The scalable Taylor/Kronecker framework for nonlinear balanced truncation (Kramer, Gugercin, Borggaard, and Balicki, CMAME 2024, whose arXiv v1 is titled 'Nonlinear Balanced Truncation: Part 1 — Computing Energy Functions', together with its successors) computes polynomial approximations of the Hamilton–Jacobi–Bellman (HJB) energy functions and balancing transformations of control-affine nonlinear systems with hundreds to thousands of states, turning Scherpen's 1993 nonlinear balancing theory into a practical model-reduction tool. The problem posed here is the framework's central open gap, explicitly flagged by the source paper and still unresolved in the April 2026 literature: it carries none of the guarantees that make linear balanced truncation trustworthy. There are no convergence rates, domains of validity, or error estimates for the truncated polynomial approximations of the true HJB/H∞\mathcal{H}_\infty energy functions as the polynomial degree grows; no a priori model-reduction error bound analogous to the linear Hankel-singular-value bound; and no exact computation of the closed-loop admissibility threshold γ0\gamma_0. The 2026 completion of the computational pipeline explicitly identifies the unknown region of validity of the polynomial approximations as the method's major drawback. The problem asks for these certification results, or rigorous negative results showing they cannot hold.

Background

Balanced truncation is a system-theoretic model-order reduction (MOR) method: given a full-order model of a dynamical system, it constructs a reduced-order model (ROM) that preserves the dominant input–output behavior. For a linear time-invariant (LTI) system, Moore (1981) defined the controllability and observability Gramians (matrix measures of how strongly inputs drive states and how strongly states affect outputs) and showed that a linear change of coordinates can make both Gramians diagonal and equal, with diagonal entries σ1≥σ2≥⋯≥σn≥0\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_n \ge 0, the Hankel singular values. Deleting the states associated with small σi\sigma_i yields a ROM with the celebrated a priori guarantee that the input–output error is at most twice the sum of the neglected Hankel singular values — the property that made balanced truncation a certified method.

Scherpen (1993) extended balancing to nonlinear control-affine systems,

x˙=f(x)+∑i=1mgi(x) ui,y=h(x),\dot{x} = f(x) + \sum_{i=1}^{m} g_i(x)\, u_i, \qquad y = h(x),

(control-affine means the control enters linearly while the drift ff, the input maps gig_i, and the output map hh may be nonlinear). The Gramians are replaced by energy functions: the controllability energy Ec(x0)E_c(x_0) is the minimal input energy needed to steer the state from the equilibrium (the origin) to x0x_0, and the observability energy Eo(x0)E_o(x_0) is the output energy released when the system starts at x0x_0 with zero input. These functions satisfy first-order nonlinear Hamilton–Jacobi (HJ) partial differential equations. Closed-loop variants — HJB balancing and H∞\mathcal{H}_\infty balancing (the latter due to Scherpen, 1996) — replace them with solutions of Hamilton–Jacobi–Bellman (HJB) equations of optimal control, parametrized by an attenuation level γ>γ0\gamma > \gamma_0, where γ0\gamma_0 is the smallest level for which a stabilizing controller achieving closed-loop L2L_2-gain (the worst-case ratio of output energy to input energy) below γ\gamma exists; the exact computation of γ0\gamma_0 is itself open, with usable bounds γ^0−1≤γ0≤γ^0+1\hat{\gamma}_0 - 1 \le \gamma_0 \le \hat{\gamma}_0 + 1 obtained from the algebraic Riccati equations of the linearized system (Mustafa and Glover, 1991). Balancing then requires a nonlinear state transformation that puts the pair of energy functions into input-normal/output-diagonal form (the nonlinear analog of simultaneously diagonalizing the Gramians), after which states whose singular value functions — the nonlinear, state-dependent analogs of Hankel singular values — are small are truncated.

