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REVIEW 3 major objections 5 minor 22 references

Optimistic planning for the near-optimal control of nonlinear switched discrete-time systems with stability guarantees

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A reworked optimistic-planning algorithm controls nonlinear switched systems near-optimally and with stability guarantees.

desk verdict The algorithm and concepts are worthwhile, but the main theorems are deferred to an unpublished companion and the paper cannot be fully verified as submitted. read the letter →

arxiv 1908.01404 v1 pith:DG3QQC5Z submitted 2019-08-04 math.OC cs.SYeess.SY

Mathieu Granzotto , Romain Postoyan , Lucian Buşoniu , Dragan Nešić , Jamal Daafouz This is my paper · ORCID
classification math.OCcs.SYeess.SY MSC 93C5593D1593D20
keywords optimisticplanningOPminnonlinearswitchedsystemsnear-optimalcontrolreceding-horizonpracticalstabilitydetectabilitydiscountedcost
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

Optimistic planning is a tree-search method from artificial intelligence that computes near-optimal inputs for systems with finitely many control choices. The paper rewrites it as OPmin so that it applies to control problems as normally posed: stage costs may be arbitrary nonnegative functions such as quadratics, discounting is optional, and the goal is cost minimization rather than reward maximization. The main claim is that, under a stabilizability-and-detectability condition, the truncated cost OPmin returns sits within an explicit state-dependent error of the infinite-horizon optimum, and that the same algorithm run in receding-horizon mode makes the closed-loop system practically stable. A strengthened version of the condition gives global exponential stability and shows that the running cost approaches the infinite-horizon value exponentially as the computational budget grows.

What carries the argument

The engine is Assumption 1, a uniform detectability/dissipation condition on every mode, together with the optimistic tree search inherited from OP. In the tree, each leaf carries the cost of the switching sequence that reaches it; OPmin always expands the leaf with the smallest such cost, so the first fully explored level yields an exact finite-horizon optimum, and the resulting horizon $d(x)$ is as large as the budget allows. The dissipation inequality is what turns the tail of the infinite-horizon cost into a bounded, state-dependent error term $v_{\gamma,d(x)}(x)$, and the same inequality serves as the Lyapunov decrement that produces the stability bound.

What would settle it

For a two-mode scalar switched linear system with quadratic stage costs, compute $V_{\gamma,\infty}$ exactly by value iteration and run OPmin with a known budget; if the measured gap $V_{\gamma,\infty}-V_{\gamma,d(x)}$ exceeds the bound $v_{\gamma,d(x)}(x)$ from Theorem 1, or the receding-horizon trajectory fails to enter the predicted $\delta$-neighborhood within the budget prescribed by Theorem 2, the central claims are false.

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Extended reading notes

Core claim

Under Assumption 1 — a Lyapunov-like dissipation inequality $W(f_u(x))-W(x) \le -\alpha_W(\sigma(x))+\ell_u(x)$ together with a stabilizability bound $V_{\gamma,\infty}(x)\le \alpha_V(\sigma(x))$ — the paper proves for every state $x$, discount factor $\gamma\in(0,1]$, and state-dependent horizon $d(x)$ that $V_{\gamma,d(x)}(x)\le V_{\gamma,\infty}(x)\le V_{\gamma,d(x)}(x)+v_{\gamma,d(x)}(x)$, where $v_{\gamma,d(x)}(x)$ is an explicit function built from the comparison functions. The same assumptions imply that the receding-horizon closed loop satisfies $\sigma(\varphi(k,x))\le \max\{\beta(\sigma(x),k),\delta\}$ for arbitrary $\delta,\Delta>0$ once the discount factor is close enough to $1$ and the budget large enough; with linear comparison functions this becomes global exponential stability. The paper further shows that the running cost of the receding-horizon scheme lies within $w_{\gamma,\bar d}\,\sigma(x)$ of the infinite-horizon optimal value, with $w_{\gamma,\bar d}$ decaying exponentially in the minimum horizon.

Load-bearing premise

The guarantees rest on Assumption 1(ii), the detectability/dissipation inequality that must hold for every mode and every stage cost, and on the correctness of the companion-paper results [6] that Theorem 1 and Theorem 2 are derived from; if that inequality fails or those results are wrong, the claims collapse.

