arXiv subjects

Alberto Bressan

Publications and source records attributed to Alberto Bressan.

At least 19 recordsLinked to original sources

arXiv2026

On the Minimum Speed for a Fire Blocking Problem

We consider a dynamic blocking problem: a fire, starting on a bounded open set $R_0\subset\mathbb R^2$, propagates in the plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed $λ>0$. A long standing conjecture by the first author states that the fire can be entirely surrounded by the barrier, in finite time, if and only if $λ>2$. The present paper improves on earlier lower bounds. Namely, if $λ\leq Λ=1.62621170228\ldots $ we show that the fire cannot be blocked. We also prove that, if $λ\leq 2$, the fire cannot be blocked by constructing a barrier which is a simple closed curve. More precisely, if the barrier does not contain purely delaying arcs, i.e.~arcs which are eventually surrounded by the fire on both sides, then for $λ<2$ the region burned by the fire at time $t>0$ has area $A(t)>(2-λ)t^2/4$.

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arXiv2026

On the Nash equilibrium for a model of traffic flow

The paper consider a conservation law model of traffic flow on a highway, where each driver chooses an optimal departure time $τ^d$, based on a departure cost $φ(τ^d)$ and an arrival cost $ψ(τ^a)$. Nash equilibria are studied, where no driver can reduce his own cost by changing his departure time. The first results provide continuity estimates on the arrival time, depending on the distribution of departures. In terms of the ${\bf L}^1$ distance, we show that the corresponding map is Hölder continuous with exponent $1/2$ in Lagrangian variables (labelling individual drivers), and Lipschitz continuous in Eulerian space variables. Concerning the regularity of the a Nash equilibrium, our results show that, when the total number of drivers is sufficiently large, a queue alway forms at the entrance of the highway and the corresponding traffic density contains a shock. Moreover, for any number of drivers (possibly small), we prove that for generic $\mathcal C^2$ cost functions $φ, ψ$ the Nash equilibrium solution still contains a shock. In the last section of the paper, assuming that the initial departure rate is not a Nash equilibrium, we consider an evolution equation for the departure rate, where each driver keeps changing his daily departure time in order to reduce his own cost. It is proved that this nonlocal integro-differential evolution equation admits a unique solution, globally in time. This provides some theoretical support to earlier numerical simulations, which indicated the dynamic instability of the Nash equilibrium and the possible existence of a chaotic attractor.

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arXiv2026

On the Value Function for a Class of Set Motion Problems

The paper considers a class of optimal set motion problems, where the control determines the velocity of boundary points in the normal direction. Results and counterexamples are given, on the continuous dependence of the minimum cost from the initial configuration. The last sections of the paper study the slicing cost of bounded open sets. For strongly Lipschitz domains, it is proved that the slicing cost varies continuously w.r.t. the Hausdorff distance of the boundaries. We conjecture that the slicing cost for a disc is maximal among all sets with the same area.

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arXiv2026

Optimal Slicing Problems

Given a bounded open set $V\subset \mathbb{R}^2$, we consider the problem of slicing this set by a family of curves, whose average length is as small as possible. The first sections of the paper prove the existence of an optimal slicing and establish a connection with the problem of eradicating an invasive biological species from an island, considered in \cite{BMS}. Namely, an optimal slicing can be obtained as a limit of minimum time eradication strategies. The second part of the paper derives various necessary conditions for optimality, describing the behavior of an optimal slicing strategy in the interior of $V$ and near the boundary $\partial V$.

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arXiv2026

A Mollification Approach to Ramified Transport and Tree Shape Optimization

The paper analyzes a mollification algorithm, for the numerical computation of optimal irrigation patterns. This provides a regularization of the standard irrigation cost functional, in a Lagrangian framework. Lower semicontinuity and Gamma-convergence results are proved. The technique is then applied to some numerical optimization problems, related to the optimal shape of tree roots and branches.

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arXiv2026

A Two-Fluxes Stochastic Model of Traffic Waves

The paper introduces a stochastic model for the spontaneous formation of traffic waves on a highway. This is formulated in terms of a conservation law with discontinuous, gradient-dependent flux. In an unstable regime, the non-uniqueness of solutions allows for the emergence of $N$-shaped spikes in the traffic density, at random points and times. Bounds are proved on the expected value of the total variation of the random solution and on the expected number of shocks. Further bounds are obtained on the average velocity and on the expected average acceleration of cars, along a given stretch of highway. Finally, it is proved that the Markov process, whose paths are random solutions to the conservation law, admits a unique stationary probability distribution and is ergodic.

