiplicativity extends them to all powers, and a finite subcover gives β>0β>0 and C with ‖Vs(θ)Ds(θ)k‖+‖Vu(θ)Du(θ)−k‖≤Ce−βk,k≥0.\|V_s(θ)D_s(θ)^k\|+\|V_u(θ)D_u(θ)^-k\|≤ Ce^-β k, k≥ 0. Uniform invertibility of Γ^N(0) _N(0) implies that Γ^∞(0)=[0(0)Vs(0)T(0)Vu(0)] _∞(0)=[ B_0(0)V_s(0)\ B_T(0)V_u(0)] is invertible. Indeed, the uniform bound gives smin(Γ^N(0))≥c0>0s_ ( _N(0))≥ c_0>0, the dichotomy bounds give Γ^N(0)→Γ^∞(0) _N(0)→ _∞(0), and continuity of singular values then gives smin(Γ^∞(0))≥c0s_ ( _∞(0))≥ c_0. Therefore Γ^∞(θ) _∞(θ) remains invertible on a neighborhood Θ1 _1, with supΘ1‖Γ^∞(θ)−1‖≤C∞ _ _1\| _∞(θ)^-1\|≤ C_∞. The uniform dichotomy bounds imply supθ∈Θ1‖Γ^N(θ)−Γ^∞(θ)‖≤Ce−βN. _θ∈ _1\| _N(θ)- _∞(θ)\|≤ Ce^-β N. Choose N∗N_* so large that C∞Ce−βN≤1/2C_∞Ce^-β N≤ 1/2 for N≥N∗N≥ N_*. Neumann’s lemma gives uniform inverses for N≥N∗N≥ N_*. For the finite set T0≤N<N∗T_0≤ N<N_*, continuity and invertibility at θ=0θ=0 give a smaller neighborhood Θ2⊂Θ1 _2⊂ _1 on which the least singular values remain bounded below. Thus Γ^N(θ) _N(θ) is uniformly invertible for every N≥T0N≥ T_0 and θ∈Θ2θ∈ _2, and Proposition 4.3 gives Definition 4.1 with constants uniform in θ. The nonlinear fixed-point argument uses endpoint weights matched to the Green kernel. For 0<α<γ/20<α<γ/2, set (4.7) wT(t)=e−αt+e−α(T−t).w_T(t)=e^-α t+e^-α(T-t). For each horizon define the weighted space XT=z=(z0,…,zT):‖z‖w,T<∞,‖z‖w,T=sup0≤t≤T‖zt‖wT(t).X_T=\z=(z_0,…,z_T):\|z\|_w,T<∞\, \|z\|_w,T= _0≤ t≤ T \|z_t\|w_T(t). All differentiability statements below concern the family of maps into these horizon-dependent spaces uniformly in T. The convolution estimate controls quadratic interior residuals. Lemma 4.5 (Weighted endpoint convolution). If 0<α<γ/20<α<γ/2, then there is a constant CconvC_ conv, independent of T and t, such that ∑s=0T−1κT(t,s)wT(s)2≤CconvwT(t)2. _s=0^T-1 _T(t,s)w_T(s)^2≤ C_ convw_T(t)^2. Moreover e−γt+e−γ(T−t)≤CconvwT(t)2e^-γ t+e^-γ(T-t)≤ C_ convw_T(t)^2, with κT _T from (4.2). Proof. Use wT(s)2≤2e−2αs+2e−2α(T−s)w_T(s)^2≤ 2e^-2α s+2e^-2α(T-s), which includes the cross term by 2ab≤a2+b22ab≤ a^2+b^2. For the interior kernel, splitting at s=ts=t gives ∑se−γ|t−s|e−2αs _se^-γ|t-s|e^-2α s ≤∑s≤te−γ(t−s)e−2αs+∑s>te−γ(s−t)e−2αs≤Ce−2αt, ≤ _s≤ te^-γ(t-s)e^-2α s+ _s>te^-γ(s-t)e^-2α s≤ Ce^-2α t, because the two ratios are e−(γ−2α)e^-(γ-2α) and e−(γ+2α)e^-(γ+2α). The change s↦T−s T-s gives ∑se−γ|t−s|e−2α(T−s)≤Ce−2α(T−t). _se^-γ|t-s|e^-2α(T-s)≤ Ce^-2α(T-t). For the endpoint kernel e−γ(t+T−s)e^-γ(t+T-s), ∑se−γ(t+T−s)e−2α(T−s)≤Ce−γt,∑se−γ(t+T−s)e−2αs≤Ce−γte−2αT. _se^-γ(t+T-s)e^-2α(T-s)≤ Ce^-γ t, _se^-γ(t+T-s)e^-2α s≤ Ce^-γ te^-2α T. Since γ>2αγ>2α, e−γt≤e−2αt≤wT(t)2e^-γ t≤ e^-2α t≤ w_T(t)^2, and the second bound is even smaller. For the other endpoint kernel, ∑se−γ(T−t+s)e−2αs≤Ce−γ(T−t),∑se−γ(T−t+s)e−2α(T−s)≤Ce−γ(T−t)e−2αT, _se^-γ(T-t+s)e^-2α s≤ Ce^-γ(T-t), _se^-γ(T-t+s)e^-2α(T-s)≤ Ce^-γ(T-t)e^-2α T, which are bounded by CwT(t)2Cw_T(t)^2 by the same use of γ>2αγ>2α; the displayed endpoint decay follows from the two endpoint comparisons, and combining the three kernel parts proves the estimate, including t=0,Tt=0,T. The convolution estimate is paired with a standard quadratic residual bound for the nonlinear equation. Lemma 4.6 (Quadratic residual from C2C^2 smoothness). Let ℱF be C2C^2 near (0,0)(0,0), assume ℱ(0,0)=0F(0,0)=0, and let L=Dℱ(0,0)L=DF(0,0). Define N(a,b)=ℱ(a,b)−L[a,b].N(a,b)=F(a,b)-L[a,b]. After shrinking the neighborhood, there is a constant CNC_N such that ∥N(a,b)∥≤CN(∥a∥+∥b∥)2 N(a,b) ≤ C_N( a + b )^2 and ∥N(a,b)−N(a~,b~)∥≤CN(∥a∥+∥b∥+∥a~∥+∥b~∥)(∥a−a~∥+∥b−b~∥). N(a,b)-N( a, b) ≤ C_N( a + b + a + b )( a- a + b- b ). Proof. Use the product norm |(a,b)|=‖a‖+‖b‖|(a,b)|=\|a\|+\|b\| and choose a convex ball U on which ‖D2ℱ‖≤M\|D^2F\|≤ M. Taylor’s formula with integral remainder gives, for z=(a,b)∈Uz=(a,b)∈ U, N(z)=∫01(1−θ)D2ℱ(θz)[z,z]θ,N(z)= _0^1(1-θ)D^2F(θ z)[z,z]\,dθ, so ‖N(a,b)‖≤CN(‖a‖+‖b‖)2\|N(a,b)\|≤ C_N(\|a\|+\|b\|)^2. Also DN(0)=0DN(0)=0 and DN(y)=∫01D2ℱ(θy)[y,⋅]θ,‖DN(y)‖≤M|y|.DN(y)= _0^1D^2F(θ y)[y,·]\,dθ, \|DN(y)\|≤ M|y|. Applying the fundamental theorem of calculus to N along the segment joining (a~,b~)( a, b) and (a,b)(a,b), which stays in the convex ball U, yields ‖N(a,b)−N(a~,b~)‖≤CN(‖a‖+‖b‖+‖a~‖+‖b~‖)(‖a−a~‖+‖b−b~‖).\|N(a,b)-N( a, b)\|≤ C_N(\|a\|+\|b\|+\| a\|+\| b\|)(\|a- a\|+\|b- b\|). We now combine the inverse estimate, the convolution bound, and the residual estimate. Theorem 4.7 (Affine endpoint reconstruction and first-order stability). Assume ℱ(0,0)=0F(0,0)=0, assume ℱF is C2C^2 and satisfies the residual estimates of Lemma 4.6, and assume that the linearization L=Dℱ(0,0)L=DF(0,0) with the rows (0,T)( B_0, B_T) satisfies the endpoint inverse property of Definition 4.1. Fix 0<α<γ/20<α<γ/2 and define wTw_T by (4.7). There are δ,ρ,r∗>0δ,ρ,r_*>0, independent of T, such that for every T≥T0T≥ T_0 and ‖Δ‖≤δ\| \|≤δ the affine nonlinear problem (4.8) ℱ(zt,zt+1)=0,0z0+TzT=ΔF(z_t,z_t+1)=0, B_0z_0+ B_Tz_T= has a unique solution in ‖z‖w,T≤ρ‖Δ‖\|z\|_w,T≤ρ\| \|, and this solution is unique in the fixed ball ‖z‖w,T≤r∗\|z\|_w,T≤ r_*. If zlin(Δ)=ℒTbdΔz^lin( )=L_T^bd , then, uniformly in T, (4.9) ‖zt(Δ)‖ \|z_t( )\| ≤CwT(t)‖Δ‖+CwT(t)2‖Δ‖2, ≤ Cw_T(t)\| \|+Cw_T(t)^2\| \|^2, (4.10) ‖zt(Δ)−ztlin(Δ)‖ \|z_t( )-z_t^lin( )\| ≤CwT(t)2‖Δ‖2. ≤ Cw_T(t)^2\| \|^2. For two admissible data Δ1,Δ2 _1, _2, (4.11) ‖zt(Δ1)−zt(Δ2)‖ \|z_t( _1)-z_t( _2)\| ≤CwT(t)‖Δ1−Δ2‖, ≤ Cw_T(t)\| _1- _2\|, (4.12) ‖[zt(Δ1)−zt(Δ2)]−[ztlin(Δ1)−ztlin(Δ2)]‖ \|[z_t( _1)-z_t( _2)]-[z_t^lin( _1)-z_t^lin( _2)]\| ≤CwT(t)2(‖Δ1‖+‖Δ2‖)‖Δ1−Δ2‖. ≤ Cw_T(t)^2(\| _1\|+\| _2\|)\| _1- _2\|. Equivalently, if ST(Δ)=z(Δ)S_T( )=z( ), then supT≥T0‖ST(Δ)−ℒTbdΔ‖w,T‖Δ‖→0(Δ→0), _T≥ T_0 \|S_T( )-L_T^bd \|_w,T\| \|→ 0 ( → 0), so the family is Fréchet differentiable at 0 uniformly in T, with derivative ℒTbdL_T^bd. Proof. Work in XTX_T and write ∥⋅∥w=∥⋅∥w,T\|·\|_w=\|·\|_w,T inside this proof. From Definition 4.1 and Lemma 4.5, if ‖fs‖≤MfwT(s)2\|f_s\|≤ M_fw_T(s)^2, then ‖(ℒTforf)t‖≤C1MfwT(t)2\|(L_T forf)_t\|≤ C_1M_fw_T(t)^2 and ‖(ℒTbdb)t‖≤C1wT(t)‖b‖\|(L_T^bdb)_t\|≤ C_1w_T(t)\|b\|, and the sharper boundary estimate ‖(ℒTbdb)t‖≤C1wT(t)2‖b‖\|(L_T^bdb)_t\|≤ C_1w_T(t)^2\|b\| holds for any boundary datum b, because e−γt+e−γ(T−t)≤CwT(t)2e^-γ t+e^-γ(T-t)≤ Cw_T(t)^2. This will be applied to quadratic endpoint residuals. The estimate wT(t+1)≤eαwT(t)w_T(t+1)≤ e^αw_T(t) and Lemma 4.6 give, on every ball ‖z‖w,‖z~‖w≤R\|z\|_w,\| z\|_w≤ R inside the C2C^2 tube, ‖N(zt,zt+1)‖≤C2R2wT(t)2,‖N(zt,zt+1)−N(z~t,z~t+1)‖≤C2RwT(t)2‖z−z~‖w.\|N(z_t,z_t+1)\|≤ C_2R^2w_T(t)^2, \|N(z_t,z_t+1)-N( z_t, z_t+1)\|≤ C_2Rw_T(t)^2\|z- z\|_w. Since wT(t)≤2w_T(t)≤ 2, choose r∗>0r_*>0 so that the radius-2r∗2r_* weighted ball keeps all pairs (zt,zt+1)(z_t,z_t+1) in the fixed C2C^2 neighborhood for every T, and so that C1C2r∗<1/4C_1C_2r_*<1/4. Then choose δ so that 2C1δ<r∗2C_1δ<r_* and 4C12C2δ<1/24C_1^2C_2δ<1/2. With R=2C1‖Δ‖R=2C_1\| \|, define Ψ(z)=y (z)=y by L[yt,yt+1]=−N(zt,zt+1),0y0+TyT=Δ.L[y_t,y_t+1]=-N(z_t,z_t+1), B_0y_0+ B_Ty_T= . For ‖z‖w≤R\|z\|_w≤ R and ‖Δ‖≤δ\| \|≤δ, ‖Ψ(z)‖w≤C1‖Δ‖+C1C2R2≤R,‖Ψ(z)−Ψ(z~)‖w≤C1C2R‖z−z~‖w<12∥z−z~∥w.\| (z)\|_w≤ C_1\| \|+C_1C_2R^2≤ R, \| (z)- ( z)\|_w≤ C_1C_2R\|z- z\|_w< 12\|z- z\|_w. Banach’s theorem gives a unique fixed point in ‖z‖w≤R\|z\|_w≤ R. If z and z~ z are any two solutions in ∥⋅∥w≤r∗\|·\|_w≤ r_*, then their difference solves the same linear problem with zero boundary datum and forcing −[N(z)−N(z~)]-[N(z)-N( z)]; hence ‖z−z~‖w≤C1C2r∗‖z−z~‖w\|z- z\|_w≤ C_1C_2r_*\|z- z\|_w, so z=z~z= z. Subtracting zlin=ℒTbdΔz^lin=L_T^bd from the fixed point leaves zero boundary datum and forcing −N(zt,zt+1)-N(z_t,z_t+1). The convolution estimate gives (4.10); (4.9) follows by adding the boundary solution. For two data, δz=z(Δ1)−z(Δ2)δ z=z( _1)-z( _2) solves L[δzt,δzt+1]=−[N(z1)−N(z2)]t,0δz0+TδzT=Δ1−Δ2.L[δ z_t,δ z_t+1]=-[N(z_1)-N(z_2)]_t, B_0δ z_0+ B_Tδ z_T= _1- _2. The residual Lipschitz bound gives an interior forcing bounded by CwT(t)2(‖Δ1‖+‖Δ2‖)‖δz‖wCw_T(t)^2(\| _1\|+\| _2\|)\|δ z\|_w. The Green estimate and Lemma 4.5 therefore give ‖δz‖w≤C‖Δ1−Δ2‖+C(‖Δ1‖+‖Δ2‖)‖δz‖w,\|δ z\|_w≤ C\| _1- _2\|+C(\| _1\|+\| _2\|)\|δ z\|_w, and the last term is absorbed by the choice of δ. This proves (4.11). Subtracting zlin(Δ1)−zlin(Δ2)z^lin( _1)-z^lin( _2) leaves zero boundary datum and the same forcing. Inserting the just-proved Lipschitz bound for ‖δz‖w\|δ z\|_w in the forcing estimate gives (4.12). Taking one datum equal to zero gives the first-order expansion; since wT(t)≤2w_T(t)≤ 2, (4.10) implies ‖z(Δ)−zlin(Δ)‖w,T≤C‖Δ‖2\|z( )-z^lin( )\|_w,T≤ C\| \|^2 uniformly in T, which is the stated uniform Fréchet differentiability. The affine boundary argument also covers smooth endpoint maps after their quadratic boundary residual is included in the fixed-point map. Theorem 4.8 (Nonlinear endpoint boundary data). Assume ℱ(0,0)=0F(0,0)=0, ℱF is C2C^2 with the residual bounds of Lemma 4.6, and the linearized equation with rows (0,T)( B_0, B_T) satisfies Definition 4.1 with exponent γ. Fix 0<α<γ/20<α<γ/2 and define wTw_T by (4.7). Replace the affine boundary condition by (4.13) ℬ(z0,zT)=Δ,B(z_0,z_T)= , where ℬ(0,0)=0B(0,0)=0, Dℬ(0,0)[a,b]=0a+TbDB(0,0)[a,b]= B_0a+ B_Tb, and ℬB is C2C^2 near the origin with bounded second derivative. Then, for sufficiently small δ>0δ>0 independent of T, each ‖Δ‖≤δ\| \|≤δ has a solution z(Δ)z( ) in ‖z‖w,T≤ρ‖Δ‖\|z\|_w,T≤ρ\| \|, and this solution is unique in a fixed local ball ‖z‖w,T≤r∗\|z\|_w,T≤ r_* independent of T and Δ . Moreover, with zlin(Δ):=ℒTbdΔz^lin( ):=L_T^bd for these linearized endpoint rows, (4.14) ‖zt(Δ)‖ \|z_t( )\| ≤CwT(t)‖Δ‖+CwT(t)2‖Δ‖2, ≤ Cw_T(t)\| \|+Cw_T(t)^2\| \|^2, (4.15) ‖zt(Δ1)−zt(Δ2)‖ \|z_t( _1)-z_t( _2)\| ≤CwT(t)‖Δ1−Δ2‖, ≤ Cw_T(t)\| _1- _2\|, (4.16) ‖[zt(Δ1)−zt(Δ2)]−ztlin(Δ1−Δ2)‖ \|[z_t( _1)-z_t( _2)]-z_t^lin( _1- _2)\| ≤CwT(t)2(‖Δ1‖+‖Δ2‖)‖Δ1−Δ2‖. ≤ Cw_T(t)^2(\| _1\|+\| _2\|)\| _1- _2\|. In particular ‖zt(Δ)−ztlin(Δ)‖≤CwT(t)2‖Δ‖2\|z_t( )-z_t^lin( )\|≤ Cw_T(t)^2\| \|^2, and the same uniform Fréchet differentiability statement as in Theorem 4.7 holds with derivative ℒTbdL_T^bd. Proof. Let RB(a,b)=ℬ(a,b)−0a−TbR_B(a,b)=B(a,b)- B_0a- B_Tb. Then RB(0,0)=DRB(0,0)=0R_B(0,0)=DR_B(0,0)=0 and, after shrinking, as in Lemma 4.6, ‖RB(a,b)‖≤C(‖a‖+‖b‖)2,‖RB(a,b)−RB(a~,b~)‖≤CΣ(‖a−a~‖+‖b−b~‖),\|R_B(a,b)\|≤ C(\|a\|+\|b\|)^2, \|R_B(a,b)-R_B( a, b)\|≤ C \,(\|a- a\|+\|b- b\|), where Σ=‖a‖+‖b‖+‖a~‖+‖b~‖ =\|a\|+\|b\|+\| a\|+\| b\|. Define Ψ(z)=y (z)=y by