Editorial extensions

If this is right

  • OPmin can be applied to optimal control problems with stage costs that are arbitrary nonnegative functions, including quadratic costs, and with discount factor $\gamma=1$; the error bound remains finite and decays as the horizon grows.
  • For any prescribed initial-state bound $\Delta$ and final tolerance $\delta$, a budget and discount factor can be chosen so that every receding-horizon trajectory converges to the $\delta$-neighborhood of the attractor, so the scheme gives certified practical stability.
  • Under linear comparison functions, the closed loop is globally exponentially stable and the mismatch between the running cost and $V_{\gamma,\infty}$ decays exponentially in the minimum horizon $\bar d$ (equivalently, in the budget).
  • The algorithm provides both an open-loop near-optimal switching sequence and a receding-horizon feedback law from the same computation, so the same tool covers planning and feedback stabilization.

Reading between the lines

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

  • Because Theorem 3 requires no terminal cost or terminal constraint, the paper suggests a model-predictive-control formulation whose only tuning knobs are budget and discount factor; a testable extension is whether the state-dependent horizon can be shortened near the attractor without losing the guarantees.
  • The exponential factor $\left(1-\frac{a_W}{\bar a_V+\bar a_W}\right)^{\bar d}$ in Corollary 1 quantifies how 'detectable' the cost is; one could turn this into a design guideline for choosing stage costs that make the error small quickly.
  • The bound in Theorem 1 vanishes as $\sigma(x)\to 0$, which the authors note; a natural extension is to use OPmin as a local stabilizer after a coarse global schedule, or to certify performance in a neighborhood of the attractor.
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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

3 major / 5 minor

Summary. The paper proposes OPmin, a modification of the optimistic planning (OP) algorithm of Hren and Munos, adapted to the minimization of discounted or undiscounted infinite-horizon costs for nonlinear switched discrete-time systems with finite input sets and arbitrary nonnegative stage costs. The algorithm is described in Algorithm 1, and the authors state that it computes, for a given budget, the exact finite-horizon value function V_{γ,d(x)}(x) for a state-dependent horizon d(x). Under a standing assumption on existence of optimal infinite-horizon sequences and under Assumption 1, which combines a stabilizability bound on V_{γ,∞} and a detectability/dissipation inequality involving a Lyapunov-like function W, the paper claims: Theorem 1, a two-sided bound between V_{γ,d(x)} and V_{γ,∞} with an explicit state- and horizon-dependent error v_{γ,d(x)}; Corollary 1, a linear-growth version of the bound that decays exponentially in d(x); Theorem 2, semiglobal practical stability of the receding-horizon closed-loop system; Corollary 2, global exponential stability under strengthened linear assumptions; Theorem 3 and Corollary 3, bounds on the running cost relative to V_{γ,∞}. A numerical example with the cubic integrator is reported. The main proofs of Theorem 1 and Corollary 2 are omitted and deferred to the authors' unpublished companion manuscript [6], and the appendix proof of Theorem 2 and Proposition 3 also import key steps from [6].

Significance. If the results are correct, the paper makes a useful contribution by adapting optimistic planning to a control-oriented setting: it removes the stage-cost normalization to [0,1], allows the undiscounted case γ=1, treats cost minimization directly, and provides stability guarantees for a receding-horizon implementation, which are absent from the original OP analysis. The state-dependent horizon appearing both in the algorithm and in the error bounds is a genuine conceptual step beyond the fixed-horizon analysis in the authors' companion work. The result relating the running cost to the infinite-horizon value function, with exponential decay in the horizon, is also valuable. However, the significance is conditional: each of these claims depends on proofs that are not present in this manuscript and are instead deferred to an unpublished companion paper. The reader cannot verify that the companion's assumptions, which may involve fixed horizons and a specific discounted setting, transfer to the state-dependent horizons, γ=1, and generic nonnegative costs treated here.

