Feedback Synthesis for Nonlinear Systems Via Convex Control Lyapunov Functions

Boris Houska; Juraj Oravec; Mario Eduardo Villanueva; Radoslav Paulen

arxiv: 2512.11256 · v2 · pith:BU6CWVP6new · submitted 2025-12-12 · 🧮 math.OC

Feedback Synthesis for Nonlinear Systems Via Convex Control Lyapunov Functions

Mario Eduardo Villanueva , Juraj Oravec , Radoslav Paulen , Boris Houska This is my paper

Pith reviewed 2026-05-16 23:36 UTC · model grok-4.3

classification 🧮 math.OC

keywords nonlinear controlpiecewise affine feedbackcontrol Lyapunov functionsHamilton-Jacobi-Bellman equationdiscrete-time systemsrobust controlexplicit controllersVan der Pol oscillator

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The pith

Optimizing a configuration-constrained piecewise affine approximation to the infinite-horizon min-max HJB value function produces explicit feedback laws that act as generalized control Lyapunov functions for nonlinear discrete-time systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops methods to synthesize explicit piecewise affine feedback laws for nonlinear discrete-time systems by approximating the value function of an infinite-horizon min-max Hamilton-Jacobi-Bellman equation. The approximation is optimized under the requirement that it functions as a generalized control Lyapunov function, which transfers robustness and performance guarantees to the resulting closed-loop system. This produces controllers whose storage complexity is fixed by a chosen template and whose evaluation time is predetermined. The approach is shown to work on a constrained Van der Pol oscillator, where an explicit controller achieves certified ergodic performance over a large domain.

Core claim

By optimizing a configuration-constrained PWA approximation of the value function of an infinite-horizon min-max Hamilton-Jacobi-Bellman equation such that the approximation remains a generalized control Lyapunov function for the nonlinear system, one obtains an explicit feedback law with configurable storage complexity, predetermined evaluation time, and inherited robustness and performance guarantees.

What carries the argument

The configuration-constrained piecewise affine approximation of the min-max HJB value function, optimized to serve as a generalized control Lyapunov function.

If this is right

Explicit PWA feedback laws are generated with storage complexity fixed by the chosen configuration template.
Robustness and performance guarantees from the infinite-horizon min-max problem transfer directly to the closed-loop system.
Controllers with predetermined evaluation times become available for constrained nonlinear discrete-time systems.
Certified ergodic performance is achieved over large operational domains, as demonstrated on the Van der Pol oscillator.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same template-based PWA construction could be applied to other nonlinear systems whose HJB value functions lack closed-form solutions.
The fixed evaluation time of the resulting laws makes them suitable for real-time embedded implementations with strict timing constraints.
The method provides an explicit alternative to numerical online optimization while retaining Lyapunov-based stability certificates.

Load-bearing premise

A configuration-constrained piecewise affine function can be optimized to approximate the HJB value function closely enough to remain a generalized control Lyapunov function for the given nonlinear system.

What would settle it

A concrete computation on the Van der Pol oscillator showing that the optimized PWA approximation satisfies the generalized CLF inequality yet the resulting closed-loop trajectories violate the certified ergodic performance bounds.

Figures

Figures reproduced from arXiv: 2512.11256 by Boris Houska, Juraj Oravec, Mario Eduardo Villanueva, Radoslav Paulen.

**Figure 1.** Figure 1: shows the generalized CLF Mz ⋆ with supply rate (L − d ⋆ ) with d ⋆ = 0.1. This CLF induces a partition of the domain {x ∈ R 2 | G1x ≤ z ⋆ 1 } into 265 polytopic regions shown in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 3.** Figure 3: Piecewise affine feedback µ interpolated from u ⋆ . a Van der Pol oscillator illustrates the framework’s practical applicability. Future research will focus on extending the theoretical results, exploring applications in economic model predictive control, and conducting numerical studies on larger-scale systems. REFERENCES Artstein, Z. (1983). Stabilization with relaxed controls. Nonlinear Analysis: Theo… view at source ↗

read the original abstract

This paper introduces computationally efficient methods for synthesizing explicit piecewise affine (PWA) feedback laws for nonlinear discrete-time systems, ensuring robustness and performance guarantees. The approach proceeds by optimizing a configuration-constrained PWA approximation of the value function of an infinite-horizon min-max Hamilton-Jacobi-Bellman equation. Here, robustness and performance are maintained by enforcing the PWA approximation to be a generalized control Lyapunov function for the given nonlinear system. This enables the generation of feedback laws with configurable storage complexity and pre-determined evaluation times, based on a selected configuration template. The framework's effectiveness is demonstrated through a constrained Van der Pol oscillator case study, where an explicit PWA controller with certified ergodic performance and specified complexity is synthesized over a large operational domain.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a synthesis method for explicit PWA feedback in nonlinear discrete-time systems by optimizing a configuration-constrained PWA approximation of the min-max HJB value function that is forced to satisfy a generalized CLF inequality, shown on a Van der Pol example.

