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Two-term small-time spectral expansions for controllability Gramians

Emmanuel Trélat, Enrique Zuazua

arXiv:2609.37077v1math.NAcs.NA

TL;DR

Known eigenvalue orders describe small-time controllability anisotropy but not the first drift-dependent correction to each eigenvalue or eigendirection. A graded Krylov analysis and midpoint centering yield intrinsic first corrections and analytic expansions for eigenvalues, eigenvectors, and spectral projectors. The eigenvalues satisfy λ_k(T)=κ_k d_k(A,b)^2 T^(2k−1)(1+a_kT+O(T^2)).

  • Problem

    Known eigenvalue orders describe small-time controllability anisotropy but not the first drift-dependent correction to each eigenvalue or eigendirection.

    Source: §1.1 · p. 2

  • Method

    A graded Krylov analysis and midpoint centering yield intrinsic first corrections and analytic expansions for eigenvalues, eigenvectors, and spectral projectors.

    Source: §1.3 · p. 3

  • Results

    The eigenvalues satisfy λ_k(T)=κ_k d_k(A,b)^2 T^(2k−1)(1+a_kT+O(T^2)).

    Source: §1.3 · p. 3

  • Takeaways & Limitations

    The smallest-eigenvalue eigenspace identifies least-controllable targets, and for symmetric dynamics the spectral data reconstruct A and determine b up to its global sign.

    Source: §3.3 · p. 12

    The analytic spectral expansions are local near T=0; the paper gives no quantitative lower bound for the uniform time interval, and positive-time collisions are not excluded.

    Source: §1.3 · p. 4

Abstract

from arXiv · show

For a controllable single-input linear system in dimension $n$, the controllability Gramian eigenvalues, ordered decreasingly, are known to have the successive small-time orders $T, T^3, \ldots, T^{2n-1}$. We refine this leading-order hierarchy by explicitly computing the next-order coefficient of every eigenvalue and the first-order variation of its eigendirection and of the nested spectral subspaces. The resulting formulas have intrinsic expressions in the orthonormal Krylov basis: eigenvalue corrections describe the action of the dynamics along each Krylov direction, while variations of the spectral subspaces describe coupling to the next direction. We also establish local joint real-analytic dependence on $(T, A, b)$ near $T=0$ for the eigenvalues divided by their leading powers of $T$, consistently oriented eigenvectors, and spectral projectors. This provides convergent expansions with remainders uniform on compact families of controllable pairs. These results yield refined asymptotics for the worst-case minimum energy required to reach a unit target from the origin, the Gramian determinant and condition number, and the Ornstein--Uhlenbeck Gaussian profile in moving principal coordinates. The energy equals the reciprocal of the smallest Gramian eigenvalue, whose eigendirection identifies the most energy-demanding targets for sufficiently small times. We further derive an exact Gramian-weighted energy identity for the forward Fokker--Planck equation and sharp anisotropic short-time smoothing asymptotics using, respectively, the Lyapunov equation and an exact Fourier norm formula combined with the graded Gramian factorization. For symmetric dynamics, the spectral data reconstruct $A$ and determine $b$ up to its global sign.

1 Introduction and main result

For controllable single-input systems, known small-time Gramian eigenvalue orders do not specify the first drift-dependent corrections or changes in eigendirections. The paper derives intrinsic Krylov-basis formulas for those changes, establishes analytic dependence, and develops consequences for control energy, covariance, and related quantities.

