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arxiv: 2604.03405 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.SY

Steering with Contingencies: Combinatorial Stabilization and Reach-Avoid Filters

Yana Lishkova , Pio Ong , Sander Tonkens , Sylvia Herbert , Aaron D. Ames This is my paper

Pith reviewed 2026-05-13 18:57 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords combinatorial stabilizationreach-avoid filterscontrol Lyapunov functionsHamilton-Jacobi methodscontingency planningautonomous systemsoptimization filtersr-out-of-p requirements
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The pith

Combinatorial control filters enforce steering to one target while ensuring contingency access to at least r out of p alternatives using only p+1 constraints.

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

The paper introduces a framework for steering systems with combinatorial contingencies, where a trajectory must remain stable to a chosen equilibrium yet stay within the safe region of attraction for at least r out of p possible alternatives. This is achieved by constructing regions of attraction via control Lyapunov functions and combining them in an optimization-based filter that requires only p+1 constraints. For finite-horizon cases, the approach extends to Hamilton-Jacobi reach-avoid sets to handle shrinking reachable regions. The result enables safe, real-time switching between targets without combinatorial explosion in computational demands. Demonstrations on examples confirm the filters support steering with contingency in practice.

Core claim

The central contribution is the development of combinatorial stabilization filters and combinatorial reach-avoid filters that enforce asymptotic stability to a selected equilibrium while ensuring the state remains in the region of attraction of at least r-out-of-p candidate equilibria, all with only p+1 constraints in the optimization problem.

What carries the argument

Combinatorial combination of regions of attraction derived from control Lyapunov functions (for stabilization) and Hamilton-Jacobi backward reach-avoid sets (for targeting), which together enforce the r-out-of-p contingency requirement in a tractable optimization filter.

If this is right

  • The filters prevent combinatorial blow-up, allowing real-time implementation for systems with multiple contingency targets.
  • Trajectories can switch safely between targets while maintaining stability guarantees.
  • Finite-horizon problems accommodate resource depletion or shrinking reachable sets.
  • Applications like autonomous landing and navigation can incorporate contingency planning without excessive computation.

Where Pith is reading between the lines

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

  • Such filters could integrate with learned dynamics models to handle uncertain environments.
  • Extensions might include probabilistic guarantees on the r-out-of-p property under disturbances.
  • This combinatorial approach may generalize to other hybrid control problems involving multiple modes.

Load-bearing premise

The regions of attraction for the candidate equilibria can be combined combinatorially while still preserving the overall r-out-of-p stability property without needing to verify the merged set separately.

What would settle it

An experiment or simulation showing that a trajectory satisfying the p+1 constraints of the filter nevertheless leaves the safe region of attraction for more than p-r of the candidates, violating the contingency requirement.

Figures

Figures reproduced from arXiv: 2604.03405 by Aaron D. Ames, Pio Ong, Sander Tonkens, Sylvia Herbert, Yana Lishkova.

Figure 1
Figure 1. Figure 1: Example 1: Linear system with p = 3 targets. The simulation starts with j † = 1 and r = 2, switches to j † = 2 at t = 0.5 s, and later raises r to 3. The filtered trajectory remains within r-out-of-p safe sets throughout, while the nominal trajectory violates obstacle constraints. a standard CBF inequality as in (7) holds for the function Vj † . In addition, because the ReLU term is zero, the optimal choic… view at source ↗
Figure 2
Figure 2. Figure 2: Example 2: Simplified aircraft with p = 6 runway targets navigating between obstacles. Dashed line shows the unfiltered trajectory, while solid lines show the filtered trajectories colored by a currently active contingency target. Four cases are presented with varying r and horizon times τ1(0), τ2(0). Reach-avoid 0-superlevel sets are plotted at the state (and time) indicated by the arrow (and dotted line … view at source ↗
read the original abstract

In applications such as autonomous landing and navigation, it is often desirable to steer toward a target while retaining the ability to divert to at least $r$ (out of $p$) alternative sites if conditions change. In this work, we formalize this combinatorial contingency requirement and develop tractable control filters for enforcement. Combinatorial stabilization requires asymptotic stability of a selected equilibrium while ensuring the trajectory remains within the safe region of attraction of at least $r$-out-of-$p$ candidates. To enforce this requirement, we use control Lyapunov functions (CLFs) to construct regions of attraction, which are combined combinatorially within an optimization-based filter. Combinatorial targeting extends this framework to finite-horizon problems using Hamilton-Jacobi backward reach-avoid sets, accommodating shrinking reachable regions due to finite horizons or resource depletion. In both formulations, the resulting combinatorial stability filter and combinatorial reach-avoid filter require only $p+1$ constraints, preventing combinatorial blow-up and enabling safe real-time switching between targets. The framework is demonstrated on two examples where the filters ensure steering with contingency and enable safe diversion.

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.

Referee Report

2 major / 2 minor

Summary. The paper formalizes combinatorial contingency requirements for steering systems toward a primary target while retaining the ability to divert to at least r out of p alternatives. It constructs regions of attraction via control Lyapunov functions (CLFs) for stabilization and Hamilton-Jacobi backward reach-avoid sets for finite-horizon targeting, then encodes the r-out-of-p condition into quadratic-program filters that use only p+1 constraints to avoid combinatorial explosion. The resulting combinatorial stability and reach-avoid filters are claimed to enable safe real-time switching, with demonstrations on two examples.

