410 research outputs found
Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes
Motivated by pedestrian modelling, we study evolution of measures in the
Wasserstein space. In particular, we consider the Cauchy problem for a
transport equation, where the velocity field depends on the measure itself.
We deal with numerical schemes for this problem and prove convergence of a
Lagrangian scheme to the solution, when the discretization parameters approach
zero. We also prove convergence of an Eulerian scheme, under more strict
hypotheses. Both schemes are discretizations of the push-forward formula
defined by the transport equation. As a by-product, we obtain existence and
uniqueness of the solution.
All the results of convergence are proved with respect to the Wasserstein
distance. We also show that spaces are not natural for such equations,
since we lose uniqueness of the solution
Mean-Field Sparse Optimal Control
We introduce the rigorous limit process connecting finite dimensional sparse
optimal control problems with ODE constraints, modeling parsimonious
interventions on the dynamics of a moving population divided into leaders and
followers, to an infinite dimensional optimal control problem with a constraint
given by a system of ODE for the leaders coupled with a PDE of Vlasov-type,
governing the dynamics of the probability distribution of the followers. In the
classical mean-field theory one studies the behavior of a large number of small
individuals freely interacting with each other, by simplifying the effect of
all the other individuals on any given individual by a single averaged effect.
In this paper we address instead the situation where the leaders are actually
influenced also by an external policy maker, and we propagate its effect for
the number of followers going to infinity. The technical derivation of the
sparse mean-field optimal control is realized by the simultaneous development
of the mean-field limit of the equations governing the followers dynamics
together with the -limit of the finite dimensional sparse optimal
control problems.Comment: arXiv admin note: text overlap with arXiv:1306.591
Control to flocking of the kinetic Cucker-Smale model
The well-known Cucker-Smale model is a macroscopic system reflecting
flocking, i.e. the alignment of velocities in a group of autonomous agents
having mutual interactions. In the present paper, we consider the mean-field
limit of that model, called the kinetic Cucker-Smale model, which is a
transport partial differential equation involving nonlocal terms. It is known
that flocking is reached asymptotically whenever the initial conditions of the
group of agents are in a favorable configuration. For other initial
configurations, it is natural to investigate whether flocking can be enforced
by means of an appropriate external force, applied to an adequate time-varying
subdomain.
In this paper we prove that we can drive to flocking any group of agents
governed by the kinetic Cucker-Smale model, by means of a sparse centralized
control strategy, and this, for any initial configuration of the crowd. Here,
"sparse control" means that the action at each time is limited over an
arbitrary proportion of the crowd, or, as a variant, of the space of
configurations; "centralized" means that the strategy is computed by an
external agent knowing the configuration of all agents. We stress that we do
not only design a control function (in a sampled feedback form), but also a
time-varying control domain on which the action is applied. The sparsity
constraint reflects the fact that one cannot act on the whole crowd at every
instant of time.
Our approach is based on geometric considerations on the velocity field of
the kinetic Cucker-Smale PDE, and in particular on the analysis of the particle
flow generated by this vector field. The control domain and the control
functions are designed to satisfy appropriate constraints, and such that, for
any initial configuration, the velocity part of the support of the measure
solution asymptotically shrinks to a singleton, which means flocking
Partially Ordered Sets and Their Invariants
We investigate how much information cardinal invariants can give on the
structure of the ordered set on which they are de�ned. We consider the basic
de�nitions of an ordered set and see how they are related to one another.
We generalize some results on cardinal invariants for ordered sets and state
some useful characterizations. We investigate how cardinal invariants can
in
uence the existence of some special suborderings. We generalize some
results on the Dilworth and Sierpinski theorems and explore the conjecture
of Miller and Sauer. We address some open problems on dominating numbers.
We investigate Model Games to �nd some characterizations on the
cardinality of a set.
