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REVIEW 5 major objections 4 minor 25 references

A Physics-Informed Learning Framework to Solve the Infinite-Horizon Optimal Control Problem

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Pith's one-line read Solving the finite-horizon HJB equation with PINNs provably recovers the infinite-horizon optimal control.

desk verdict Finite-horizon PINN solver is a good idea, but the main convergence proof has a gap at Lemma 2 that must be fixed before publication. read the letter →

arxiv 2505.21842 v1 pith:T6KXNIM7 submitted 2025-05-28 eess.SY cs.LGcs.SY

Filippos Fotiadis , Kyriakos G. Vamvoudakis This is my paper
classification eess.SYcs.LGcs.SY MSC 49L2049K1568T07
keywords physics-informedneuralnetworksoptimalcontrolHamilton-Jacobi-Bellmanequationfinite-horizonapproximationnonlinearsystemsvaluefunctionhorizonextensionerror
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a way to make physics-informed neural networks (PINNs) solve the infinite-horizon optimal control problem for nonlinear systems without running into the multiple-solution issue of the steady-state Hamilton-Jacobi-Bellman (HJB) equation. The idea is to train the PINN on the time-dependent, finite-horizon version of the HJB equation, which has a unique solution, instead of on the steady-state equation. The paper's central claim is that as the horizon grows, the finite-horizon value function at time zero converges uniformly to the true optimal value function, and the associated feedback control converges uniformly to the optimal policy, for any positive semidefinite terminal cost. It also proves a horizon-extension mechanism that reuses an already-computed value function and argues that approximation errors need not accumulate under repeated extensions, together with a residual-based test for deciding whether a chosen horizon is long enough. The paper presents the framework as a principled, guess-free route to learned optimal stabilization that requires neither an initial stabilizing controller nor iterative policy evaluation.

What carries the argument

The central object is the finite-horizon value function $V_T(x,t)$, defined as the unique solution of the terminal-value HJB equation (5) and equivalently as the optimal value of the finite-horizon cost (6) with terminal penalty $\phi$. It carries the argument because the steady-state HJB (4) is its formal stationary limit and because, unlike (4), equation (5) has a unique continuously differentiable solution, so a PINN trained on its residual (8) has a single target. The proof mechanism is the sandwich inequality (9): for any positive semidefinite $\phi$, the value $V_T(\cdot,0;\phi)$ lies between the value with zero terminal cost and the value with terminal cost $\phi+V^*$, both of which are shown to converge uniformly to $V^*$. The horizon-extension machinery in Theorem 2 is a second key element: it shows that the value for horizon $T'$ can be obtained from the value for horizon $T$ by solving a shorter companion PDE (24) over $[0,T'-T]$, which is a direct application of the principle of optimality.

What would settle it

Solve the finite-horizon HJB for a system with a known optimal value function (for instance the quartic-value system of Section 6.2) for a sequence of increasing horizons $T$, and record both the tail cost $\int_{T/2}^{T}(q(x_T(\tau)) + u_T(\tau)^\top R u_T(\tau))\,d\tau$ and the endpoint state $x_T(T)$. If the tail cost tends to zero while $x_T(T)$ does not approach the origin, Lemma 2 is false and the uniform-convergence theorem needs extra assumptions. A concrete search target is a positive-definite running cost with dynamics whose closed-loop trajectories hover near a nonzero state for a long stretch and then decay sharply near the horizon, so that the windowed integral vanishes without the endpoint converging.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1: for an arbitrary positive semidefinite terminal cost, the finite-horizon value function $V_T(\cdot,0)$ of the time-dependent HJB equation (5) converges uniformly to the infinite-horizon optimal value function $V^*$ on any compact set $\Omega$ as $T\to\infty$, and the corresponding control $u_T(\cdot,0)$ converges uniformly to $u^*$. The argument funnels through a comparison inequality (9) that sandwiches the finite-horizon value for any terminal cost between the value with zero terminal cost and the value with terminal cost $\phi+V^*$. The zero-terminal-cost side uses monotonicity in $T$ plus Dini's theorem to get uniform convergence; the $\phi+V^*$ side uses the stabilizing property of the optimal policy to control the tail integral. This justifies replacing the multi-solution steady-state HJB (4) with the unique-solution finite-horizon HJB (5) inside a PINN training loop, and the paper adds a residual (22) to certify the horizon and a dynamic-programming extension (Theorem 2) to grow it without restarting.

