Kernel-Based LMI Approaches to Solving the Hamilton-Jacobi-Bellman Equation and Nonlinear Optimal Control

Boumediene Hamzi; Umesh Vaidya

arxiv: 2603.01084 · v2 · pith:YMM5XUKInew · submitted 2026-03-01 · 🧮 math.DS · cs.NA· math.NA· math.OC

Kernel-Based LMI Approaches to Solving the Hamilton-Jacobi-Bellman Equation and Nonlinear Optimal Control

Boumediene Hamzi , Umesh Vaidya This is my paper

Pith reviewed 2026-05-21 12:07 UTC · model grok-4.3

classification 🧮 math.DS cs.NAmath.NAmath.OC

keywords Hamilton-Jacobi-Bellman equationnonlinear optimal controllinear matrix inequalityreproducing kernel Hilbert spaceRiccati equationsuboptimality bound

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The pith

An explicit Riccati-Hessian equality constraint at equilibrium lets a kernel LMI formulation approximate solutions to the Hamilton-Jacobi-Bellman equation without collapsing to the trivial solution.

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

The paper develops a convex optimization method for finding approximate solutions to the Hamilton-Jacobi-Bellman equation that governs nonlinear optimal control. The gradient of the value function is represented in a reproducing kernel Hilbert space, and the quadratic Hamilton-Jacobi-Bellman inequality is rewritten as a linear matrix inequality that remains linear in the kernel coefficients. Adding a Riccati-Hessian equality constraint at the equilibrium forces the approximation to agree with the algebraic Riccati equation of the linearized system. This construction produces a semidefinite program whose solution yields a feedback law whose closed-loop cost stays within a bound of the optimal cost; the bound depends on the residual of the inequality and on problem data rather than on further details of the approximant.

Core claim

Representing the gradient of the value function in a reproducing kernel Hilbert space and converting the Hamilton-Jacobi-Bellman inequality into a Schur-complement linear matrix inequality yields a convex semidefinite program in the kernel coefficients. The novel Riccati-Hessian equality constraint imposed at the equilibrium removes the trivial zero solution and enforces consistency with the algebraic Riccati equation of the linearized dynamics. The resulting approximation satisfies the suboptimality relation J(x0; û) − V*(x0) ≤ ε T(x0), where T(x0) is determined solely by the system data and the working domain.

What carries the argument

The Riccati-Hessian equality constraint at the equilibrium point, which forces the Hessian of the kernel approximant to satisfy the algebraic Riccati equation of the linearized system.

If this is right

The semidefinite program is convex and can be solved by standard interior-point solvers.
The suboptimality bound remains valid even when the exact value function is not contained in the reproducing kernel Hilbert space.
On benchmark problems the method recovers exact polynomial solutions to machine precision when they lie in the chosen space and produces residuals smaller than those of several competing techniques 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 LMI-plus-constraint structure could be applied to other reproducing kernel spaces or to polynomial sum-of-squares bases for systems whose value functions admit known structure.
Because the bound T(x0) depends only on data and domain size, the approach may remain useful for real-time receding-horizon implementations where exact dynamic programming is intractable.
Scaling the number of kernel centers while monitoring both residual and actual closed-loop cost offers a practical diagnostic for when further refinement ceases to improve performance.

Load-bearing premise

The gradient of the value function lies sufficiently close to the chosen reproducing kernel Hilbert space that the linear matrix inequality and the Riccati-Hessian constraint together produce an approximation whose suboptimality stays bounded.

What would settle it

Compute the closed-loop cost starting from a chosen initial state and verify whether it exceeds the predicted quantity ε T(x0) by more than numerical tolerance; separately, check whether the Hessian of the obtained approximant at the equilibrium matches the algebraic Riccati solution of the linearized dynamics.

Figures

Figures reproduced from arXiv: 2603.01084 by Boumediene Hamzi, Umesh Vaidya.

**Figure 2.** Figure 2: Comprehensive numerical results for the 2D radially symmetric system with multiple [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Figure 3: Comprehensive numerical results for the Van der Pol oscillator ( [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

read the original abstract

We present a kernel-based linear matrix inequality (LMI) approach for the approximate solution of Hamilton--Jacobi--Bellman (HJB) equations arising in nonlinear optimal control. The method represents the gradient of the value function in a reproducing kernel Hilbert space (RKHS) and uses a Schur-complement reformulation to convert the quadratic HJB inequality into an LMI that is linear in the kernel coefficients, yielding a convex semidefinite program. The novel ingredient is an explicit Riccati--Hessian \emph{equality} constraint at the equilibrium, which removes the trivial solution and forces the Hessian of the approximation to match the algebraic Riccati equation solution of the linearised system. We give a suboptimality bound $J(x_0;\hat u) - V^*(x_0)\le \varepsilon\,T(x_0)$ in which $T(x_0)$ depends only on the problem data and the working domain (not on the approximation), and an RKHS approximation rate. Numerical experiments on a corrected 1D polynomial benchmark and on the Van der Pol oscillator measure $\varepsilon$, the RKHS approximation error, and the closed-loop cost $J(x_0;\hat u)$ versus the optimal value $V^*(x_0)$. On the 1D problem with $V^*$ in the polynomial-kernel RKHS the method recovers $V^*$ to within $3\times10^{-7}$ and achieves $0.000\%$ suboptimality. On Van der Pol it achieves the smallest HJB residual ($\varepsilon\approx 2.62$) of any method tested, beats LQR on every initial condition, and is within $0.42\%$ of the best per-IC cost (Albrekht order 6). When $V^*$ is not in the chosen RKHS, the method degrades gracefully: residuals stop improving with more centres but suboptimality remains bounded ($\le 13\%$ on the 1D test).