major comments (3)
  1. [Section IV-B, Theorem 1 and Eq. (8)] The central near-optimality inequality is not proved in this manuscript; the text states that 'Theorem 1 follows from [6, Theorem 3], therefore the proof is omitted.' Reference [6] is described only as 'submitted for journal publication' and is not an archival, publicly citable source. A reader therefore cannot check whether [6] covers γ=1, nonnegative stage costs outside [0,1], and state-dependent horizons, all of which are explicit claimed improvements over OP. Please provide a self-contained proof of Theorem 1, or make the companion paper publicly verifiable and show explicitly that its assumptions and proof steps extend to OPmin with state-dependent horizons.
  2. [Section IV-C, Corollary 2 and Eq. (12)] The global exponential stability result is stated with 'the proof is omitted' and with justification that it follows from [6, Corollary 2] plus 'the modifications given in the appendix.' Because the receding-horizon implementation uses a state-dependent horizon d(x), uniformity of the Lyapunov decay over all horizon sequences generated by OPmin is a load-bearing issue. The full proof must appear in this paper, including verification that the Lyapunov decrease in Proposition 3 holds uniformly over the possible state-dependent horizons.
  3. [Appendix, proof of Theorem 2 and Proposition 3] The appendix proof of Theorem 2 is not self-contained: it explicitly imports 'item (i) of Theorem 1 in [6]', 'the same manipulations as in the proof of Theorem 1 in [6]', and 'the steps of [6]'. Proposition 3, which is the key Lyapunov property used for Theorem 2, is therefore not verifiable from the manuscript alone. Please provide a complete proof of Proposition 3 and of Theorem 2 that does not rely on unpublished companion results.
minor comments (5)
  1. [Section IV-B, Theorem 1] The definitions of the class-K functions in the error term are garbled: the text reads 'αY = αY := αW, αY := αV + αW', which defines αY twice and leaves the intended upper and lower functions unclear. Please introduce αY and αY with distinct, consistent definitions.
  2. [Section III-B, Algorithm 1] In the paragraph defining leaves, there is a typo: 'the the set of all leaves' should be 'the set of all leaves'.
  3. [Remark 1] The verb 'chosing' should be 'choosing'.
  4. [Section V, Table I] Table I reports 'estimated running cost' computed over 200 simulation steps, whereas Theorem 3 concerns the infinite-horizon running cost. A sentence on the truncation error or on why 200 steps is sufficient would help the reader interpret the numerical comparison.
  5. [Section IV-B, Remark 1] The remark notes that the error bound v_{γ,d(x)} is not monotonic in γ because of competing terms; the discussion would be clearer if the two terms were displayed explicitly.

Circularity Check

4 steps flagged · score 8.0 of 10

Central near-optimality and stability theorems are not proved in the manuscript: Theorem 1 and Corollary 2 are explicitly deferred to the authors' unpublished companion [6], and Theorem 2's appendix imports [6]; the main claims therefore reduce to an unverifiable self-citation chain.

  1. self citation load bearing [Section IV-B, Theorem 1 (Eq. 8)]
    "Theorem 1 follows from [6, Theorem 3], therefore the proof is omitted."

    The paper's principal near-optimality guarantee, inequality (8), is not derived in this manuscript; it is imported from [6, Theorem 3]. Reference [6] is the authors' own companion submission, not published and available only through a reviewer URL, so the citation is not an independent, checkable proof. The paper's advertised adaptations (generic stage costs, gamma = 1, state-dependent horizon d(x)) are exactly where the transfer from [6] would need to be demonstrated, but the proof is omitted.

  2. self citation load bearing [Section IV-C, Corollary 2]
    "Since it follows the same arguments as in [6, Corollary 2] and the modifications given in the appendix, the proof is omitted."

    This is the paper's global exponential stability result for the OPmin closed loop. The proof is not present; it is asserted to follow from the same authors' [6, Corollary 2] plus modifications in the appendix. The condition (12) and the constants K and lambda come from [6]. Thus the strongest stability claim is load-bearing on an unverified companion rather than being established in this paper.

2 more flagged steps
  1. self citation load bearing [Appendix, Proof of Theorem 2]
    "The proof of Theorem 2 follows the same steps as the proof of [6, Theorem 2]. The difference is that the horizon in cost (4) is not fixed as in [6], but depends on the state. Nevertheless, as noted in [6, Remark 3], the results can be modified to hold for varying horizons, provided that considered horizons are lower-bounded by a sufficiently large constant, ¯d in our case."

    The semiglobal practical stability theorem, a central advertised result, is justified by following [6, Theorem 2] and by invoking [6, Remark 3] for precisely the state-dependent-horizon generalization that the paper claims as a novelty over [6,7]. The only part proved in the appendix, Proposition 3, is itself obtained by following [6, proof of Theorem 1]. Without [6], Theorem 2 is unsupported.

  2. self citation load bearing [Appendix, Proposition 3 proof]
    "The proof works by following the steps in [6, proof of Theorem 1], only one step needs to be carefully modified."

    Proposition 3 provides the Lyapunov decrease inequality on which Theorem 2 relies. Its proof is not self-contained: it adopts the framework, inequalities, and iteration arguments of [6, proof of Theorem 1] and modifies one step. Since [6] is unpublished, the validity of the Lyapunov construction and therefore of Theorem 2 depends entirely on the same authors' unavailable companion.