1.1 Problem setting and motivation

  • The setting is a controllable single-input linear system; the known hierarchy T, T^3, …, T^(2n−1) leaves the first drift-dependent eigenvalue and direction corrections unspecified.The smallest-eigenvalue eigenspace identifies unit targets requiring the greatest minimum energy.Source: §1.1 · pp. 1–2

1.3 Main result and immediate consequences

  • The worst-case minimum control energy is the reciprocal of the smallest Gramian eigenvalue, whose eigenspace gives the least-controllable directions.The paper also derives small-time asymptotics for the determinant and conditioning.Source: §1.1 · p. 2, §1.3 · pp. 3–4

1.2 Krylov coordinates and notation

  • The orthonormal Krylov basis is obtained by Gram–Schmidt, and controllability makes each filtration layer add exactly one direction; a_k records ⟨q_k,Aq_k⟩.The paper uses this basis to express the spectral corrections intrinsically.Source: §1.2 · p. 2, §1.1 · p. 2

1.3 Main result and immediate consequences

  • Each eigenvalue has leading scale κ_k d_k(A,b)^2 T^(2k−1) and relative correction a_kT, while the spectral data admit analytic expansions with uniform remainders on compact controllable families.The first variation of each nested spectral subspace is toward the next Krylov direction.Source: §1.3 · p. 3

1.4 Relation to previous work

  • The coefficients also describe hypoelliptic derivative layers: a_k corrects the k-th layer, while β_k produces the first rotation of the spectral flag.The paper relates β_k to drift commutation generating the next Krylov derivative.Source: §1.4 · p. 5
  • The authors position the simultaneous two-term Euclidean formulas and their joint analytic, compact-family uniformity as contributions, while not claiming the leading exponents or limiting Krylov flag as new.They attribute the explicit first corrections to the moment structure and midpoint parity.Source: §1.4 · p. 5

1.5 Proof strategy and organization of the paper

  • The proof uses graded Gramian factorization to establish spectral scales and analyticity, then centers the time interval and transports the Krylov frame to obtain a_k and β_k.Midpoint translation is a proof device.Source: §1.5 · p. 6

2 Proof of the main result

The proof uses a graded factorization, then centers and transports the Krylov frame to obtain the two-term spectral expansions. These steps also establish analytic spectral data near zero and identify the first changes in eigenvalues and spectral subspaces.

2.1 A graded factorization

  • The graded factorization yields λk(T) = κkdk(A, b)^2T^(2k−1)(1 + O(T)) and implies simple eigenvalues for sufficiently small positive T.It also makes the normalized eigenvalues, flag projectors, and oriented eigenvectors jointly real analytic near T = 0.Source: §2.1 · p. 6
  • In the Krylov basis, each leading principal Gramian minor approximates the product of the first k eigenvalues with relative error O(T^2), uniformly locally on controllable pairs.For k = n, the identity is exact.Source: §2.1 · pp. 6–7

2.2 Centering and the gain of one order

  • Centering on a symmetric time interval gives flag projectors and oriented eigenvectors within O(T^2) of the Krylov flag and basis, with normalized spectral data analytic in T^2.The original Gramian is related to the centered covariance by bG_T(A,b) = G_T(A,b_T), where b_T = e^(TA/2)b.Source: §2.2 · pp. 8, 10

2.3 Expansion of the transported Krylov data

  • The passages give λk(T) = κk dk(A, bT)^2 T^(2k−1)(1 + O(T^2)) and Πk(T) = ΠKk(A,bT) + O(T^2), with dk(A, bT)^2 = dk(A, b)^2(1 + akT + O(T^2)).Source: §2.3 · p. 10
  • Differentiating consecutive flag projectors yields the rank-one eigenprojector expansion, and the normalized spectral data retain joint real-analytic dependence on (T, A, b) near zero.The rank-one projector onto ψk(T) is Πk(T) − Πk−1(T).Source: §2.3 · p. 10

3 Spectral consequences and an exact energy identity

The section develops refined Gaussian and spectral consequences, including inverse recovery for symmetric dynamics, then derives an exact forward energy identity and sharp directional smoothing results.