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arXiv2026

Local Asymptotic Patterns for Viscous Approximations of Conservation Laws

Solutions to hyperbolic conservation laws can be approximated in many different ways: by vanishing viscosity, relaxations, discrete or semi-discrete numerical schemes, approximation with a nonlocal flux, etc$\ldots$ For some of these methods, general ${\bf L}^1$ convergence results are available. Aim of this paper is to understand the local behavior of these approximations, in a neighborhood of point where the hyperbolic solution has a singularity. Specifically: a point along a shock, or where two shocks interact, or where a new shock is formed. Given a sequence of $ε$-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as $ε\to 0$ a well defined limit is obtained. This corresponds to a specific ``eternal solution" (globally defined both in space and in time) to the approximating equation. Precise results this direction are here given, in the case of vanishing viscosity.

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arXiv2026

A Uniqueness Condition for Conservation Laws with Discontinuous Gradient-Dependent Flux

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. In the stable case where $f(u)<g(u)$ for all $u\in R$, it was proved in [1] that the limits of vanishing viscosity approximations form a contractive semigroup w.r.t. the $L^1$ distance. Further, they coincide with the limits of a suitable family of front tracking approximations. In the present paper we introduce a simple condition that guarantees that every weak, entropy admissible solution of a Cauchy problem coincides with the corresponding semigroup trajectory, and hence is unique.

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arXiv2026

Conservation Laws with Discontinuous Gradient-Dependent Flux: the Unstable Case

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the unstable case where $f(u)>g(u)$ for all $u\in {\mathbb R}$. Assuming that both $f$ and $g$ are strictly convex, solutions to the Riemann problem are constructed. Even for a smooth initial data, examples show that the Cauchy problem can have infinitely many solutions. For an initial data which is piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a solution can be constructed globally in time. It is proved that such solution is unique under the additional requirement that the number of interfaces, where the flux switches between $f$ and $g$, remains as small as possible.

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arXiv2025

Uniqueness Domains for ${\bf L}^\infty$ Solutions of $2 \times 2$ Hyperbolic Conservation Laws

For a genuinely nonlinear $2\times 2$ hyperbolic system of conservation laws, assuming that the initial data have small ${\bf L}^\infty$ norm but possibly unbounded total variation, the existence of global solutions was proved in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like $t^{-1}$. Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with faster decay rate: $\mathrm{Tot. Var. }\bigl\{u(t,\cdot)\bigr\}\leq C t^{α-1}$. For these solutions, a uniqueness theorem is proved. Indeed, as the initial data range over a domain of functions with $\|\bar u\|_{{\bf L}^\infty} \leq\varepsilon_1$ small enough, solutions with fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to $1$ by further shrinking the value of $\varepsilon_1>0$. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.

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arXiv2025

Optimally Controlled Moving Sets with Geographical Constraints

The paper is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a region $V\subset \R^2$ bounded by geographical barriers. If no control is applied, the contaminated set $Ω(t)\subset V$ expands with unit speed in all directions. By implementing a control, a region of area $M$ can be cleared up per unit time. Given an initial set $Ω(0)=Ω_0\subseteq V$, three main problems are studied: (1) Existence of an admissible strategy $t\mapstoΩ(t)$ which eradicates the contamination in finite time, so that $Ω(T)=\emptyset$ for some $T>0$. (2) Optimal strategies that achieve eradication in minimum time. (3) Strategies that minimize the average area of the contaminated set on a given time interval $[0,T]$. For these optimization problems, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions $t\mapsto Ω(t)$ are explicitly constructed in a number of cases. \end{abstract}

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arXiv2025

Results and open questions on the boundary control of moving sets

These notes provide a survey of recent results and open problems on the boundary control of moving sets. Motivated by the control of an invasive biological species, we consider a class of optimization problems for a moving set $t\mapsto Ω(t)$, where the goal is to minimize the area of the contaminated set $Ω(t)$ over time, plus a cost related to the control effort. Here the control function is the inward normal speed, assigned along the boundary $\partial Ω(t)$. We also consider problems with geographical constraints, where the invasive population is restricted within an island. Existence and structure of eradication strategies, which entirely remove the invasive population in minimum time, is also discussed.