L[yt,yt+1]=−N(zt,zt+1),0y0+TyT=Δ−RB(z0,zT).L[y_t,y_t+1]=-N(z_t,z_t+1), B_0y_0+ B_Ty_T= -R_B(z_0,z_T). Since wT(0),wT(T)≤2w_T(0),w_T(T)≤ 2, on ‖z‖w≤R\|z\|_w≤ R the boundary residual is O(R2)O(R^2) and contributes at time t at most C(e−γt+e−γ(T−t))R2≤CwT(t)2R2C(e^-γ t+e^-γ(T-t))R^2≤ Cw_T(t)^2R^2, while the interior residual contributes CwT(t)2R2Cw_T(t)^2R^2 by Lemma 4.5; the difference estimates carry the same kernels with R2R^2 replaced by R‖z−z~‖wR\|z- z\|_w. Hence ‖Ψ(z)‖w≤C‖Δ‖+CR2,‖Ψ(z)−Ψ(z~)‖w≤CR‖z−z~‖w,\| (z)\|_w≤ C\| \|+CR^2, \| (z)- ( z)\|_w≤ CR\|z- z\|_w, and the contraction scheme of Theorem 4.7, with R=2C‖Δ‖R=2C\| \| and δ small, gives a fixed point in ‖z‖w≤ρ‖Δ‖\|z\|_w≤ρ\| \|; the same difference estimate on a fixed ball ‖z‖w≤r∗\|z\|_w≤ r_* with Cr∗<1Cr_*<1 gives uniqueness there. Let r=z−ℒTbdΔr=z-L_T^bd . Then L[rt,rt+1]=−N(zt,zt+1),0r0+TrT=−RB(z0,zT),L[r_t,r_t+1]=-N(z_t,z_t+1), B_0r_0+ B_Tr_T=-R_B(z_0,z_T), with ‖N(zt,zt+1)‖≤CwT(t)2‖Δ‖2\|N(z_t,z_t+1)\|≤ Cw_T(t)^2\| \|^2 and ‖RB(z0,zT)‖≤C‖Δ‖2\|R_B(z_0,z_T)\|≤ C\| \|^2 by the fixed-point bound. Lemma 4.5 handles the interior part, and the endpoint part is multiplied by e−γt+e−γ(T−t)≤CwT(t)2e^-γ t+e^-γ(T-t)≤ Cw_T(t)^2, which proves the size and first-order estimates. For stability, put zi=z(Δi)z_i=z( _i) and δz=z1−z2δ z=z_1-z_2, with forcing −[N(z1)−N(z2)]-[N(z_1)-N(z_2)] and boundary datum Δ1−Δ2−[RB(z1,0,z1,T)−RB(z2,0,z2,T)] _1- _2-[R_B(z_1,0,z_1,T)-R_B(z_2,0,z_2,T)]. Because ‖zi‖w≤C‖Δi‖\|z_i\|_w≤ C\| _i\|, the residual differences obey ‖RB(z1,0,z1,T)−RB(z2,0,z2,T)‖ \|R_B(z_1,0,z_1,T)-R_B(z_2,0,z_2,T)\| ≤C(‖Δ1‖+‖Δ2‖)‖δz‖w, ≤ C(\| _1\|+\| _2\|)\|δ z\|_w, ‖N(z1,t,z1,t+1)−N(z2,t,z2,t+1)‖ \|N(z_1,t,z_1,t+1)-N(z_2,t,z_2,t+1)\| ≤CwT(t)2(‖Δ1‖+‖Δ2‖)‖δz‖w, ≤ Cw_T(t)^2(\| _1\|+\| _2\|)\|δ z\|_w, so the Green estimate gives ‖δz‖w≤C‖Δ1−Δ2‖+C(‖Δ1‖+‖Δ2‖)‖δz‖w\|δ z\|_w≤ C\| _1- _2\|+C(\| _1\|+\| _2\|)\|δ z\|_w, and absorption after shrinking δ proves (4.15). Finally, r=δz−zlin(Δ1−Δ2)r=δ z-z^lin( _1- _2) solves the linear problem whose forcing and boundary datum are the two residual differences above; inserting the just-proved bound ‖δz‖w≤C‖Δ1−Δ2‖\|δ z\|_w≤ C\| _1- _2\| bounds them by CwT(t)2(‖Δ1‖+‖Δ2‖)‖Δ1−Δ2‖Cw_T(t)^2(\| _1\|+\| _2\|)\| _1- _2\| and C(‖Δ1‖+‖Δ2‖)‖Δ1−Δ2‖C(\| _1\|+\| _2\|)\| _1- _2\|, the interior part is handled by Lemma 4.5, the endpoint part is multiplied by e−γt+e−γ(T−t)≤CwT(t)2e^-γ t+e^-γ(T-t)≤ Cw_T(t)^2, and (4.16) follows. Setting Δ2=0 _2=0 gives the stated first-order expansion. Lemma 4.9 (Uniform perturbation of endpoint rows). Assume Definition 4.1 holds for L[ht,ht+1]=ftL[h_t,h_t+1]=f_t with rows (0,T)( B_0, B_T). There is η>0η>0 such that rows (~0,~T)( B_0, B_T) satisfying ‖~0−0‖+‖~T−T‖≤η\| B_0- B_0\|+\| B_T- B_T\|≤η have the same endpoint inverse property, with constants uniform over this class. Consequently, if nonlinear boundary maps ℬθB_θ have uniformly bounded C2C^2 remainders and Dℬθ(0,0)DB_θ(0,0) is sufficiently close to Dℬ0(0,0)DB_0(0,0), then Theorem 4.8 applies uniformly in θ. Proof. Work in ℓ∞(0,…,T;ℝn) ^∞(\0,…,T\;R^n). Write the perturbed rows as 0+E0 B_0+E_0 and T+ET B_T+E_T, with ε=‖E0‖+‖ET‖ =\|E_0\|+\|E_T\|, and let ℒTbdL_T^bd denote the unperturbed boundary inverse. A solution of the perturbed problem is exactly a fixed point of h=ℒTforf+ℒTbd(b−E0h0−EThT)h=L_T^forf+L_T^bd(b-E_0h_0-E_Th_T). Since ∑s=0T−1κT(t,s)≤C _s=0^T-1 _T(t,s)≤ C uniformly in T and t, the Green estimate gives ‖ℒTbdc‖∞≤C‖c‖\|L_T^bdc\|_∞≤ C\|c\| and ‖ℒTforf‖∞≤Csups‖fs‖\|L_T^forf\|_∞≤ C _s\|f_s\|. The map is therefore well defined and is a contraction when Cε<1/2C <1/2, uniformly in T, so existence and uniqueness follow for all horizons. Evaluating the fixed-point identity at the endpoints and using the sharper endpoint part of the unperturbed Green estimate gives ‖h0‖+‖hT‖≤C(‖b‖+ε(‖h0‖+‖hT‖)+∑s(e−γs+e−γ(T−s))‖fs‖),\|h_0\|+\|h_T\|≤ C (\|b\|+ (\|h_0\|+\|h_T\|)+ _s(e^-γ s+e^-γ(T-s))\|f_s\| ), and after decreasing η so that Cε<1/2C <1/2 the middle term is absorbed. Substituting the resulting endpoint bound in the fixed-point identity gives the unperturbed Green estimate plus products e−γ(t+s)e^-γ(t+s), e−γ(t+T−s)e^-γ(t+T-s), e−γ(T−t+s)e^-γ(T-t+s), and e−γ(2T−t−s)e^-γ(2T-t-s), multiplied by ‖fs‖\|f_s\|. Since t+s≥|t−s|t+s≥|t-s| and 2T−t−s≥|t−s|2T-t-s≥|t-s|, the first and fourth are bounded by e−γ|t−s|e^-γ|t-s|; the middle two are precisely the endpoint-image kernels in (4.2). Hence the perturbed rows satisfy Definition 4.1 with uniform constants. Uniform C2C^2 boundary remainders then satisfy the same quadratic and Lipschitz bounds used in Theorem 4.8, giving uniform applicability. 