arXiv subjects

Alberto Bressan

Publications and source records attributed to Alberto Bressan.

At least 37 records · Page 2Linked to original sources

arXiv2023

Optimal Solutions for a Class of Set-Valued Evolution Problems

The paper is concerned with a class of optimization problems for moving sets $t\mapstoΩ(t)\subset\mathbb{R}^2$, motivated by the control of invasive biological populations. Assuming that the initial contaminated set $Ω_0$ is convex, we prove that a strategy is optimal if an only if at each given time $t\in [0,T]$ the control is active along the portion of the boundary $\partial Ω(t)$ where the curvature is maximal. In particular, this implies that $Ω(t)$ is convex for all $t\geq 0$. The proof relies on the analysis of a one-step constrained optimization problem, obtained by a time discretization.

math.OC↗

arXiv2023

A remark on the uniqueness of solutions to hyperbolic conservation laws

Given a strictly hyperbolic $n\times n$ system of conservation laws, it is well known that there exists a unique Lipschitz semigroup of weak solutions, defined on a domain of functions with small total variation, which are limits of vanishing viscosity approximations. Aim of this note is to prove that every weak solution taking values in the domain of the semigroup, and whose shocks satisfy the Liu admissibility conditions, actually coincides with a semigroup trajectory.

math.AP↗

arXiv2023

Unique Solutions to Hyperbolic Conservation Laws with a Strictly Convex Entropy

Consider a strictly hyperbolic $n\times n$ system of conservation laws, where each characteristic field is either genuinely nonlinear or linearly degenerate. In this standard setting, it is well known that there exists a Lipschitz semigroup of weak solutions, defined on a domain of functions with small total variation. If the system admits a strictly convex entropy, we give a short proof that every entropy weak solution taking values within the domain of the semigroup coincides with a semigroup trajectory. The result shows that the assumptions of ``Tame Variation" or ``Tame Oscillation", previously used to achieve uniqueness, can be removed in the presence of a strictly convex entropy.

math.AP↗

arXiv2023

Controlled Traveling Profiles for Models of Invasive Biological Species

We consider a family of controlled reaction-diffusion equations, describing the spatial spreading of an invasive biological species. For a given propagation speed $c\in{I\!\!R}$, we seek a control with minimum cost, which achieves a traveling profile with speed $c$. For various nonlinear models, the existence of a (possibly measure valued) optimal control is proved, together with necessary conditions for optimality. In the last section we study a case where the wave speed cannot be modified by any control with finite cost. The present analysis is motivated by the recent results in arXiv:2201.01723 and arXiv:2108.09321, showing how a control problem for a reaction-diffusion equation can be approximated by a simpler problem of optimal control of a moving set.

math.OC↗

arXiv2022

Generic Properties of First Order Mean Field Games

We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of view of generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for nearly all mean field games(in the Baire category sense) the best reply map is single valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.

math.OC↗

arXiv2022

Optimal Shapes for Tree Roots

The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $μ$ describing the distribution of root hair cells, we seek to maximize a harvest functional $\mathcal{H}$, computing the total amount of water and nutrients gathered by the roots, subject to a cost for transporting these nutrients from the roots to the trunk. Earlier papers had established the existence of an optimal measure, and a priori bounds. Here we derive necessary conditions for optimality. Moreover, in space dimension $d=2$, we prove that the support of an optimal measure is nowhere dense.

math.OC↗

arXiv2022

Shock interactions for the Burgers-Hilbert Equation

This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with $H^2$ regularity away from the shocks plus a corrector term having an asymptotic behavior like |x|ln|x| close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.

math.AP↗

arXiv2022

Optimal Control of Moving Sets

Motivated by the control of invasive biological populations, we consider a class of optimization problems for moving sets $t\mapsto Ω(t)\subset\mathbb{R}^2$. Given an initial set $Ω_0$, 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 along the boundary $\partial Ω(t)$. We prove the existence of optimal solutions, within a class of sets with finite perimeter. Necessary conditions for optimality are then derived, in the form of a Pontryagin maximum principle. Additional optimality conditions show that the sets $Ω(t)$ cannot have certain types of outward or inward corners. Finally, some explicit solutions are presented.

math.OC↗

arXiv2021

Numerical Study of Non-uniqueness for 2D Compressible Isentropic Euler Equations

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.