full rationale

The OPmin algorithm itself, Propositions 1 and 2 on the budget/horizon relation, and Theorem 3's running-cost argument are genuinely developed in the text. However, the two central guarantees that justify the algorithm are not proved here: the near-optimality inequality (8) is explicitly stated to follow from [6, Theorem 3], and the global exponential stability result of Corollary 2 is explicitly stated to follow from [6, Corollary 2]. Theorem 2's proof consists of following [6, Theorem 2] with a state-dependent-horizon patch supported by [6, Remark 3], and Proposition 3's proof follows [6, proof of Theorem 1]. Reference [6] is a companion submission by the same five authors, described as 'submitted for journal publication' and available only to reviewers, so it is not machine-checked, code-reproduced, or independently falsifiable within the paper. This is load-bearing self-citation rather than cosmetic citation. The paper does contain original contributions, so it is not equivalent to its inputs by definition, but the central derivation chain reduces to an unverifiable companion, warranting a high circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. Its theoretical results are conditional on Assumption 1 and on unproved companion theorems in [6], which is the main burden for a reader trying to verify the claims.

assumptions (5)
  • domain assumption Standing Assumption (SA): for every x and gamma in (0,1], an optimal infinite-horizon input sequence exists and the value function is finite.
    Invoked in Section II and IV-A; needed to define V_gamma,infty and for the lower bounds in Theorems 1 and 3. Standard in optimal control but nontrivial for the undiscounted case gamma = 1.
  • domain assumption Assumption 1(i): there exist alpha_V and sigma such that V_gamma,infty(x) <= alpha_V(sigma(x)) for all x and gamma.
    Stabilizability with respect to the stage cost; imported from MPC literature [6,8,14]. Needed for the near-optimality error bound in Theorem 1 and for the Lyapunov bound in Proposition 3.
  • domain assumption Assumption 1(ii): existence of W, alpha_W, alpha_W such that W(x) <= alpha_W(sigma(x)) and W(f_u(x)) - W(x) <= -alpha_W(sigma(x)) + ell_u(x) for all x and u.
    Detectability and dissipation inequality for every mode; the central structural premise for all stability and running-cost results. Not automatically satisfied; the example verifies it for one cubic-integrator system.
  • ad hoc to paper Validity of the companion results in [6]: Theorem 3 (near-optimality bound), Theorem 2 and Corollary 2 (stability) of [6].
    The proofs of Theorem 1 and Corollary 2 in this paper are omitted and delegated to [6], an unpublished same-author manuscript not accessible to readers. The paper's central claims stand or fall with [6].
  • domain assumption Linear growth conditions in Corollaries 1 and 2: alpha_V(s) <= bar_a_V s, alpha_W(s) <= bar_a_W s, and alpha_W(s) >= a_W s for all s >= 0.
    Used to derive explicit exponential bounds that are uniform in gamma; a strengthening of Assumption 1 required for the global exponential stability result.

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Pith. "Pith review of Optimistic planning for the near-optimal control of nonlinear switched discrete-time systems with stability guarantees." pith.science (2026). https://pith.science/paper/DG3QQC5Z

@misc{pith2026190801404,
  author       = {Pith},
  title        = {Pith review of: Optimistic planning for the near-optimal control of nonlinear switched discrete-time systems with stability guarantees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DG3QQC5Z}},
  note         = {Machine review of arXiv:1908.01404}
}
read the original abstract

Originating in the artificial intelligence literature, optimistic planning (OP) is an algorithm that generates near-optimal control inputs for generic nonlinear discrete-time systems whose input set is finite. This technique is therefore relevant for the near-optimal control of nonlinear switched systems, for which the switching signal is the control. However, OP exhibits several limitations, which prevent its application in a standard control context. First, it requires the stage cost to take values in [0,1], an unnatural prerequisite as it excludes, for instance, quadratic stage costs. Second, it requires the cost function to be discounted. Third, it applies for reward maximization, and not cost minimization. In this paper, we modify OP to overcome these limitations, and we call the new algorithm OPmin. We then make stabilizability and detectability assumptions, under which we derive near-optimality guarantees for OPmin and we show that the obtained bound has major advantages compared to the bound originally given by OP. In addition, we prove that a system whose inputs are generated by OPmin in a receding-horizon fashion exhibits stability properties. As a result, OPmin provides a new tool for the near-optimal, stable control of nonlinear switched discrete-time systems for generic cost functions.

Figures

Figures reproduced from arXiv: 1908.01404 by the authors.

Figure 1
Figure 1. State evolution for B = 3000 and x = [−1, 1.5]>. Budget 30 300 3000 [ 10, 15]> 199015 13757 12609 Initial States [−1 , 1.5]> 314 28 22 [−15, −10]> 128184477 46875 42952 [ 10, −15]> 14180 2802 2615 TABLE I: Estimated running cost for various budgets and initial conditions. VI. CONCLUSION We have modified the optimistic planning algorithm in [10] to be applicable for the near-optimal, stable control of nonlinear switc… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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