5 Local two-point expansions for Pontryagin branches We now apply the reconstruction framework to the reduced Pontryagin equation (2.6). The main theorem states separately the affine-row and nonlinear-row expansions. These are local stationary-branch statements, not optimality claims. Theorem 5.1 (Local branch). Assume Condition 2.2 and the hypotheses of Lemma 2.1. Let (x∞,u∞,p∞)(x_∞,u_∞,p_∞) be an interior stationary triple, and let ν be the local stationarity graph from that lemma. Put z=(ξ,π)z=(ξ,π) and ℱ~(z,z+):=ℱ((x∞,p∞)+z,(x∞,p∞)+z+). F(z,z^+):=F((x_∞,p_∞)+z,(x_∞,p_∞)+z^+). Let ℬB be either ℬ(a,b)=0a+TbB(a,b)= B_0a+ B_Tb, or a C2C^2 endpoint map with ℬ(0,0)=0,Dℬ(0,0)[a,b]=0a+Tb,B(0,0)=0, DB(0,0)[a,b]= B_0a+ B_Tb, and bounded second derivative near the origin. Assume that the linearized two-point problem with rows (0,T)( B_0, B_T) satisfies Definition 4.1 with exponent γ. Then for every 0<α<γ/20<α<γ/2 there are constants δ,ρ,r∗,C>0δ,ρ,r_*,C>0, independent of T, such that for all T≥T0T≥ T_0 and ‖Δbd‖≤δ\| _bd\|≤δ the problem ℱ~(zt,zt+1)=0,ℬ(z0,zT)=Δbd F(z_t,z_t+1)=0, (z_0,z_T)= _bd has a solution satisfying ‖z‖w,T≤ρ‖Δbd‖\|z\|_w,T≤ρ\| _bd\|. This solution is unique in the fixed ball ‖z‖w,T≤r∗\|z\|_w,T≤ r_*. The reconstructed controls ut(Δbd)=ν(x∞+ξt(Δbd),p∞+πt+1(Δbd)),0≤t<T,u_t( _bd)=ν(x_∞+ _t( _bd),p_∞+ _t+1( _bd)), 0≤ t<T, belong to intUintU after shrinking δ if necessary. Let zlin(η)=(ξlin(η),πlin(η))z^lin(η)=(ξ^lin(η),π^lin(η)) solve the linearized problem with endpoint datum η, and define utlin(η):=Dν(x∞,p∞)[ξtlin(η),πt+1lin(η)].u_t^lin(η):=Dν(x_∞,p_∞)[ _t^lin(η), _t+1^lin(η)]. Uniformly in T, for 0≤t<T0≤ t<T, (5.1) ‖zt(Δbd)‖+‖ut(Δbd)−u∞‖ \|z_t( _bd)\|+\|u_t( _bd)-u_∞\| ≤CwT(t)‖Δbd‖, ≤ Cw_T(t)\| _bd\|, (5.2) ‖zt(Δbd)−ztlin(Δbd)‖+‖(ut(Δbd)−u∞)−utlin(Δbd)‖ \|z_t( _bd)-z_t^lin( _bd)\|+\|(u_t( _bd)-u_∞)-u_t^lin( _bd)\| ≤CwT(t)2‖Δbd‖2, ≤ Cw_T(t)^2\| _bd\|^2, (5.3) ‖zt(Δbd1)−zt(Δbd2)‖+‖ut(Δbd1)−ut(Δbd2)‖ \|z_t( _bd^1)-z_t( _bd^2)\|+\|u_t( _bd^1)-u_t( _bd^2)\| ≤CwT(t)‖Δbd1−Δbd2‖. ≤ Cw_T(t)\| _bd^1- _bd^2\|. The same estimates for the z-terms hold also at t=Tt=T. The branch is a local stationary Pontryagin branch; finite-horizon optimality is not asserted. Proof. After translation, the reduced equation has the quadratic residual bounds from Condition 2.2. The endpoint residual is zero in the affine case and quadratic in the C2C^2 case. Hence Theorem 4.8 applies with linearized rows (0,T)( B_0, B_T) and gives a solution of size O(‖Δbd‖)O(\| _bd\|) and uniqueness in the fixed weighted ball, together with the three estimates in (5.1)–(5.3) for the z-terms; the quadratic part in the size bound is absorbed into CwT(t)‖Δbd‖Cw_T(t)\| _bd\| after shrinking δ, since wT(t)≤2w_T(t)≤ 2. Shrinking δ also keeps this ball inside the tube where ν is defined and the controls lie in intUintU. Since ν satisfies Hu(x,u,p+)=0H_u(x,u,p_+)=0, substituting ut=ν(xt,pt+1)u_t=ν(x_t,p_t+1) into the reduced equation recovers the interior Pontryagin equations. Thus the branch is stationary. No second-order sufficient condition or global optimality argument is used. It remains only to transfer the estimates to u. Set ζt(Δbd)=(ξt(Δbd),πt+1(Δbd)) _t( _bd)=( _t( _bd), _t+1( _bd)). Since wT(t+1)≤eαwT(t)w_T(t+1)≤ e^αw_T(t), the estimates for ztz_t control ζt _t with the same weight. Taylor’s formula gives ut(Δbd)−u∞=Dν(x∞,p∞)ζt(Δbd)+O(‖ζt(Δbd)‖2)u_t( _bd)-u_∞=Dν(x_∞,p_∞) _t( _bd)+O(\| _t( _bd)\|^2), with a constant independent of T. Applying the same expansion to ζtlin=(ξtlin,πt+1lin) _t^lin=( _t^lin, _t+1^lin), and using the estimate for ζt−ζtlin _t- _t^lin together with ‖ζt‖2≤CwT(t)2‖Δbd‖2\| _t\|^2≤ Cw_T(t)^2\| _bd\|^2, gives the first-order control remainder. The mean-value formula for ν gives the Lipschitz estimate. These bounds yield the u-parts of (5.1)–(5.3). Corollary 5.2 (Original Pontryagin endpoint rows). Assume the hypotheses of Theorem 5.1, except for its endpoint-row hypothesis. Let y∞∈Yy_∞∈ Y satisfy p∞=rx(x∞,y∞)p_∞=r_x(x_∞,y_∞). Assume that r is C3C^3 in x, uniformly for y near y∞y_∞, and that y↦rx(x,y)y r_x(x,y) and y↦rxx(x,y)y r_x(x,y) are continuous uniformly in x on the terminal tube. For y near y∞y_∞ define ℬy(z0,zT) _y(z_0,z_T) =(ξ0πT−(rx(x∞+ξT,y)−rx(x∞,y))), = _0 _T- (r_x(x_∞+ _T,y)-r_x(x_∞,y) ), Δoc(xin,y) _ oc(x_ in,y) =(xin−x∞rx(x∞,y)−p∞), = x_ in-x_∞r_x(x_∞,y)-p_∞, where z=(ξ,π)=(x−x∞,p−p∞)z=(ξ,π)=(x-x_∞,p-p_∞). Then ℬy(z0,zT)=Δoc(xin,y)B_y(z_0,z_T)= _ oc(x_ in,y) is equivalent to x0=xin,pT=rx(xT,y).x_0=x_ in, p_T=r_x(x_T,y). If the rows Dℬy∞(0,0)DB_y_∞(0,0) satisfy Definition 4.1, then there are neighborhoods Y1Y_1 of y∞y_∞ and X1X_1 of x∞x_∞, and a δ>0δ>0, such that for every T≥T0T≥ T_0 and every (xin,y)∈X1×Y1(x_ in,y)∈ X_1× Y_1 with ‖Δoc(xin,y)‖≤δ\| _ oc(x_ in,y)\|≤δ, the original Pontryagin boundary value problem has the local stationary branch and the estimates of Theorem 5.1, uniformly in T and y. The branch is locally Lipschitz in Δoc _ oc, uniformly in y. If, in addition, y↦rx(x,y)y r_x(x,y) and y↦rxx(x,y)y r_x(x,y) are locally Lipschitz uniformly in x on the terminal tube, then the branch is jointly locally Lipschitz in (xin,y)(x_ in,y) on this endpoint-data neighborhood. Proof. Since ξ0=x0−x∞ _0=x_0-x_∞ and πT=pT−p∞ _T=p_T-p_∞, the boundary equation ℬy(z0,zT)=Δoc(xin,y)B_y(z_0,z_T)= _ oc(x_ in,y) is exactly x0=xinx_0=x_ in, pT=rx(xT,y)p_T=r_x(x_T,y). Set Dy:=Dℬy(0,0)D_y:=DB_y(0,0) and Ry:=ℬy−DyR_y:=B_y-D_y. Then Dy[z0,zT]=(ξ0πT−rxx(x∞,y)ξT),D_y[z_0,z_T]= _0 _T-r_x(x_∞,y) _T, so at y=y∞y=y_∞ the endpoint rows are 0=(I000),T=(00−rxx(x∞,y∞)I), B_0= pmatrixI&0\\ 0&0 pmatrix, B_T= pmatrix0&0\\ -r_x(x_∞,y_∞)&I pmatrix, and the boundary residual has only the terminal component, given by Taylor’s formula as Ry(z0,zT)=−(0∫01[rxx(x∞+θξT,y)−rxx(x∞,y)]ξTθ).R_y(z_0,z_T)=- 0 _0^1 [r_x(x_∞+θ _T,y)-r_x(x_∞,y) ] _T\,dθ. The uniform C3C^3 assumption in x gives