math.AP↗

arXiv2021

On the Optimal Control of Propagation Fronts

We consider a controlled reaction-diffusion equation, motivated by a pest eradication problem. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. In this direction, the first part of the paper studies the optimal control of 1-dimensional traveling wave profiles. Using Stokes' formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. In the last section we introduce a family of optimization problems for a moving set. We show how these can be derived from the original parabolic problems, by taking a sharp interface limit.

math.OC↗

arXiv2021

Diffusion Approximations of Markovian Solutions to Discontinuous ODEs

In a companion paper, the authors have characterized all deterministic semigroups, and all Markov semigroups, whose trajectories are Carathe'odory solutions to a given ODE x'=f(x), with f possibly discontinuous. The present paper establishes two approximation results. Namely, every deterministic semigroup can be obtained as the pointwise limit of the flows generated by a sequence of ODEs $x'=f_n(x) with smooth right hand sides. Moreover, every Markov semigroup can be obtained as limit of a sequence of diffusion processes with smooth drifts and with diffusion coefficients approaching zero.

math.CA↗

arXiv2020

On the Regularity of Optimal Dynamic Blocking Strategies

The paper studies a dynamic blocking problem, motivated by a model of optimal fire confinement. While the fire can expand with unit speed in all directions, barriers are constructed in real time. An optimal strategy is sought, minimizing the total value of the burned region, plus a construction cost. It is well known that optimal barriers exists. In general, they are a countable union of compact, connected, rectifiable sets. The main result of the present paper shows that optimal barriers are nowhere dense. The proof relies on new estimates on the reachable sets and on optimal trajectories for the fire, solving a minimum time problem in the presence of obstacles.

math.AP↗

arXiv2020

Entropy Admissibility of the Limit Solution for a Nonlocal Model of Traffic Flow

We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $ρ$ ahead. The averaging kernel is of exponential type: $w_\varepsilon(s)=\varepsilon^{-1} e^{-s/\varepsilon}$. For any decreasing velocity function $v$, we prove that, as $\varepsilon\to 0$, the limit of solutions to the nonlocal equation coincides with the unique entropy-admissible solution to the scalar conservation law $ρ_t + (ρv(ρ))_x=0$.

math.AP↗

arXiv2020

A posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws

The paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for $n\times n$ hyperbolic conservation laws in one space dimension. These estimates are achieved by a "post-processing algorithm", checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained by the Godunov or the Lax-Friedrichs scheme, backward Euler approximations, and the method of periodic smoothing. Some numerical implementations are presented.

math.NA↗

arXiv2020

Markovian Solutions to Discontinuous ODEs

Given a possibly discontinuous, bounded function $f:\mathbb{R}\mapsto\mathbb{R}$, we consider the set of generalized flows, obtained by assigning a probability measure on the set of Carathéodory solutions to the ODE ~$\dot x = f(x)$. The paper provides a complete characterization of all such flows which have a Markov property in time. This is achieved in terms of (i) a positive, atomless measure supported on the set $f^{-1}(0)$ where $f$ vanishes, (ii) a countable number of Poisson random variables, determining the waiting times at points in $f^{-1}(0)$, and (iii) a countable set of numbers $θ_k\in [0,1]$, describing the probability of moving up or down, at isolated points where two distinct trajectories can originate.

math.CA↗

arXiv2020

On a Shape Optimization Problem for Tree Branches

This paper is concerned with a shape optimization problem, where the functional to be maximized describes the total sunlight collected by a distribution of tree leaves, minus the cost for transporting water and nutrient from the base of the trunk to all the leaves. In the case of 2 space dimensions, the solution is proved to be unique, and explicitly determined.

math.OC↗

arXiv2020

Soil searching by an artificial root

We model an artificial root which grows in the soil for underground prospecting. Its evolution is described by a controlled system of two integro-partial differential equations: one for the growth of the body and the other for the elongation of the tip. At any given time, the angular velocity of the root is obtained by solving a minimization problem with state constraints. We prove the existence of solutions to the evolution problem, up to the first time where a "breakdown configuration" is reached. Some numerical simulations are performed, to test the effectiveness of our feedback control algorithm.

math.AP↗

arXiv2020

A Posteriori Error Estimates for Self-Similar Solutions to the Euler Equations

The main goal of this paper is to analyze a family of "simplest possible" initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.

math.AP↗