Load-bearing premise

The proof assumes that because the tail integral of the running cost over $[T/2,T]$ vanishes as $T\to\infty$, the endpoint state $x_T(T)$ of the finite-horizon optimal trajectory must converge to the origin; this inference from a vanishing moving-window integral to pointwise endpoint convergence is asserted in Lemma 2 but not established.

Editorial extensions

If this is right

  • A PINN can be trained on the finite-horizon HJB (5) with any positive semidefinite terminal cost, and for a sufficiently large horizon $T$ the resulting value and control estimates are uniformly close to the true infinite-horizon optimal value and control on the compact set $\Omega$.
  • The residual $E$ defined in (22) gives a computable test of whether a chosen horizon is large enough: a large residual signals either that $T$ should be increased or that the network architecture cannot represent the finite-horizon value accurately.
  • If the horizon is too small, the already-computed value function $V_T(\cdot,0)$ can be used as a terminal cost in the shorter companion problem (23), whose solution coincides with the solution of the original longer-horizon problem (Theorem 2), so the horizon can be extended without solving from scratch.
  • Approximation errors from the initial solve do not necessarily accumulate when the horizon is extended multiple times: with a bounded error that vanishes at the origin, the extended-horizon error stays within the same bound (Theorem 3 and the discussion following it).
  • The approach avoids the three standard restrictions of policy iteration: it needs no prior knowledge of a stabilizing controller, does not require polynomial basis functions, and performs no iterative policy evaluations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The unproved endpoint-convergence step in Lemma 2 suggests that the uniform-convergence theorem as written requires an extra regularity assumption, such as uniform exponential decay of the finite-horizon optimal trajectories; the simulations may satisfy this informally even if the proof does not.
  • The residual-based horizon check (22) could be used online during training as a stopping or architecture-selection criterion, letting the network itself signal when the horizon or the network capacity must be increased.
  • If the endpoint-convergence gap is filled, the same template—embedding a multi-solution steady-state PDE in a unique-solution time-dependent companion equation—applies to other inverse and optimal-control problems beyond the specific HJB setting.
  • A natural next test is to extend the argument to zero-sum differential games, since the Isaacs equation suffers the same non-uniqueness pathology; the authors list exactly this as future work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The manuscript proposes a PINN-based method for the infinite-horizon optimal control problem by training on the finite-horizon HJB equation (5) rather than the steady-state HJB equation (4), on the grounds that (5) has a unique solution. Section 4 claims that, as the horizon T tends to infinity, the finite-horizon value function V_T(·,0) and the associated controller u_T(·,0) converge uniformly to the infinite-horizon value function V* and optimal controller u* on a compact set Omega, for an arbitrary positive semidefinite terminal cost. Section 5 proposes a residual-based test for whether T is large enough, a horizon-extension procedure based on the dynamic programming principle, and an error-propagation analysis for approximate value functions. The paper closes with three numerical examples: a torsional pendulum, a system with known quartic value function, and a third-order system.

Significance. If the convergence theorem were rigorously established, the paper would offer a practical alternative to policy iteration and to direct PINN solution of the steady-state HJB: it would exploit the uniqueness of the finite-horizon HJB, would not require a stabilizing initial policy, and would work with non-polynomial basis functions. The horizon-extension idea in Theorem 2 and the error bound in Theorem 3 are useful contributions that appear largely correct under suitable regularity assumptions. However, the central convergence result currently rests on a gap in Lemma 2 and an underproved controller-convergence step in Theorem 1; these issues must be repaired before the main claim is acceptable.