the quadratic and Lipschitz estimates ‖Ry(z)‖≤C‖z‖2,‖Ry(z)−Ry(z~)‖≤C(‖z‖+‖z~‖)‖z−z~‖,\|R_y(z)\|≤ C\|z\|^2, \|R_y(z)-R_y( z)\|≤ C(\|z\|+\| z\|)\|z- z\|, with constants uniform for y in a smaller neighborhood of y∞y_∞, and continuity of y↦rxx(x,y)y r_x(x,y) gives Dy→Dy∞D_y→ D_y_∞. Hence Lemma 4.9 yields the endpoint Green property for DyD_y, uniformly for y∈Y1y∈ Y_1 after shrinking Y1Y_1, and Theorem 5.1 with endpoint map ℬ=ℬyB=B_y and boundary datum Δoc(xin,y) _ oc(x_ in,y) proves the uniform local branch, the stated estimates, and the fixed-y Lipschitz dependence in the boundary datum. For the joint Lipschitz statement, let ziz^i be the branches of (xini,yi)(x_ in^i,y_i) and write Di=DyiD_i=D_y_i, Ri=RyiR_i=R_y_i, Δi=Δoc(xini,yi) _i= _ oc(x_ in^i,y_i), δz=z1−z2δ z=z^1-z^2. Subtracting the boundary equations and writing both with the linear part D1D_1 gives D1[δz0,δzT]=Δ1−Δ2−[R1(z1)−R1(z2)]−(D1−D2)[z02,zT2]−[R1(z2)−R2(z2)],D_1[δ z_0,δ z_T]= _1- _2-[R_1(z^1)-R_1(z^2)]-(D_1-D_2)[z^2_0,z^2_T]-[R_1(z^2)-R_2(z^2)], while the interior equation has forcing −[N(z1)−N(z2)]-[N(z^1)-N(z^2)]. On a sufficiently small endpoint-data neighborhood, where ‖zi‖w,T≤ε\|z^i\|_w,T≤ , the nonlinear residual differences are bounded by CεwT(t)2‖δz‖w,TC w_T(t)^2\|δ z\|_w,T for 0≤t<T0≤ t<T and by Cε‖δz‖w,TC \|δ z\|_w,T at the endpoints, and the uniform Lipschitz assumptions in y give ‖Δ1−Δ2‖≤C(‖xin1−xin2‖+‖y1−y2‖)\| _1- _2\|≤ C(\|x_ in^1-x_ in^2\|+\|y_1-y_2\|). Moreover D1−D2D_1-D_2 contains only the terminal block −[rxx(x∞,y1)−rxx(x∞,y2)]ξT-[r_x(x_∞,y_1)-r_x(x_∞,y_2)] _T, and the integral formula shows that R1(z2)−R2(z2)R_1(z^2)-R_2(z^2) is controlled by the Lipschitz constant of rxxr_x in y times ‖ξT‖\| _T\| of z2z^2; since ‖z2‖w,T≤ε\|z^2\|_w,T≤ and wT(0),wT(T)≤2w_T(0),w_T(T)≤ 2, ‖(D1−D2)[z02,zT2]‖+‖R1(z2)−R2(z2)‖≤Cε‖y1−y2‖.\|(D_1-D_2)[z^2_0,z^2_T]\|+\|R_1(z^2)-R_2(z^2)\|≤ C \|y_1-y_2\|. The uniform Green estimate for D1D_1 therefore yields ‖δz‖w,T≤C(‖xin1−xin2‖+‖y1−y2‖)+Cε‖δz‖w,T\|δ z\|_w,T≤ C(\|x_ in^1-x_ in^2\|+\|y_1-y_2\|)+C \|δ z\|_w,T, and shrinking the endpoint neighborhood so that Cε<1/2C <1/2 absorbs the last term. The same conclusion for the controls follows from the local Lipschitz continuity of ν. 6 Data-level symplectic transversality and LQ data Before verifying the abstract inverse property, we fix the Riccati sign convention used by the maximization Hamiltonian and then use the standard symplectic and Lagrangian subspace viewpoint for Riccati equations and Hamiltonian and symplectic pencils [arnold1989mathematical, anderson2007optimal, bittanti1991riccati, lancaster1995algebraic, laub1979schur]. For notational simplicity we set λ=1λ=1 and a positive λ can be absorbed into Q and R. Here B may be rectangular. The LQ verification first treats the unconstrained stationarity graph; compact control sets containing the reference control in their interior are imposed afterward by shrinking the endpoint ball. For the LQ dynamics xt+1=Axt+Butx_t+1=Ax_t+Bu_t with running term −12xt⊤Qxt−12ut⊤Rut- 12x_t Qx_t- 12u_t Ru_t, Q⪰0Q 0, R≻0R 0, the Hamiltonian stationarity equation is ut=R−1B⊤pt+1u_t=R^-1B p_t+1 and the state-costate equations are (6.1) xt+1=Axt+BR−1B⊤pt+1,pt=A⊤pt+1−Qxt.x_t+1=Ax_t+BR^-1B p_t+1, p_t=A p_t+1-Qx_t. The resulting recursion is the backward Riccati equation written with p=−Pxp=-Px: if pt+1=−Pt+1xt+1p_t+1=-P_t+1x_t+1 and Pt+1=Pt+1⊤⪰0P_t+1=P_t+1 0, then substitution gives Pt=Q+A⊤Pt+1A−A⊤Pt+1B(R+B⊤Pt+1B)−1B⊤Pt+1A,P_t=Q+A P_t+1A-A P_t+1B(R+B P_t+1B)^-1B P_t+1A, with ut=−(R+B⊤Pt+1B)−1B⊤Pt+1Axtu_t=-(R+B P_t+1B)^-1B P_t+1Ax_t. Here Kt=(R+B⊤Pt+1B)−1B⊤Pt+1AK_t=(R+B P_t+1B)^-1B P_t+1A, and the identity Pt=Q+(A−BKt)⊤Pt+1(A−BKt)+Kt⊤RKtP_t=Q+(A-BK_t) P_t+1(A-BK_t)+K_t RK_t gives Pt=Pt⊤⪰0P_t=P_t 0. This fixes the sign convention used below. For the structural verification it is more convenient to eliminate pt+1p_t+1 in (6.1). Assume A is invertible and put (6.2) M(A,B,Q,R)=(A+BR−1B⊤A−⊤QBR−1B⊤A−⊤A−⊤QA−⊤).M(A,B,Q,R)= pmatrixA+BR^-1B A^- Q&BR^-1B A^- \\ A^- Q&A^- pmatrix. Then (xt+1pt+1)=M(A,B,Q,R)(xtpt) x_t+1p_t+1=M(A,B,Q,R) x_tp_t, and let =(0I−I0)J= ( smallmatrix0&I\\ -I&0 smallmatrix ). Lemma 6.1 (Symplectic LQ matrix). For every A,B,Q,RA,B,Q,R with A invertible, Q=Q⊤Q=Q , and R=R⊤≻0R=R 0, the matrix M(A,B,Q,R)M(A,B,Q,R) in (6.2) is symplectic: M⊤M=M JM=J. Hence its spectrum is reciprocal, counting algebraic multiplicities, and hyperbolicity is equivalent to absence of eigenvalues on the unit circle. Proof. Write G=BR−1B⊤G=BR^-1B , so G=G⊤G=G . The LQ equations imply p+=A−⊤(p+Qx)p^+=A^- (p+Qx) and x+=Ax+Gp+x^+=Ax+Gp^+, which is (6.2). Factor M=(IG0I)(A00A−⊤)(I0QI).M= pmatrixI&G\\ 0&I pmatrix pmatrixA&0\\ 0&A^- pmatrix pmatrixI&0\\ Q&I pmatrix. The first and third factors are symplectic shears because G and Q are symmetric, and the middle factor is symplectic for every invertible A. Hence their product satisfies M⊤M=M JM=J. The next result is the structural form of the endpoint hypothesis. It is stated for an arbitrary hyperbolic symplectic transition matrix, before imposing that the matrix comes from LQ matrix data. We use only standard symplectic facts on Lagrangian subspaces and matrix pencils [arnold1989mathematical]. The symplectic