Control of reaction-diffusion equations on time-evolving manifolds
Among the main actors of organism development there are morphogens, which are
signaling molecules diffusing in the developing organism and acting on cells to
produce local responses. Growth is thus determined by the distribution of such
signal. Meanwhile, the diffusion of the signal is itself affected by the
changes in shape and size of the organism. In other words, there is a complete
coupling between the diffusion of the signal and the change of the shapes. In
this paper, we introduce a mathematical model to investigate such coupling. The
shape is given by a manifold, that varies in time as the result of a
deformation given by a transport equation. The signal is represented by a
density, diffusing on the manifold via a diffusion equation. We show the
non-commutativity of the transport and diffusion evolution by introducing a new
concept of Lie bracket between the diffusion and the transport operator. We
also provide numerical simulations showing this phenomenon
Mean-field sparse Jurdjevic-Quinn control
International audienceWe consider nonlinear transport equations with non-local velocity, describing the time-evolution of a measure, which in practice may represent the density of a crowd. Such equations often appear by taking the mean-field limit of finite-dimensional systems modelling collective dynamics. We first give a sense to dissipativity of these mean-field equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, we address the problem of controlling such equations by means of a time-varying bounded control action localized on a time-varying control subset with bounded Lebesgue measure (sparsity space constraint). Finite-dimensional versions are given by control-affine systems, which can be stabilized by the well known Jurdjevic–Quinn procedure. In this paper, assuming that the uncontrolled dynamics are dissipative, we develop an approach in the spirit of the classical Jurdjevic–Quinn theorem, showing how to steer the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function, and enjoy sparsity properties in the sense that the control support is small. Finally, we show that our result applies to a large class of kinetic equations modelling multi-agent dynamics
Generalized solutions to bounded-confidence models
Bounded-confidence models in social dynamics describe multi-agent systems,
where each individual interacts only locally with others. Several models are
written as systems of ordinary differential equations with discontinuous
right-hand side: this is a direct consequence of restricting interactions to a
bounded region with non-vanishing strength at the boundary. Various works in
the literature analyzed properties of solutions, such as barycenter invariance
and clustering. On the other side, the problem of giving a precise definition
of solution, from an analytical point of view, was often overlooked. However, a
rich literature proposing different concepts of solution to discontinuous
differential equations is available. Using several concepts of solution, we
show how existence is granted under general assumptions, while uniqueness may
fail even in dimension one, but holds for almost every initial conditions.
Consequently, various properties of solutions depend on the used definition and
initial conditions
Generalized Wasserstein distance and its application to transport equations with source
In this article, we generalize the Wasserstein distance to measures with
different masses. We study the properties of such distance. In particular, we
show that it metrizes weak convergence for tight sequences.
We use this generalized Wasserstein distance to study a transport equation
with source, in which both the vector field and the source depend on the
measure itself. We prove existence and uniqueness of the solution to the Cauchy
problem when the vector field and the source are Lipschitzian with respect to
the generalized Wasserstein distance
Unmaking capitalism through community empowerment: Findings from Italian agricultural experiences
Capitalism – in the theorizations of sustainability transformation – has been largely taken for granted for its misleadingly assumed stability and homogeneity, thus limiting the scope for defining alternative futures, policy options and strategies for transformative change. Theorizations regarding sustainable transformative pathways have often overshadowed a nuanced landscape of normative and ontological pluralism thus contributing to generating techno-centric and top-down responses to issues such as access to food, farmers' control over the food-chain and global environmental change. The expansion of capital, under a mechanism of production-reproduction, with a constant attempt to subsume different forms of production into the global market, generates manifold temporal frictions that, on the one hand, contribute to the consolidation of the capitalist model and, on the other hand, give rise to conflicting elements and re-orientation of modernity in a process of “unmaking” of capitalism. This article, drawing upon empirical work conducted in Northern Italy, presents two experiences emerging from the scenario of local food networks, namely the “C’è Campo” Participatory Guarantee System and the “Ortazzo” Community Supported Agriculture project. These show elements and mechanisms of local community empowerment for unmaking capitalism from the inside, as steps for a sustainable and bottom-up transformation which do not necessarily imply the generation of socio-economic novelties ex-nihilo. The “conventionalization” of organic agriculture has pushed those actors who participate in local food networks to reconfigure their self-regulation towards a “bottom-up” approach driven by the adoption of PGS or CSA instruments as an attempt to secure or reacquire control over the market and the construction of quality. Convivial tools, in particular, are crucial for understanding - and finding responses to - the social, economic, cultural and environmental crisis that contemporary society is now facing
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