major comments (5)
  1. [Section 4.1, Lemma 2] The proof of Lemma 2 shows only that the tail integral from T/2 to T of q(x_T(tau)) + u_T^T R u_T vanishes as T tends to infinity, and then concludes that the endpoint x_T(T) converges to 0. This pointwise endpoint inference does not follow from a vanishing L1 tail over a moving interval: a boundary layer near t=T, or a trajectory spending most of the interval near the origin and then moving away, is not excluded by the stated assumptions. Positive definiteness of q does not provide coercivity, and no uniform bounds on x_T or u_T are assumed. Since Lemma 3 uses V*(x_T(T)) to 0 and Eq. (16), the pointwise convergence, the Dini-based uniform convergence in Lemma 4, and hence Theorem 1 all inherit this gap. The authors should either prove endpoint convergence under additional explicit assumptions or replace the argument.
  2. [Section 4.3, Theorem 1] The uniform convergence of u_T(·,0) to u* is asserted by saying that 'following the same arguments as in [4]' the trajectories converge uniformly and hence the controllers converge uniformly. Uniform convergence of the value functions does not imply uniform convergence of their gradients, and the controller is defined as u_T(·,0) = -(1/2)R^{-1}g^T grad V_T(·,0). A separate argument is needed to show that grad V_T(·,0) converges uniformly to grad V*, or otherwise to prove the uniform convergence of u_T directly. As written, the control-convergence half of Theorem 1 is not proved.
  3. [Section 4.2, Lemma 5 and Eq. (20)] The proof of Lemma 5 uses the step that the maximum over x0 in Omega of gamma(x*_T(T)) tends to 0, i.e., uniform-in-initial-condition convergence of the optimal trajectories to the origin. Definition 1 only requires that the policy asymptotically stabilizes the system for each x0 in Omega, which is pointwise convergence and does not imply uniformity over the compact set Omega without an additional assumption such as uniform asymptotic stability. Since Lemma 5 is needed for the arbitrary-terminal-cost part of Theorem 1, this uniformity assumption should be stated explicitly or proved.
  4. [Section 5.3, after Theorem 3] The paragraph following Theorem 3 states that 'by Lemma 2, one has the limit as T' tends to infinity of x_{T'}(T'-T) equal to 0'. This is not a consequence of Lemma 2, which concerns the endpoint x_T(T) of the zero-terminal-cost problem with the same horizon. For a fixed T, x_{T'}(T'-T) is not an endpoint of a zero-terminal-cost optimal control problem, since the optimal policy over [0,T'] is not generally the optimal policy for the restricted interval [0,T'-T]. The claimed non-accumulation of approximation errors therefore requires a separate proof or an additional assumption.
  5. [Section 5.1, Eqs. (21)-(22)] The residual E in Eq. (22) is proposed as a way to verify that V_T(·,0) is close to V*. However, the steady-state HJB equation (4) is known to have multiple solutions, as the paper itself emphasizes in Sections 1 and 3. A small value of E only indicates that V_T(·,0) nearly solves some solution of (4), not necessarily the optimal value V*. Thus the verification criterion needs an additional uniqueness or certificate argument, or it should be explicitly presented as a heuristic rather than a guaranteed test.
minor comments (4)
  1. [Section 3, Eq. (5)] The terminal condition is written as V_T(x,T)=psi(x), but the paper consistently uses V_T(·,0); it would help to state explicitly that V_T(x,t) is the value function with t time units remaining, so that the terminal condition and the evaluation at time 0 are unambiguous.
  2. [Definition 1 and Eq. (6)] Definition 1 defines admissibility for time-invariant feedback policies, whereas the finite-horizon problem (6) is minimized over time-varying policies u_T(x,t). The admissible class for the finite-horizon problem should be defined separately or the notation should be adjusted.
  3. [Section 5.3, Theorem 3] The bound in Eq. (27) involves the approximation error e_T evaluated at states x_{T'}(T'-T) and hat x_{T'}(T'-T), which may lie outside the set Omega on which the PINN approximation error is controlled. Unless the trajectories are shown to remain in Omega, the bound is not directly usable as a quantitative error estimate.
  4. [Section 6, Table 1 and Figures 1-7] The numerical comparison in Table 1 reports the mean squared residual (22) for different horizon lengths and for the direct steady-state HJB approach; it would be helpful to state the number of independent training runs and the variance of the reported MSE values, and to add axis labels and color bars to the figure panels.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finite-horizon HJB objective is an independent equation, and the convergence proof does not assume its own conclusion.

full rationale

The paper trains on the time-dependent HJB equation (5), which is a distinct PDE from the target steady-state HJB (4), so the central approximation is not fitted to the quantity it claims to predict. The convergence argument is a genuine derivation chain: Lemma 1 establishes monotonicity and boundedness of V_T(·,0), Lemma 2 attempts to show endpoint convergence x_T(T)→0, Lemma 3 derives pointwise convergence through the comparison (16), Lemma 4 upgrades to uniform convergence via Dini's theorem, Lemma 5 treats the ψ+V* terminal-cost case, and inequality (9) sandwiches arbitrary terminal costs. None of these steps assumes the desired conclusion V_T→V*. The horizon-verification residual E in (22) checks whether V_T(·,0) approximately satisfies the steady-state HJB (4), but the paper itself acknowledges that (4) has multiple solutions, so this check is at best a heuristic necessary condition rather than an equivalence with optimality; this is a logical insufficiency, not a construction-level circularity. The only self-citation, [17], is a provenance note describing a preliminary version and is not load-bearing in any proof. The proof does contain real mathematical gaps: Lemma 2's inference from a vanishing tail integral over [T/2,T] to pointwise endpoint convergence x_T(T)→0 is not established under the stated assumptions, and Theorem 1's control-convergence claim requires uniform convergence of gradients, not just of value functions. These are correctness risks, not circularity, because the statements are not assumed or defined in terms of each other within the derivation.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard optimal-control regularity assumptions (local Lipschitz dynamics, admissibility, positive definiteness of q and R), plus unproved regularity of the finite-horizon HJB solution and the endpoint-convergence lemma used to establish uniform approximation. The horizon length T and neural-network hyperparameters are user-chosen degrees of freedom that shape the approximation error, but are not fitted to the target value function. No invented physical entities are introduced.