read the letter

This paper shows how to build explicit piecewise affine feedback laws for nonlinear discrete-time systems. They approximate the infinite-horizon min-max HJB value function with a PWA function whose form is fixed by a chosen configuration template, then tune the parameters so the result meets the generalized CLF decrease condition along the nonlinear dynamics. The payoff is a controller whose storage size and evaluation time are fixed ahead of time, plus some robustness and performance claims that follow from the CLF property. The Van der Pol oscillator example covers a large domain and reports certified ergodic performance at a chosen complexity level. That combination of fixed-template approximation plus direct CLF enforcement is the concrete technical step they add. The soft spot sits in the performance transfer. Satisfying the CLF inequality while staying near the true value function does not by itself guarantee a uniform error bound. Without an explicit estimate on how far the optimized PWA can drift from the actual HJB value function, the ergodic performance certificate rests on the optimizer happening to keep the approximation tight enough over the whole domain. The abstract leaves that step implicit. The work is aimed at researchers in nonlinear control and explicit MPC who already work with Lyapunov or value-function methods and want offline-computed controllers with predictable runtime. A reader looking for new synthesis tools in that area will find the setup and the numerical case useful to examine. It deserves peer review because the core construction is coherent and the example supplies something concrete to check, even if the error-bound question needs tightening in revision.

Referee Report

1 major / 1 minor

Summary. The paper introduces computationally efficient methods for synthesizing explicit piecewise affine (PWA) feedback laws for nonlinear discrete-time systems. It optimizes a configuration-constrained PWA approximation of the infinite-horizon min-max Hamilton-Jacobi-Bellman value function while enforcing the approximation to be a generalized control Lyapunov function (CLF) for the nonlinear dynamics. This is claimed to transfer robustness and performance guarantees, enabling feedback laws with configurable storage complexity and fixed evaluation times. The approach is demonstrated on a constrained Van der Pol oscillator, producing an explicit PWA controller with certified ergodic performance over a large domain.

Significance. If the central construction holds, the work provides a template-based route to explicit, certifiably robust controllers for nonlinear systems whose storage and runtime costs are fixed a priori by the chosen configuration. This addresses a practical bottleneck in real-time nonlinear control where full HJB solutions are intractable, and the Van der Pol case study suggests the method can operate over sizable state domains.

major comments (1)

[Abstract] Abstract: the claim of 'certified ergodic performance' for the synthesized PWA controller requires either that the generalized CLF inequality alone implies the ergodic bound independently of approximation quality, or that an explicit a priori error bound between the optimized PWA V and the true min-max HJB value function is derived and propagated to the performance certificate. Neither is supplied; the large operational domain makes uniform approximation under a fixed configuration template non-trivial, so the performance transfer rests on an unverified closeness assumption.

minor comments (1)

The abstract would benefit from a concise statement of the precise assumptions (e.g., on the configuration template or the min-max formulation) needed for the CLF enforcement step to be feasible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below and have revised the manuscript accordingly to ensure the claims accurately reflect the provided guarantees.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of 'certified ergodic performance' for the synthesized PWA controller requires either that the generalized CLF inequality alone implies the ergodic bound independently of approximation quality, or that an explicit a priori error bound between the optimized PWA V and the true min-max HJB value function is derived and propagated to the performance certificate. Neither is supplied; the large operational domain makes uniform approximation under a fixed configuration template non-trivial, so the performance transfer rests on an unverified closeness assumption.