The computational obstacle is that the HJ(B) equations are PDEs in the full state dimension. A power-series method going back to Al'brekht (1961) expands the energy functions in a Taylor series around the equilibrium; Fujimoto and Tsubakino (2008) applied it to nonlinear balancing on toy examples. Kramer, Gugercin, Borggaard, and Balicki (Computer Methods in Applied Mechanics and Engineering, 2024) made this scalable: they formulated a single parametrized HJB family (with γ=1\gamma = 1 recovering open-loop balancing and γ>γ0\gamma > \gamma_0 the closed-loop variants, via η=1−γ−2\eta = 1 - \gamma^{-2}) whose Taylor coefficients are determined by structured linear tensor (Kronecker-product) systems solvable with billions of unknowns, demonstrated on semi-discretized Burgers and Kuramoto–Sivashinsky equations. Successors extended the framework to polynomial control-affine systems (Corbin and Kramer, IEEE Transactions on Automatic Control, 2025), proved solvability and gave scalable algorithms for the input-normal/output-diagonal balancing transformation (Corbin, Sarkar, Scherpen, and Kramer, Systems & Control Letters, 2025), and completed the pipeline with explicit polynomial balanced realizations, nonlinear ROMs, and open-source software working up to roughly n=768n = 768 states (Corbin and Kramer, 2026).

What is missing is exactly the theory that made the linear method trustworthy. All accuracy evidence for the nonlinear pipeline is numerical (tabulated convergence in polynomial degree, HJB residuals); Taylor's theorem guarantees only that the power series converges in some unspecified neighborhood of the expansion point. Hence the source paper and its successors state, as of 2026: no convergence rates, domains of validity, or error estimates for the polynomial energy-function approximations; no a priori ROM error bound in terms of neglected energy contributions analogous to the Hankel-singular-value bound; and no exact computation of γ0\gamma_0. A 2026 MOR survey (Kramer and Qian) confirms that rigorous error bounds remain 'usually not available for strongly nonlinear models'.

Problem Statement

The objective is to replace the currently purely numerical validation of polynomial-approximation-based nonlinear balanced truncation with provable guarantees, for control-affine systems with polynomial dynamics (the class handled by the scalable framework). The source paper poses this as three linked open questions, each an independent deliverable: a rigorous answer to any one of them — positive or negative — resolves that sub-question, and leaves the remaining questions open.

(1) Convergence of the energy-function approximations. For the parametrized HJB equations defining the past and future energy functions Eγ±E^{\pm}_\gamma (with γ=1\gamma = 1 giving the open-loop and γ>γ0\gamma > \gamma_0 the closed-loop balancing variants), establish a rigorous convergence theory for the degree-dd truncated Taylor/polynomial approximations Eγ±,dE^{\pm,d}_\gamma computed by the tensor/Kronecker framework: rates of convergence, explicit or computably estimable domains of validity around the expansion equilibrium as functions of the system data, the degree dd, and γ\gamma, and error bounds for ∥Eγ±−Eγ±,d∥\|E^{\pm}_\gamma - E^{\pm,d}_\gamma\| on such domains.

(2) A priori truncation error bound. Prove an error bound for nonlinear balanced truncation: an inequality bounding the input–output error between the full-order system and the truncated balanced reduced-order model in terms of the neglected higher-order energy contributions / truncated singular value functions (the nonlinear, state-dependent analogs of Hankel singular values), which reduces to the classical bound (twice the sum of the neglected Hankel singular values) when the system is linear.

(3) Admissibility threshold. Compute or exactly characterize γ0\gamma_0, the smallest attenuation level for which a stabilizing controller exists, in the nonlinear parametrized setting, beyond the linear-Riccati-based bounds currently used.