3.1 Gaussian profile in moving principal coordinates

  • In moving principal coordinates at natural Kalman scales, the Gaussian profile resolves the correction at z = 0, −tr(A)T/2, into individual spectral contributions away from zero.The profile is observed in T-dependent coordinates, and its first rotation is described in the fixed Krylov basis.Source: §3.1 · p. 11

3.2 An even expansion for skew-symmetric dynamics

  • For skew-symmetric A, every odd relative coefficient of each normalized eigenvalue vanishes, although fixed-coordinate spectral projectors still undergo orthogonal conjugation.The normalized eigenvalues extend real-analytically near T = 0.Source: §3.2 · p. 11

3.3 An inverse consequence for symmetric dynamics

  • For symmetric A, the scalar eigenvalue coefficients determine its Jacobi representation up to orthogonal coordinates and the global sign of b; ambient projectors recover A in the original coordinates.The Gramian cannot distinguish b from −b, and this is the only ambiguity stated for the reconstruction.Source: §3.3 · p. 12
  • For nonsymmetric A, the first spectral jet supplies ak and βk but does not determine the remaining Krylov coefficients.The moment–orthogonal-polynomial–Jacobi correspondence is classical; here the spectral jets directly supply recurrence coefficients.Source: §3.3 · p. 12

3.4 An exact energy identity and sharp directional smoothing for the forward equation

  • For damped Langevin dynamics, et tr(A)Eθ(t) is nonincreasing for every initial datum when θ ≥ 1, and θ = 1 is the smallest universal coefficient when b ≠ 0.Damping appears at first order on the large covariance axis but not the small one; frequency enters only at higher order.Source: §3.4 · pp. 13, 15
  • The forward Fokker–Planck energy identity is exact whenever its solution and displayed integrals are defined, requires no Kalman condition, stability assumption, or invariant measure, and supports layerwise analysis.In each layer, ak corrects diagonal energy while adjacent Krylov ratios couple neighboring layers; truncation must respect the distinct leading scales t^(2k−1).Source: §3.4 · pp. 12–13
  • The exact directional smoothing norm equals e^(−1/2)e^(−t tr(A)/2) times the square root of the control cost in the same direction; the weakest spectral direction attains both suprema for small t.The smoothing blow-up exponents are one half of the corresponding control-energy exponents.Source: §3.4 · p. 15
  • Fixed-Krylov leading smoothing constants exceed moving-principal constants for k < n, with ratio n at k = 1 and ratio 1 at k = n.The powers t^(−k+1/2) are optimal, while constants obtained from monotonicity are larger by e^(1/2).Source: §3.4 · p. 15

4 Conclusion and perspectives

The conclusion summarizes the first-order spectral refinements and their analytic expansions, then identifies scalar-drift behavior and boundaries for higher-order, multi-input, and growing-dimension extensions.

4 Conclusion and perspectives

  • Theorem 1 gives convergent expansions for higher coefficients, but no closed invariant formula as simple as the first-order formula is known.Higher coefficients can be computed from orbit Taylor series and analytic perturbation equations; at relative order T^2, several correction terms interact.Source: §4 · p. 15
  • The first-order correction is special because the normalized Gramian spectral data are even in T before the input is transformed by e^(TA/2).For fixed c, the first-order term therefore arises entirely from b ↦ e^(TA/2)b.Source: §4 · p. 15
  • Adding scalar drift ζ shifts every a_k by ζ while leaving Krylov spaces, β_k, and the first variation of the spectral flag unchanged.Each eigenvalue acquires the same relative factor 1 + ζT + O(T^2).Source: §4 · p. 16
  • For multi-input systems, leading spectral exponents form blocks of multiplicity m_j, but a two-term coordinate-free block formula is a separate problem.Within-layer eigenvalues need not be simple, and their first projectors may depend on internal spectral splittings.Source: §4 · p. 16
  • The analysis does not control the joint limit T → 0+ and n → +∞ because the last Krylov height and κ_n may deteriorate with dimension.Uniform bounds in n would require observability and fast-control estimates uniform in the discretization.Source: §4 · p. 16
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