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 new piece is the Riccati-Hessian equality at equilibrium that turns the kernel LMI into a non-trivial convex program with a suboptimality bound depending only on data and domain.

read the letter

The paper's main move is to represent the gradient of the value function in an RKHS, convert the HJB inequality to an LMI via Schur complement, and then add an explicit equality constraint that forces the Hessian at the origin to match the algebraic Riccati solution of the linearised system. That constraint is what stops the zero solution and links the approximation to the local linear behaviour. The resulting SDP is convex in the kernel coefficients, which is clean. They also claim a suboptimality bound J(x0; û) - V*(x0) ≤ ε T(x0) where T depends only on the problem data and the working domain, not on how well the approximation fits. If the derivation holds, that is useful for applications that need a priori guarantees rather than post-hoc checks. Numerically the results are straightforward: near-machine precision on the 1D polynomial case where the true V* lies in the RKHS, and on Van der Pol the method gives the smallest residual among the methods they compare while staying within 0.42 % of the best reported cost and beating LQR on every tested initial condition. When V* is outside the RKHS the residuals plateau but the closed-loop cost degradation stays bounded (up to 13 % in their test), which is at least consistent with the claimed graceful behaviour. The soft spot is the standing assumption that the gradient can be represented well enough in the chosen RKHS for the bound to remain informative in practice; kernel and centre selection still look like tuning parameters even if the program itself is convex. The examples are low-dimensional, so scaling behaviour is not yet shown. Overall the formulation is internally consistent and the numerics line up with the claims. This is the kind of paper that control theorists working on approximate nonlinear synthesis would want to see in a journal. It has enough structure and evidence to go to a serious referee rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The paper proposes a kernel-based LMI approach to approximately solve the Hamilton-Jacobi-Bellman equation for nonlinear optimal control. The gradient of the value function is represented in an RKHS, the quadratic HJB inequality is converted to a convex LMI via Schur complement, and a novel Riccati-Hessian equality constraint is imposed at the equilibrium to exclude the trivial zero solution and enforce consistency with the linearized algebraic Riccati equation. A suboptimality bound J(x0; û) - V*(x0) ≤ ε T(x0) is derived where T(x0) depends only on problem data and domain, together with an RKHS approximation rate. Numerical results are reported on a corrected 1D polynomial system (recovery to 3e-7, 0% suboptimality) and the Van der Pol oscillator (smallest HJB residual among tested methods, within 0.42% of best per-IC cost).

Significance. If the central derivations hold, the work supplies a convex SDP formulation for nonlinear optimal control with an explicit, data-dependent suboptimality guarantee that remains valid even when the true value function lies outside the chosen RKHS. This combination of convexity, a non-triviality constraint that anchors the approximation to the linearization, and a bound independent of the particular kernel coefficients is a meaningful contribution to approximate dynamic programming and could enable reliable controller synthesis for systems where exact HJB solution is intractable.

minor comments (3)

[Abstract] Abstract: the phrase 'corrected 1D polynomial benchmark' should briefly indicate what the original benchmark was and what correction was applied, so that readers can immediately place the reported 3e-7 recovery in context.
[Numerical experiments] Numerical experiments section: the specific kernel (e.g., polynomial degree or Gaussian bandwidth) and the placement rule for the kernel centers should be stated explicitly, together with the number of centers used in each example, to support reproducibility.
[Numerical experiments] The suboptimality percentages (0.000% on the 1D case, 0.42% on Van der Pol) are given relative to different references; a short clarifying sentence on how each percentage is computed would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, for recognizing the combination of convexity, the Riccati-Hessian anchoring constraint, and the data-dependent suboptimality bound as a meaningful contribution, and for recommending minor revision. The referee's description of the method and results is accurate.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives its LMI formulation by representing the value-function gradient in an RKHS, applying a Schur-complement transformation to the HJB inequality, and imposing an explicit linear Riccati-Hessian equality at the origin to exclude the zero solution. The suboptimality bound is stated to depend only on problem data and domain (not on the kernel coefficients or approximation error), and the numerical examples measure residuals and costs directly against known optima without reducing any central claim to a fitted input or self-referential definition. No load-bearing self-citation chains, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work are used to justify the core steps; the approach remains convex and internally consistent even when V* lies outside the RKHS.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard properties of reproducing kernel Hilbert spaces and the existence of a solution to the linearised algebraic Riccati equation; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)

domain assumption The gradient of the optimal value function lies in the reproducing kernel Hilbert space spanned by the chosen kernel and centres.
Invoked when representing the gradient in RKHS to obtain the LMI in kernel coefficients.
standard math The linearised system around the equilibrium admits a stabilising solution to the algebraic Riccati equation.
Required for the Riccati-Hessian equality constraint to be well-defined.

pith-pipeline@v0.9.0 · 5918 in / 1546 out tokens · 52840 ms · 2026-05-21T12:07:52.425358+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.

The novel ingredient is an explicit Riccati-Hessian equality constraint at the equilibrium, which removes the trivial solution and forces the Hessian of the approximation to match the algebraic Riccati equation solution of the linearised system.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

We give a suboptimality bound J(x0;û) - V*(x0) ≤ ε T(x0) in which T(x0) depends only on the problem data and the working domain.

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

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work page 2020

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Batlle, Y

P. Batlle, Y. Chen, B. Hosseini, H. Owhadi, and A. M. Stuart. Error analysis of kernel/GP methods for nonlinear and parametric PDEs. arXiv preprint arXiv:2305.04962, 2023

work page arXiv 2023

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work page 2022

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