free parameters (3)
  • Horizon length T = T=1 extended to 4 (pendulum); T=0.5 extended to 4 (quartic)
    Choice of finite horizon trades off approximation accuracy and training cost; in simulations T is chosen small and extended until the steady-state HJB residual falls below a threshold. It is a user-tuned degree of freedom, not fitted to data.
  • Terminal cost psi = psi = 0 in all simulations
    The theory allows arbitrary positive semidefinite terminal cost, but experiments use psi=0, which affects the convergence rate and the observed residual values.
  • Neural network architecture and training hyperparameters = e.g., 3 layers x 100 nodes, tanh, Adam lr 0.005 with decay 0.002, termination loss 1e-3
    Chosen by hand for each experiment; affects the approximation error e_T and the measured residuals. No sensitivity analysis is provided.
assumptions (6)
  • domain assumption Existence of an admissible stabilizing policy on Ω with finite infinite-horizon cost for every x0 in Ω.
    Needed for V* to be finite and for the upper bound in Lemma 1. Introduced in Section 2 after Definition 1 and before equation (4).
  • standard math Local Lipschitz continuity of f and g, with f(0)=0, ensures existence and uniqueness of trajectories of system (1).
    Stated in Section 2 and used throughout the trajectory arguments in Lemmas 1-3 and Theorem 1.
  • domain assumption The finite-horizon HJB equation (5) admits a unique sufficiently smooth solution V_T for each T.
    The paper asserts uniqueness when V_T is continuously differentiable (Section 3), but does not prove existence or regularity of V_T under the stated assumptions.
  • ad hoc to paper The endpoint state x_T(T) converges to 0 as T grows, used to conclude V*(x_T(T))->0.
    This is the content of Lemma 2, which the paper attempts to prove from tail-integral decay. The inference from averaged energy decay to endpoint convergence is not rigorously established and is load-bearing for Lemma 3 and Theorem 1.
  • standard math Principle of optimality (dynamic programming) for the finite-horizon cost (6) and its decomposition in Theorem 2.
    Used in Lemma 5 and Theorem 2 to show that the tail of the horizon can be optimized independently given the value at the intermediate time.
  • standard math Dini's theorem for uniform convergence of an increasing sequence of continuous functions on a compact set.
    Invoked in Lemma 4 to upgrade pointwise convergence of V_T(·,0) to uniform convergence on Ω.

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Cite this review

Pith. "Pith review of A Physics-Informed Learning Framework to Solve the Infinite-Horizon Optimal Control Problem." pith.science (2026). https://pith.science/paper/T6KXNIM7

@misc{pith2026250521842,
  author       = {Pith},
  title        = {Pith review of: A Physics-Informed Learning Framework to Solve the Infinite-Horizon Optimal Control Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6KXNIM7}},
  note         = {Machine review of arXiv:2505.21842}
}
read the original abstract

We propose a physics-informed neural networks (PINNs) framework to solve the infinite-horizon optimal control problem of nonlinear systems. In particular, since PINNs are generally able to solve a class of partial differential equations (PDEs), they can be employed to learn the value function of the infinite-horizon optimal control problem via solving the associated steady-state Hamilton-Jacobi-Bellman (HJB) equation. However, an issue here is that the steady-state HJB equation generally yields multiple solutions; hence if PINNs are directly employed to it, they may end up approximating a solution that is different from the optimal value function of the problem. We tackle this by instead applying PINNs to a finite-horizon variant of the steady-state HJB that has a unique solution, and which uniformly approximates the optimal value function as the horizon increases. An algorithm to verify if the chosen horizon is large enough is also given, as well as a method to extend it -- with reduced computations and robustness to approximation errors -- in case it is not. Unlike many existing methods, the proposed technique works well with non-polynomial basis functions, does not require prior knowledge of a stabilizing controller, and does not perform iterative policy evaluations. Simulations are performed, which verify and clarify theoretical findings.

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