Authors: We agree that the manuscript does not supply an explicit a priori error bound between the configuration-constrained PWA approximation and the true min-max HJB value function, nor does the generalized CLF inequality alone yield a rigorous ergodic performance certificate independent of approximation quality. The CLF property certifies closed-loop stability and robustness to disturbances for the nonlinear system, but the specific ergodic performance claim in the abstract is not fully supported by a propagated bound. We will revise the abstract (and corresponding statements in the introduction and conclusions) to remove the unqualified 'certified ergodic performance' phrasing and instead state that the synthesized controller provides certified stability and robustness via the generalized CLF, with ergodic performance illustrated numerically on the Van der Pol example. This revision ensures the claims are precisely aligned with what is proven. revision: yes

Circularity Check

0 steps flagged

No significant circularity; optimization enforces CLF property independently

full rationale

The paper's derivation optimizes a configuration-constrained PWA approximation to satisfy the generalized control Lyapunov function inequality V(f(x,u*)) - V(x) ≤ -ℓ(x,u*) directly as a constraint in the optimization problem. This enforcement is independent of the target ergodic performance metric and does not reduce to fitting parameters to the output or assuming the CLF property by construction. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the abstract or described chain; the method relies on standard min-max HJB theory and convex optimization without circular reduction to its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard domain assumptions from optimal control theory plus the novel but unverified premise that the chosen PWA template class is rich enough to approximate a CLF for the target nonlinear dynamics.

free parameters (1)

configuration template parameters
The selected configuration template for the PWA approximation introduces parameters that define the partition and affine pieces; these are chosen by the designer and affect the resulting controller.

axioms (1)

domain assumption Existence of a generalized control Lyapunov function for the nonlinear discrete-time system that can be approximated by the chosen PWA class.
Invoked when the optimization is required to produce a function that certifies stability and performance.

pith-pipeline@v0.9.0 · 5426 in / 1377 out tokens · 41968 ms · 2026-05-16T23:36:29.741824+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

optimizing a configuration-constrained PWA approximation of the value function of an infinite-horizon min-max Hamilton-Jacobi-Bellman equation... enforcing the PWA approximation to be a generalized control Lyapunov function
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat induction / embed_strictMono unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

M(x)+d ≥ min_u max_w L(x,u)+I_X(x)+M(f(x,u)+w)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Artstein, Z. (1983). Stabilization with relaxed controls.Nonlinear Analysis: Theory, Methods & Applications, 7(11), 1163–1173. Aubin, J. and Frankowska, H. (2009).Set-valued analysis. Springer Science & Business Media. Bardi, M. and Capuzzo-Dolcetta, I. (1997).Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkh¨ auser. Bell...

work page 1983
[2]

and Goulart, P

Fabiani, F. and Goulart, P. (2022). Reliably-stabilizing piecewise- affine neural network controllers.IEEE Transactions on Auto- matic Control, 68(9), 5201–5215. Fukuda, K. (2020).Polyhedral Computation. ETH Z¨ urich Research Collection. Ganguly, S. and Chatterjee, D. (2025). Explicit feedback synthesis driven by quasi-interpolation for nonlinear model pr...

work page doi:10.1109/tac.2025.3538767 2022
[3]

Gr¨ une, L., Sperl, M., and Chatterjee, D. (2025). Representation of practical nonsmooth control lyapunov functions by piecewise affine functions and neural networks.Systems & Control Letters, 202, 106103. He, K., Shi, S., van den Boom, T., and De Schutter, B. (2024). Ap- proximate dynamic programming for constrained linear systems: A piecewise quadratic ...

work page arXiv 2025

[1] [1]

Artstein, Z. (1983). Stabilization with relaxed controls.Nonlinear Analysis: Theory, Methods & Applications, 7(11), 1163–1173. Aubin, J. and Frankowska, H. (2009).Set-valued analysis. Springer Science & Business Media. Bardi, M. and Capuzzo-Dolcetta, I. (1997).Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkh¨ auser. Bell...

work page 1983

[2] [2]

and Goulart, P

Fabiani, F. and Goulart, P. (2022). Reliably-stabilizing piecewise- affine neural network controllers.IEEE Transactions on Auto- matic Control, 68(9), 5201–5215. Fukuda, K. (2020).Polyhedral Computation. ETH Z¨ urich Research Collection. Ganguly, S. and Chatterjee, D. (2025). Explicit feedback synthesis driven by quasi-interpolation for nonlinear model pr...

work page doi:10.1109/tac.2025.3538767 2022

[3] [3]

Gr¨ une, L., Sperl, M., and Chatterjee, D. (2025). Representation of practical nonsmooth control lyapunov functions by piecewise affine functions and neural networks.Systems & Control Letters, 202, 106103. He, K., Shi, S., van den Boom, T., and De Schutter, B. (2024). Ap- proximate dynamic programming for constrained linear systems: A piecewise quadratic ...

work page arXiv 2025