assumption is not needed for the algebraic Green construction itself, but it makes the stable and unstable spaces Lagrangian and gives the data-level subspace interpretation used in the LQ corollaries. Theorem 6.2 (Symplectic transversality). Let M∈Sp(2d,ℝ)M (2d,R) be hyperbolic, with stable and unstable subspaces EsE_s and EuE_u. Let 0,T∈ℝ2d×2d B_0, B_T ^2d× 2d define a linear two-endpoint condition 0h0+ThT=b B_0h_0+ B_Th_T=b for sequences ht∈ℝ2dh_t ^2d. Choose any bases VsV_s and VuV_u of EsE_s and EuE_u. If the limiting boundary matrix (6.3) Γ∞=[0VsTVu] _∞=[ B_0V_s B_TV_u] is nonsingular, then the forced finite-horizon problem ht+1=Mht+gth_t+1=Mh_t+g_t, 0h0+ThT=b B_0h_0+ B_Th_T=b, 0≤t<T0≤ t<T, satisfies the endpoint inverse estimate of Definition 4.1 for all sufficiently large T. If the finitely many scaled matrices Γ^N _N for T0≤N<N∗T_0≤ N<N_* are invertible, where N∗N_* is such a large-horizon cutoff, the same estimate holds uniformly for all T≥T0T≥ T_0. Moreover, the condition is open: if M(θ)∈Sp(2d,ℝ)M(θ) (2d,R), 0(θ) B_0(θ), and T(θ) B_T(θ) depend continuously on θ, M(0)=M(0)=M, and the hyperbolic splitting persists, then (6.3) remains nonsingular and the inverse constants are uniform for all small θ on all sufficiently large horizons. If the finitely many scaled matrices for T0≤N<N∗T_0≤ N<N_* are invertible at θ=0θ=0, the uniformity holds for all T≥T0T≥ T_0 after shrinking the parameter neighborhood. Proof. Let ω(z,w)=z⊤wω(z,w)=z Jw. The spectrum of a symplectic matrix is reciprocal; because M has no unit-circle spectrum, the algebraic dimensions inside and outside the unit circle are both d. If z,w∈Esz,w∈ E_s, then ω(Mkz,Mkw)=ω(z,w)ω(M^kz,M^kw)=ω(z,w) while Mkz,Mkw→0M^kz,M^kw→ 0 even on generalized stable eigenspaces, so ω(z,w)=0ω(z,w)=0. Hence EsE_s is an isotropic d-plane and is Lagrangian; applying the same argument to M−1M^-1 gives the result for EuE_u. The Green estimate itself uses only the hyperbolic splitting and the boundary transversality. Choose bases so that MVs=VsDsMV_s=V_sD_s and MVu=VuDuMV_u=V_uD_u, and let Γ^N _N be the scaled boundary matrix (4.5). The dichotomy bounds give ‖TVsDsN‖+‖0VuDu−N‖≤Ce−γN\| B_TV_sD_s^\,N\|+\| B_0V_uD_u^-N\|≤ Ce^-γ N, hence Γ^N→[0VsTVu]=Γ∞ _N→[ B_0V_s\ B_TV_u]= _∞. If Γ∞ _∞ is nonsingular, a Neumann-series argument makes Γ^N _N uniformly invertible for all sufficiently large N. The finitely many shorter horizons are covered by the stated invertibility assumption. Proposition 4.3 therefore gives the endpoint inverse estimate after relabeling N as T. For openness, hyperbolicity and the Riesz projectors are open and continuous. On a small compact parameter neighborhood the spectral gap, the dichotomy constants, and the least singular value of Γ∞(θ) _∞(θ) are bounded uniformly away from zero. Hence the Neumann cutoff and large-horizon Green constants are uniform. If the finitely many shorter-horizon scaled matrices are invertible at θ=0θ=0, their least singular values remain positive after shrinking the neighborhood, giving uniform constants down to T0T_0. For endpoint rows x0=ξx_0=ξ and pT−SxT=πp_T-Sx_T=π, Theorem 6.2 becomes a graph-transversality test. Corollary 6.3 (Riccati graph transversality). Let M∈Sp(2d,ℝ)M (2d,R) be hyperbolic, with stable and unstable Lagrangian subspaces Es,EuE_s,E_u. Write bases as Vs=(XsPs),Vu=(XuPu)V_s= X_sP_s,V_u= X_uP_u. Let S=S⊤S=S be a prescribed terminal Hessian. If XsX_s and Pu−SXuP_u-SX_u are invertible, equivalently Es∩(0×ℝd)=0E_s∩(\0\×R^d)=\0\ and Eu∩(x,Sx):x∈ℝd=0E_u∩\(x,Sx):x ^d\=\0\, then the boundary problem with x0=ξx_0=ξ and pT−SxT=πp_T-Sx_T=π satisfies Definition 4.1 uniformly for all sufficiently large horizons, and uniformly on compact subsets of this transversality region. The canonical condition pT=πp_T=π is the special case S=0S=0. Proof. For x0=ξx_0=ξ and pT−SxT=πp_T-Sx_T=π take 0=(I000),T=(00−SI) B_0= pmatrixI&0\\ 0&0 pmatrix, B_T= pmatrix0&0\\ -S&I pmatrix. Then [0VsTVu]=(Xs00Pu−SXu),[ B_0V_s B_TV_u]= pmatrixX_s&0\\ 0&P_u-SX_u pmatrix, which is invertible. These invertibility conditions do not depend on the chosen bases, since a basis change right-multiplies the relevant blocks by invertible matrices. Theorem 6.2 gives the estimate; uniformity on compact subsets follows from openness and a positive lower bound on the least singular value of the limiting matrix. Combining Lemma 6.1 with this corollary gives data-level matrix criteria for LQ systems. Corollary 6.4 (LQ endpoint inverse region). Let A,Q∈ℝd×dA,Q ^d× d, B∈ℝd×mB ^d× m, and R∈ℝm×mR ^m× m, with A invertible, Q=Q⊤Q=Q , and R=R⊤≻0R=R 0. Let M=M(A,B,Q,R)M=M(A,B,Q,R) be the reduced LQ Hamiltonian matrix in (6.2). Suppose that M is hyperbolic and write bases of its stable and unstable subspaces as Vs=(XsPs),Vu=(XuPu)V_s= X_sP_s,V_u= X_uP_u. If XsX_s and PuP_u are invertible, then, for some T0T_0, the canonical LQ rows x0=ξx_0=ξ, pT=πp_T=π satisfy Definition 4.1 for all T≥T0T≥ T_0. More generally, for any S=S⊤S=S , the rows x0=ξx_0=ξ, pT−SxT=πp_T-Sx_T=π are covered whenever Pu−SXuP_u-SX_u is invertible. The canonical region is relatively open in A invertible,Q=Q⊤,R=R⊤≻0\A invertible,\ Q=Q ,\ R=R 0\; the shifted region is relatively open after including S=S⊤S=S . For every compact