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On the well posedness of vanishing viscosity limits

Alberto Bressan — 2002

Journées équations aux dérivées partielles

This paper provides a survey of recent results concerning the stability and convergence of viscous approximations, for a strictly hyperbolic system of conservation laws in one space dimension. In the case of initial data with small total variation, the vanishing viscosity limit is well defined. It yields the unique entropy weak solution to the corresponding hyperbolic system.

Hyperbolic systems of conservation laws.

Alberto Bressan — 1999

Revista Matemática Complutense

This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions....

Sistemi iperbolici di leggi di conservazione

Alberto Bressan — 2000

Bollettino dell'Unione Matematica Italiana

This survey paper provides a brief introduction to the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After reviewing some basic concepts, we describe the fundamental theorem of Glimm on the global existence of BV solutions. We then outline the more recent results on uniqueness and stability of entropy weak solutions. Finally, some major open problems and research directions are discussed in the last section.

Some remarks on multidimensional systems of conservation laws

Alberto Bressan — 2004

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

This note is concerned with the Cauchy problem for hyperbolic systems of conservation laws in several space dimensions. We first discuss an example of ill-posedness, for a special system having a radial symmetry property. Some conjectures are formulated, on the compactness of the set of flow maps generated by vector fields with bounded variation.

Selections and representations of multifunctions in paracompact spaces

Alberto BressanGiovanni Colombo — 1992

Studia Mathematica

Let (X,T) be a paracompact space, Y a complete metric space, F : X 2 Y a lower semicontinuous multifunction with nonempty closed values. We prove that if T + is a (stronger than T) topology on X satisfying a compatibility property, then F admits a T + -continuous selection. If Y is separable, then there exists a sequence ( f n ) of T + -continuous selections such that F ( x ) = f n ( x ) ; n 1 ¯ for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.

Nash equilibria for a model of traffic flow with several groups of drivers

Alberto BressanKe Han — 2012

ESAIM: Control, Optimisation and Calculus of Variations

Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are groups of drivers, The -th group consists of drivers, sharing the same departure and arrival costs (), (). For any given population sizes ,, , we prove the existence of a Nash equilibrium solution, where no...

Stability rates for patchy vector fields

Fabio AnconaAlberto Bressan — 2004

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the stability of the set of trajectories of a patchy vector field, in the presence of impulsive perturbations. Patchy vector fields are discontinuous, piecewise smooth vector fields that were introduced in Ancona and Bressan (1999) to study feedback stabilization problems. For patchy vector fields in the plane, with polygonal patches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude...

Global optimality conditions for a dynamic blocking problem

Alberto BressanTao Wang — 2012

ESAIM: Control, Optimisation and Calculus of Variations

The paper is concerned with a class of optimal blocking problems in the plane. We consider a time dependent set () ⊂ ℝ, described as the reachable set for a differential inclusion. To restrict its growth, a barrier can be constructed, in real time. This is a one-dimensional rectifiable set which blocks the trajectories of the differential inclusion. In this paper we introduce a definition of “regular strategy”, based on a careful classification of blocking arcs. Moreover, we derive local and global...

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