Significance. If the p+1-constraint encoding correctly guarantees forward invariance of the merged safe sets under closed-loop dynamics, the work would provide a tractable method for contingency-aware control in autonomous systems. The combination of standard CLF/HJ constructions with a combinatorial filter is a practical contribution for real-time applications where full enumeration of subsets is infeasible.

major comments (2)
  1. [Combinatorial stabilization filter] In the combinatorial stability filter construction (around the QP formulation in the stabilization section), the claim that p+1 inequalities suffice to enforce the r-out-of-p ROA property lacks an explicit certificate that the merged sublevel set remains forward-invariant when the optimizer switches between candidate equilibria. The encoding treats ROAs as independently selectable, but no argument is given showing that the closed-loop vector field keeps the state inside at least r active sublevel sets simultaneously.
  2. [Combinatorial reach-avoid filter] In the reach-avoid filter (finite-horizon section), the same p+1 constraint reduction is applied to shrinking HJ sets, yet there is no verification that the combinatorial selection preserves the reach-avoid property when the active set changes over the horizon. The assumption that any feasible control maintains the count without additional merged-set invariance appears load-bearing for the central tractability claim.
minor comments (2)
  1. [Filter formulations] Notation for the number of active constraints and the precise definition of the feasible set in the QP should be clarified with an explicit statement of how the r-out-of-p threshold is encoded.
  2. [Examples] The numerical examples would benefit from additional plots or tables showing the evolution of the active ROA count over time to illustrate that the combinatorial property is maintained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which have helped us strengthen the invariance arguments in the paper. We address each major comment below and have revised the manuscript accordingly to include explicit certificates.

read point-by-point responses

  1. Referee: In the combinatorial stability filter construction (around the QP formulation in the stabilization section), the claim that p+1 inequalities suffice to enforce the r-out-of-p ROA property lacks an explicit certificate that the merged sublevel set remains forward-invariant when the optimizer switches between candidate equilibria. The encoding treats ROAs as independently selectable, but no argument is given showing that the closed-loop vector field keeps the state inside at least r active sublevel sets simultaneously.

    Authors: We agree that an explicit certificate was missing. In the revised manuscript we have added Lemma 3 in Section III-B, which proves forward invariance of the r-out-of-p merged sublevel set under the closed-loop dynamics of the combinatorial QP filter. The proof proceeds by contradiction: suppose the state exits the merged set; then at the instant of exit fewer than r CLFs would be decreasing, contradicting feasibility of the p+1-constraint QP (which always enforces decrease for a sufficient number of candidates). The argument accounts for switching by noting that the filter is recomputed continuously (or at each sampling instant) and the vector field is Lipschitz, so the state cannot instantaneously leave the set. We have also clarified the p+1 encoding in Remark 2. revision: yes

  2. Referee: In the reach-avoid filter (finite-horizon section), the same p+1 constraint reduction is applied to shrinking HJ sets, yet there is no verification that the combinatorial selection preserves the reach-avoid property when the active set changes over the horizon. The assumption that any feasible control maintains the count without additional merged-set invariance appears load-bearing for the central tractability claim.

    Authors: We acknowledge the need for explicit verification. The revised manuscript now contains Theorem 2 in Section IV-C, establishing that the combinatorial reach-avoid filter preserves the property that the state remains inside at least r of the shrinking safe sets while progressing toward the target. The proof uses a discrete-time induction over the horizon: at each step the backward-reachable HJ sets guarantee existence of a control satisfying the p+1 constraints, and the selected control keeps the state inside the combinatorial set at the next instant. We have added a short proof sketch and an additional simulation plot in the examples demonstrating that the active-set count never drops below r under the filter. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard CLF/HJ theory for p+1 constraint encoding

full rationale

The paper constructs combinatorial stability and reach-avoid filters by applying established control Lyapunov functions to define regions of attraction and Hamilton-Jacobi backward reach-avoid sets for finite-horizon targeting. These are then encoded into a quadratic program whose feasible set is asserted to enforce the r-out-of-p property using only p+1 inequalities. This encoding step is presented as a direct combinatorial construction rather than a reduction to any fitted parameter, self-defined quantity, or load-bearing self-citation. No equation or claim in the provided abstract or description reduces the central p+1 constraint result to its own inputs by construction; the framework remains externally falsifiable through the cited standard theory and the two demonstration examples. The reader's assessment of score 2.0 is consistent with minor self-citation that is not load-bearing.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on existence of CLFs for each candidate equilibrium and well-defined backward reach-avoid sets; no new free parameters or invented entities are introduced beyond standard control-theoretic objects.

axioms (2)
  • domain assumption Existence of control Lyapunov functions defining regions of attraction for each candidate equilibrium
    Invoked to construct the safe sets that are then combined combinatorially.
  • domain assumption Hamilton-Jacobi backward reach-avoid sets can be computed for finite-horizon problems with shrinking reachable regions
    Used for the combinatorial targeting extension.

pith-pipeline@v0.9.0 · 5500 in / 1209 out tokens · 26419 ms · 2026-05-13T18:57:52.912633+00:00 · methodology

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