subset of these regions, a single cutoff horizon and a single set of Green constants can be chosen. Proof. By Lemma 6.1, M(A,B,Q,R)M(A,B,Q,R) is symplectic. Corollary 6.3 applies with terminal Hessian S, where S=0S=0 for the canonical rows, and the limiting transversality matrix is [0VsTVu]=(Xs00Pu−SXu),[ B_0V_s B_TV_u]= pmatrixX_s&0\\ 0&P_u-SX_u pmatrix, invertible precisely under the stated graph assumptions; the conditions are basis independent because a basis change right-multiplies XsX_s and Pu−SXuP_u-SX_u by invertible matrices. Thus Definition 4.1 holds for all sufficiently large horizons. The map (A,B,Q,R)↦M(A,B,Q,R)(A,B,Q,R) M(A,B,Q,R) is continuous where A is invertible and R≻0R 0, hyperbolicity is open, and the Riesz projectors vary continuously, so local continuous frames for EsE_s and EuE_u make XsX_s and Pu−SXuP_u-SX_u continuous; their invertibility is therefore a relative open condition. On a compact subset of the region, finitely many such frames give a uniform spectral gap and a positive uniform lower bound for the limiting matrices, and the large-horizon Neumann argument in Theorem 6.2, equivalently Proposition 4.4, gives one cutoff T0T_0 and one set of Green constants for the whole compact subset. 7 A classical LQ class We now exhibit a nonperturbative data class for Corollary 6.4. Recall that (A,B)(A,B) is stabilizable if rank[A−λIB]=drank[A-λ I\ \ B]=d over ℂC for every λ∈ℂλ with |λ|≥1|λ|≥ 1, equivalently if ρ(A−BK)<1ρ(A-BK)<1 for some feedback K [anderson2007optimal]. The boundary data are (x0,pT)(x_0,p_T). Lemma 7.1 (Energy identity). Along every complex solution of (6.1), (7.1) pt+1∗xt+1−pt∗xt=pt+1∗Gpt+1+xt∗Qxt,G=BR−1B⊤.p_t+1^*x_t+1-p_t^*x_t=p_t+1^*Gp_t+1+x_t^*Qx_t, G=BR^-1B . Proof. By (6.1), pt∗xt=(A⊤pt+1−Qxt)∗xt=pt+1∗Axt−xt∗Qxtp_t^*x_t=(A p_t+1-Qx_t)^*x_t=p_t+1^*Ax_t-x_t^*Qx_t and pt+1∗xt+1=pt+1∗Axt+pt+1∗Gpt+1p_t+1^*x_t+1=p_t+1^*Ax_t+p_t+1^*Gp_t+1, then subtract. If Q≻0Q 0 and R≻0R 0, both right-hand terms in (7.1) are nonnegative, x∗Qx=0x^*Qx=0 forces x=0x=0, and p∗Gp=‖R−1/2B⊤p‖2=0p^*Gp=\|R^-1/2B p\|^2=0 forces B⊤p=0B p=0. The conclusions of the next theorem are classical in substance: under stabilizability and definiteness, the symplectic pencil has no unit-circle eigenvalues and the relevant invariant subspaces are graphs [lancaster1995algebraic, bittanti1991riccati, laub1979schur]. The point here is a short self-contained energy proof in the sign conventions of (6.2), producing exactly the transversality certificate consumed by Corollary 6.4. Theorem 7.2 (Stabilizable definite LQ data). Let A∈ℝd×dA ^d× d be invertible, B∈ℝd×mB ^d× m, Q=Q⊤≻0Q=Q 0, R=R⊤≻0R=R 0, and let (A,B)(A,B) be stabilizable. Then M=M(A,B,Q,R)M=M(A,B,Q,R) is hyperbolic and the blocks XsX_s and PuP_u of Corollary 6.4 are invertible for every choice of bases. Consequently there is T0T_0 such that the rows x0=ξx_0=ξ, pT=πp_T=π satisfy Definition 4.1 for all T≥T0T≥ T_0, and for every compact subset of this data class one cutoff T0T_0 and one set of Green constants suffice; since the unconstrained stationarity graph u=R−1B⊤p+u=R^-1B p_+ is global, Theorem 5.1 applies after shrinking endpoint data for any compact U with 0∈intU0 . The class is relatively open and, for d≥2d≥ 2, contains coupled data: every invertible A and every Q≻0Q 0 with [A,Q]≠0[A,Q]≠ 0 are admissible with B=IB=I, R=IR=I. Proof. Step 1: hyperbolicity. Let Mz0=λz0Mz_0=λ z_0 with |λ|=1|λ|=1 and z0=(x0,p0)≠0z_0=(x_0,p_0)≠ 0, and put zt=λtz0z_t=λ^tz_0. Then pt∗xt≡p0∗x0p_t^*x_t≡ p_0^*x_0, so each increment in (7.1) vanishes; hence x0=0x_0=0 and B⊤p1=0B p_1=0, and p1=λp0p_1=λ p_0 with λ≠0λ≠ 0 gives B⊤p0=0B p_0=0, while p0≠0p_0≠ 0. The second block row of (6.2) gives A−⊤p0=λp0A^- p_0=λ p_0, hence A⊤p0=λ−1p0A p_0=λ^-1p_0 and p0∗A=λp0∗p_0^*A=λ p_0^* because |λ|=1|λ|=1. Then p0∗[A−λIB]=0p_0^*[A-λ I\ \ B]=0 contradicts stabilizability, so M has no unit-circle spectrum; since the spectrum is reciprocal (Lemma 6.1), dimEs=dimEu=d E_s= E_u=d and the blocks XsX_s, PuP_u are square. Step 2: XsX_s is invertible. By Corollary 6.3 it suffices to show Es∩(0×ℝd)=0E_s∩(\0\×R^d)=\0\. Let v=(0,p)∈Esv=(0,p)∈ E_s be real and zt=Mtv=(xt,pt)z_t=M^tv=(x_t,p_t) for t≥0t≥ 0, so zt→0z_t→ 0 exponentially. Telescoping (7.1) over 0≤t<N0≤ t<N and letting N→∞N→∞ gives ∑t=0∞(pt+1⊤Gpt+1+xt⊤Qxt)=limN→∞pN⊤xN−p0⊤x0=0, _t=0^∞ (p_t+1 Gp_t+1+x_t Qx_t )= _N→∞p_N x_N-p_0 x_0=0, because x0=0x_0=0 and zN→0z_N→ 0. Hence xt=0x_t=0 for all t≥0t≥ 0 and B⊤pt=0B p_t=0 for all t≥1t≥ 1, and the costate equation in (6.1) reduces to pt+1=A−⊤ptp_t+1=A^- p_t. Suppose p≠0p≠ 0 and let V=spanpt:t≥1≠0V=span\p_t:t≥ 1\≠\0\. Then V is A−⊤A^- -invariant, and injectivity of A−⊤A^- on the finite-dimensional V gives A−⊤V=VA^- V=V, hence A⊤V=VA V=V; B⊤B vanishes on V; and every vector of V is a finite combination of the decaying orbit, so (A−⊤)s→0(A^- )^s→ 0 pointwise on V. If an eigenvalue of A−⊤|VA^- |_V had modulus at least one, the powers could not tend to zero on the real and imaginary parts of the corresponding eigenvector in V⊗ℂV ; hence ρ(A−⊤|V)<1ρ(A^- |_V)<1 and every eigenvalue of A⊤|VA |_V has modulus larger than one. An eigenvector w∈V⊗ℂw∈ V of A⊤|VA |_V with A⊤w=μwA w=μ w, |μ|>1|μ|>1, satisfies w∗B=0w^*B=0 and