Bressan's Fire Conjecture: Spiral Barrier Dynamics

• Bressan's Fire Conjecture is a geometric motion-planning problem that investigates constructing a non-self-intersecting, locally convex spiral barrier to confine a fire spreading at unit speed.
• The analysis uses a retarded differential equation and variational reformulation to determine a sharp critical construction speed (approximately 2.614) necessary for successful containment.
• Numerical evaluations and a homotopy argument confirm the uniqueness of the optimal spiral design and establish its relevance to dynamic control in resource-constrained systems.

Bressan's Fire Conjecture is a geometric motion-planning problem concerning the optimal construction of barriers to block an isotropically expanding fire in the plane. Originating in the paper of dynamic systems and control under resource constraints, the conjecture postulates a sharp threshold for the minimum speed at which a single firefighter must build a barrier (specifically of spiral-like form) in order to eventually contain a fire that grows at unit speed. The central question is to determine, both constructively and analytically, whether a barrier exists that confines the fire in a bounded region of $\mathbb{R}^2$ for a given construction speed, and to identify the critical speed below which such a barrier cannot exist.

1. Precise Definition and Geometric Context

In the classical formulation, a single firefighter constructs a continuous wall (barrier) in real time within $\mathbb{R}^2$, tasked with preventing an expanding circular fire (spreading at speed $1$) from escaping indefinitely. Admissible strategies are those in which the barrier $\zeta$ is a locally convex, simple (i.e. non-self-intersecting) curve parameterized by arc-length: $\zeta : [0,S] \rightarrow \mathbb{R}^2$. The barrier construction obeys a speed constraint $\sigma>1$, interpreted as the maximum length that can be built per unit time. The conjecture distinguishes spiral-like (locally convex, outward-growing) barriers from more general curves, focusing on the realization that locally convex geometry is optimal for single-wall containment.

Key geometric conditions include:

• The barrier trace is simple, i.e. no self-intersections.
• The “burn time” function $u \circ \zeta$ along the curve is monotone increasing, guaranteeing outward construction.
• Local convexity: Each small segment is the graph of a convex function up to rigid motion.

The directionality and convexity of the barrier enable a parametrization in angle–ray coordinates for both analytic and constructive methods.

2. Spiral Barrier Representation and Angle–Ray Formalism

Each spiral barrier can be expressed in an angle–ray representation: $$\zeta(s^+(\phi)) = \zeta(s^-(\phi)) + r(\phi)e^{i\phi}$$ where $s^- (\phi)$ is the base point of the start of round $\phi$, $r(\phi)$ is the ray length, and $\phi$ is the rotation angle of the tangent vector. The control variable $\beta(\phi)$ (the angle between tangent and constructed ray) may vary along the spiral, but is constant for “saturated” spirals.

This parametrization is crucial for modeling the evolution of the barrier and for establishing delay differential equations governing the spiral’s closure properties. The spiral barrier thus advances in rounds, with each round defined by an angular increment and corresponding local convex evolution.

Locally convex spiral barriers serve as the preferred strategy for containment due to their geometric efficiency in maximizing the barrier’s reach from the origin.

3. Critical Speed and Retarded Differential Equation (RDE) Analysis

Bressan's Fire Conjecture asserts the existence of a sharp critical speed $\bar{\sigma}$ for spiral-like barriers, given by: $$\bar{\sigma} = 1/\cos(\bar{\alpha}) \approx 2.61443...$$ where $\bar{\alpha}$ solves the transcendental equation: $$(2\pi + \bar{\alpha})\left(\cot(\bar{\alpha}) + \frac{\ln[(2\pi + \bar{\alpha})/\sin(\bar{\alpha})]}{2\pi + \bar{\alpha}}\right) = 1$$

For “saturated” spirals (i.e. those fully exhausting available construction per round with constant control $\beta(\phi) = \alpha$), the ray length $r(\phi)$ satisfies a retarded differential equation: $$\frac{d}{d\phi} r(\phi) = \cot(\alpha) r(\phi) - \frac{r(\phi - [2\pi + \alpha])}{\sin(\alpha)}$$

Analysis of the associated spectrum via the characteristic equation

$\lambda + e^{-\lambda} - a = 0$

with $a = (2\pi + \alpha)(\cot(\alpha) - c)$ and $$c = \frac{\ln[(2\pi + \alpha)/\sin(\alpha)]}{2\pi + \alpha},$$ shows:

• For $\sigma > \bar{\sigma}$, the saturated spiral eventually “closes,” i.e. $r(\phi) = 0$ for some finite $\phi$.
• For $\sigma \leq \bar{\sigma}$, the spiral fails to close, and the fire is not contained.