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arXiv2025

Generic uniqueness and conjugate points for optimal control problems

The paper is concerned with an optimal control problem on $\mathbb{R}^n$, where the dynamics is linear w.r.t.~the control functions. For a terminal cost $ψ$ in a $mathcal{G}_δ$ set of $\mathcal{C}^4(\mathbb{R}^n)$ (i.e., in a countable intersection of open dense subsets), two main results are proved.Namely: the set $Γ_ψ\subset\mathbb{R}^n$ of conjugate points is closed, with locally bounded $(n-2)$-dimensional Hausdorff measure. Moreover, the set of initial points $y\in \mathbb{R}^n\setminusΓ_ψ$, which admit two or more globally optimal trajectories, is contained in the union of a locally finite family of embedded manifolds. In particular, the value function is continuously differentiable on an open, dense subset of $\mathbb{R}^n$.

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arXiv2024

Conservation Laws with Discontinuous Gradient-Dependent Flux: the Stable Case

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the stable case where $f(u)<g(u)$ for all $u\in {\mathbb R}$, with $f,g$ smooth but possibly not convex. A front tracking algorithm is introduced, proving that piecewise constant approximations converge to the trajectories of a contractive semigroup on $\mathbf{L}^1({\mathbb R})$. In the spatially periodic case, we prove that semigroup trajectories coincide with the unique limits of a suitable class of vanishing viscosity approximations.

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arXiv2024

Generic Solutions to Controlled Balance Laws

The paper is concerned with a scalar balance law, where the source term depends on a control function $α(t)$. Given a control $α\in \mathbf{L}^\infty\bigl([0,T]\bigr)$, it is proved that, for generic initial data $\bar u \in \mathcal{C}^3(\mathbb{R})$, the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time $T$ no shock interaction occurs, and no new shock is formed. In addition, a family of optimal control problems is considered, including a running cost and a terminal cost. An example is constructed where the optimal solution contains two shocks merging exactly at the terminal time $T$. Such behavior persists under any suitably small perturbation of the flux, source, and cost functions, and of the initial data. This shows that generic solutions of optimization problems have different qualitative properties, compared with generic solutions to Cauchy problems.

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arXiv2024

The Initial Stages of a Generic Singularity for a 2D Pressureless Gas

We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time $t_0$, where both the the density and the velocity gradient become unbounded. Aim of this paper is to provide an asymptotic description of the solution beyond the time of singularity formation. For $t>t_0$ we show that a singular curve is formed, where the mass has positive density w.r.t.1-dimensional Hausdorff measure. The system of equations describing the behavior of the singular curve is not hyperbolic. Working within a class of analytic data, local solutions can be constructed using a version of the Cauchy-Kovalevskaya theorem. For this purpose, by a suitable change of variables we rewrite the evolution equations as a first order system of Briot-Bouquet type, to which a general existence-uniqueness theorem can then be applied.

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arXiv2024

Intermediate Domains for Scalar Conservation Laws

For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data $\overline{u}\in \bf{L}^\infty(\mathbb R)$ decays like $t^{-1}$. This paper introduces a class of intermediate domains $\mathcal P_α$, $0<α<1$, such that for $\overline u\in \mathcal P_α$ a faster decay rate is achieved: $\mathrm{Tot.Var.}\bigl\{ u(t,\cdot)\bigr\}\sim t^{α-1}$. A key ingredient of the analysis is a ``Fourier-type" decomposition of $\overline u$ into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting.

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arXiv2024

Generic Properties of Conjugate Points in Optimal Control Problems

The first part of the paper studies a class of optimal control problems in Bolza form, where the dynamics is linear w.r.t.~the control function. A necessary condition is derived, for the optimality of a trajectory which starts at a conjugate point. The second part is concerned with a classical problem in the Calculus of Variations, with free terminal point. For a generic terminal cost $ψ\in \C^4(\mathbb{R}^n)$, applying the previous necessary condition we show that the set of conjugate points is contained in the image of an $(n-2)$-dimensional manifold, and has locally bounded $(n-2)$-dimensional Hausdorff measure.

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