w∗A=μ¯w∗w^*A= μw^*, so w∗[A−μ¯IB]=0w^*[A- μI\ \ B]=0 contradicts stabilizability over ℂC. Hence p=0p=0. Step 3: PuP_u is invertible. It suffices to show Eu∩(ℝd×0)=0E_u∩(R^d×\0\)=\0\. Let v=(x,0)∈Euv=(x,0)∈ E_u be real; then zt=Mtvz_t=M^tv is defined for all t≤0t≤ 0 and z−N→0z_-N→ 0 exponentially as N→∞N→∞. Telescoping (7.1) over −N≤t<0-N≤ t<0 and letting N→∞N→∞ gives ∑t=−∞−1(pt+1⊤Gpt+1+xt⊤Qxt)=p0⊤x0−limN→∞p−N⊤x−N=0, _t=-∞^-1 (p_t+1 Gp_t+1+x_t Qx_t )=p_0 x_0- _N→∞p_-N x_-N=0, because p0=0p_0=0 and z−N→0z_-N→ 0. Hence xt=0x_t=0 for all t≤−1t≤-1 and Gp0=0Gp_0=0, and the state equation at t=−1t=-1 gives x0=Ax−1+Gp0=0x_0=Ax_-1+Gp_0=0, so v=0v=0. This step does not use stabilizability. Steps 1–3 verify the hypotheses of Corollary 6.4, which gives Definition 4.1 for all T≥T0T≥ T_0, with uniform constants on compact subsets of the class. The class is relatively open: invertibility of A and definiteness of Q and R are open, and stabilizability is open because a feedback K with ρ(A−BK)<1ρ(A-BK)<1 keeps ρ(A′−B′K)<1ρ(A -B K)<1 for all nearby (A′,B′)(A ,B ). Finally, B=IB=I is stabilizable for every A (take K=A−12IK=A- 12I), so every invertible A and Q≻0Q 0 with [A,Q]≠0[A,Q]≠ 0 give admissible coupled data when d≥2d≥ 2. 8 Numerical illustration Both certificates are finite matrix computations. All experiments use the coupled stabilizable data (8.1) A=(1.50.400.6),B=(10.5),Q=(10.30.30.6),R=1,A= pmatrix1.5&0.4\\ 0&0.6 pmatrix, B= pmatrix1\\ 0.5 pmatrix, Q= pmatrix1&0.3\\ 0.3&0.6 pmatrix, R=1, with rectangular input: λmin(Q)≈0.439 _ (Q)≈ 0.439, ‖[A,Q]‖≈0.334\|[A,Q]\|≈ 0.334, and the PBH matrix [A−1.5IB][A-1.5I\ \ B] has least singular value ≈0.980≈ 0.980, so Theorem 7.2 applies. The reduced matrix (6.2) has eigenvalue moduli 0.367,0.556,1.798,2.723\0.367,0.556,1.798,2.723\, hence spectral gap γ≈0.587γ≈ 0.587, and with unit-norm eigenvector bases σmin(Xs)≈0.188 _ (X_s)≈ 0.188 and σmin(Pu)≈0.325 _ (P_u)≈ 0.325: the graph transversality of Corollary 6.4 holds with a computable margin. Table 1 (left) tracks the scaled boundary matrices of Proposition 4.3 for the canonical rows: ‖Γ^N−Γ^∞‖\| _N- _∞\| decays at the predicted rate e−γNe^-γ N, σmin(Γ^N) _ ( _N) stays bounded below, and the empirical Green constant CGemp(N)C_G emp(N), the maximum over all unit impulse forcings, all unit boundary data, and all times t of the ratio between ‖ht‖\|h_t\| and the matching kernel in (4.3), saturates near 4.94.9. By linearity, impulse ratios bound the constant for general data up to a fixed dimensional factor, so the table is an empirical certificate of Definition 4.1 on the tested horizons, not a proof. For the nonlinear branch test the dynamics are perturbed to F(x,u)=Ax+Bu+εφ(x)F(x,u)=Ax+Bu+ (x) with φ(x)=(x22,x1x2) (x)=(x_2^2,\ x_1x_2) and ε=0.1 =0.1, with unchanged costs; Dφ(0)=0D (0)=0, so the linearization at the origin is the LQ system above. The reduced two-point problem with rows x0=Δx_0= _x, pT=Δp_T= _p is solved by Newton’s method for Δ=sΔ =s , ‖Δ^‖=1\| \|=1, and compared with the linearized solution. Table 1 (right) reports the weighted remainder C^(T,s)=sup0≤t≤T‖zt(Δ)−ztlin(Δ)‖wT(t)2s2,α=0.05<γ/2. C(T,s)= _0≤ t≤ T \|z_t( )-z_t^lin( )\|w_T(t)^2s^2, α=0.05<γ/2. The values are essentially constant in s and bounded in T, consistent with the quadratic remainder in (5.2) and with its horizon-uniform constant. Table 1: Left: certificates for the data (8.1) with canonical rows; the differences Γ^N−Γ^∞ _N- _∞ are computed from the matrix powers in (4.5), which enter otherwise zero blocks, so no cancellation occurs. Right: weighted first-order remainder C^(T,s) C(T,s) for the nonlinear branch. N ‖Γ^N−Γ^∞‖\| _N- _∞\| σmin(Γ^N) _ ( _N) CGemp(N)C_G emp(N) 5 3.6×10−23.6× 10^-2 0.18840.1884 3.983.98 10 1.9×10−31.9× 10^-3 0.18790.1879 4.824.82 20 5.4×10−65.4× 10^-6 0.18790.1879 4.884.88 40 4.3×10−114.3× 10^-11 0.18790.1879 4.884.88 80 2.8×10−212.8× 10^-21 0.18790.1879 4.884.88 160 1.2×10−411.2× 10^-41 0.18790.1879 4.884.88 T s=10−1s=10^-1 s=10−2s=10^-2 s=10−3s=10^-3 20 0.0520.052 0.0510.051 0.0510.051 40 0.0750.075 0.0740.074 0.0740.074 80 0.0930.093 0.0920.092 0.0920.092 160 0.0970.097 0.0960.096 0.0950.095 9 Concluding remarks The chain developed here—endpoint-corrected Green estimates from scaled transversality, horizon-uniform weighted contractions, and symplectic data-level certificates—gives a verifiable route to local stationary branches of finite-horizon Pontryagin systems with constants independent of the horizon. Natural extensions are references on a face of a polyhedral control set, via a fixed-active-face reduction under strict complementarity; nonstationary references, via nonautonomous dichotomies with projector-based limits in place of (4.6); and singular L1L_1, for which Definition 4.1 needs a data-level verification route of its own. All quantitative hypotheses used here are checkable at the level of the matrix data, as Section 8 illustrates. Declaration on AI-assisted tools AI-assisted tools were used for inspiration, language polishing and translating, limited editorial suggestions, and drafting of the numerical verification scripts of Section 8. The authors verified all proofs and numerical results and assume responsibility for all content. References