Thus, $\sigma > \bar{\sigma}$ is both necessary and sufficient for spiral barrier containment. The existence of the critical speed threshold is validated by analysis of the RDE’s eigenvalues and their transition from real to complex conjugate, yielding oscillations that drive the spiral closure for supercritical speeds but fail as $\sigma \to \bar{\sigma}$.

4. Variational Reformulation and Optimal Spiral Construction

The containment problem is equivalently posed as a minimization problem for a prescribed functional over the class of admissible spirals. Define the admissibility functional

$\mathcal{M}(x) = u(x) - \frac{1}{\sigma} \mbox{length}(Z \cap \{u \leq u(x)\})$

where $u$ is the time function for fire propagation, and $Z$ denotes the barrier.

The spiral closes (i.e. confines the fire) iff

$$\min \left\{r_{\mathcal{Z}}(\bar{\phi}) : \mathcal{Z} \text{ is a spiral continuation}\right\} = 0$$

where $r_{\mathcal{Z}}(\bar{\phi})$ is the final fire–ray length at angular point $\bar{\phi}$ corresponding to full rotation.

Constructively, the optimal closing spiral beyond a fixed initial arc is achieved by:

• Suitable continuation with level sets of $u$ (arc segments of constant burn time),
• Tangential departure via a segment (exploiting convexity),
• Attachment of a saturated spiral segment.

The chain of construction uses explicit analytic formulas involving the Green kernel for the RDE, assembled to ensure tangent junctions and continuity of the admissibility functional.

5. Homotopy Argument and Uniqueness of Minimizer

To demonstrate optimality and uniqueness, the paper introduces a homotopy: a differentiable family of admissible spirals parameterized by the breaking point $s_0$ after the initial construction. The derivative of the final ray length $r(\bar{\phi})$ with respect to $s_0$ is shown, via analysis of the corresponding retarded differential equation and Green kernel, to be strictly positive except at the optimal spiral.

The homotopy argument yields:

• Any admissible spiral deviating from the optimal minimizer must incur strictly higher final ray length, precluding closure at the same or lower construction speed.
• The optimal closing spiral is therefore unique.
• Evaluation of the sign of derivatives is performed numerically, using tools such as Mathematica, to confirm strict positivity on relevant intervals (see Figures 6–8 in the original paper).

This framework both establishes sharpness of the critical threshold and excludes the existence of subcritical admissible spirals.

6. Numerical Evaluation of Critical Quantities

Because RDEs and transcendental equations arise in the analytic characterization (e.g. critical angle, eigenvalues and Green kernel asymptotics), numerical approaches underpin key conclusions:

• The precise value for $\bar{\sigma}$ is computed via solution of the transcendental equation over angles.
• Eigenvalue distributions for the delay operator are visualized to distinguish between closing and non-closing regimes.
• Numerical plots validate theoretical assertions about incremental derivative ratios and uniqueness of minimizers.

Numerical confirmation is essential especially in demonstrating positivity of derivatives along homotopy paths and verifying behavior of saturated spiral evolution near the critical speed, where analytic intractability prompts computational investigation.

7. Broader Implications and Connections

Bressan's Fire Conjecture integrates geometric motion planning, delay differential systems, and variational analysis within dynamic resource-constrained systems. The spiral strategy case, conclusively resolved in (Bianchini et al., 7 Aug 2025), establishes $\bar{\sigma} \approx 2.614$ as the sharp threshold for admissible locally convex barrier construction—a value also corroborated by earlier studies using logarithmic spiral segments (Klein et al., 2014). The result frames both necessary and sufficient conditions for containment and elucidates the precise mathematical structure governing optimal barrier design.

The methodologies and results connect deeply to analogous dynamic containment problems, combinatorial burning processes on graphs, and optimal transport in mixing scenarios, as explored in (Zhou, 4 Mar 2024, Kim et al., 2019), and related work. The underlying principle is the emergence of sharp geometric thresholds determined by global properties of the evolving system and strict optimization over admissible strategies.

These findings are foundational for future research in dynamic control of spreading processes, optimal strategy design under speed or resource constraints, and extension to multi-agent and multi-barrier scenarios. They also highlight the importance of integrating analytic and computational approaches in resolving complex geometric conjectures.