References

  1. Boris Kramer, Serkan Gugercin, Jeff Borggaard, Linus Balicki, 'Scalable computation of energy functions for nonlinear balanced truncation', Computer Methods in Applied Mechanics and Engineering 427:117011, 2024. DOI: 10.1016/j.cma.2024.117011; arXiv:2209.07645 (v1 titled 'Nonlinear Balanced Truncation: Part 1 — Computing Energy Functions'). https://arxiv.org/abs/2209.07645
  2. Boris Kramer, Serkan Gugercin, Jeff Borggaard, 'Nonlinear Balanced Truncation: Part 2 — Model Reduction on Manifolds', arXiv:2302.02036, 2023. https://arxiv.org/abs/2302.02036
  3. Nicholas A. Corbin, Boris Kramer, 'Scalable Computation of H∞ Energy Functions for Polynomial Control-Affine Systems', IEEE Transactions on Automatic Control 70(5):3088–3100, 2025. DOI: 10.1109/TAC.2024.3494472; arXiv:2408.08970. https://arxiv.org/abs/2408.08970
  4. Nicholas A. Corbin, Arijit Sarkar, Jacquelien M. A. Scherpen, Boris Kramer, 'Scalable computation of input-normal/output-diagonal balanced realization for control-affine polynomial systems', Systems & Control Letters 204:106178, 2025. DOI: 10.1016/j.sysconle.2025.106178; arXiv:2410.22435. https://arxiv.org/abs/2410.22435
  5. Nicholas A. Corbin, Boris Kramer, 'Nonlinear balanced truncation model reduction through scalable Taylor series', arXiv:2604.23044, 2026. https://arxiv.org/abs/2604.23044
  6. Hamza Adjerid, Jeff Borggaard, 'Energy function approximations for differential algebraic polynomial systems of Stokes-type', arXiv:2507.18909, 2025. https://arxiv.org/abs/2507.18909
  7. Hamza Adjerid, Jeff Borggaard, 'Sum of squares approximations to energy functions', arXiv:2408.12478, 2024 (American Control Conference). https://arxiv.org/abs/2408.12478
  8. Tobias Breiten, Attila Karsai, 'Nonlinear systems and passivity: feedback control, model reduction, and time discretization', arXiv:2502.04987, 2025. https://arxiv.org/abs/2502.04987
  9. J. M. A. Scherpen, 'Balancing for nonlinear systems', Systems & Control Letters 21(2):143–153, 1993. DOI: 10.1016/0167-6911(93)90117-O. https://doi.org/10.1016/0167-6911(93)90117-O
  10. Jacquelien M. A. Scherpen, 'H∞ balancing for nonlinear systems', International Journal of Robust and Nonlinear Control 6(7):645–668, 1996. DOI: 10.1002/(SICI)1099-1239(199608)6:7<645::AID-RNC179>3.0.CO;2-X. https://doi.org/10.1002/(SICI)1099-1239(199608)6:7%3C645::AID-RNC179%3E3.0.CO;2-X
  11. B. C. Moore, 'Principal component analysis in linear systems: Controllability, observability, and model reduction', IEEE Transactions on Automatic Control 26(1):17–32, 1981. DOI: 10.1109/TAC.1981.1102568. https://doi.org/10.1109/TAC.1981.1102568
  12. K. Fujimoto, D. Tsubakino, 'Computation of nonlinear balanced realization and model reduction based on Taylor series expansion', Systems & Control Letters 57:283–289, 2008. DOI: 10.1016/j.sysconle.2007.08.015. https://doi.org/10.1016/j.sysconle.2007.08.015
  13. D. Mustafa, K. Glover, 'Controller reduction by H∞-balanced truncation', IEEE Transactions on Automatic Control 36(6):668–682, 1991. DOI: 10.1109/9.86941. https://doi.org/10.1109/9.86941
  14. E. G. Al'brekht, 'On the optimal stabilization of nonlinear systems', Journal of Applied Mathematics and Mechanics 25:1254–1266, 1961. DOI: 10.1016/0021-8928(61)90005-3. https://doi.org/10.1016/0021-8928(61)90005-3
  15. Boris Kramer, Elizabeth Qian, 'Reduced-order modeling for engineering systems: survey and opportunities for digital twins', Structural and Multidisciplinary Optimization 69(6):141, 2026. DOI: 10.1007/s00158-026-04310-0. https://doi.org/10.1007/s00158-026-04310-0