13 research outputs found
A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks
In this paper we propose a LWR-like model for traffic flow on networks which
allows one to track several groups of drivers, each of them being characterized
only by their destination in the network. The path actually followed to reach
the destination is not assigned a priori, and can be chosen by the drivers
during the journey, taking decisions at junctions.
The model is then used to describe three possible behaviors of drivers,
associated to three different ways to solve the route choice problem: 1.
Drivers ignore the presence of the other vehicles; 2. Drivers react to the
current distribution of traffic, but they do not forecast what will happen at
later times; 3. Drivers take into account the current and future distribution
of vehicles. Notice that, in the latter case, we enter the field of
differential games, and, if a solution exists, it likely represents a global
equilibrium among drivers.
Numerical simulations highlight the differences between the three behaviors
and suggest the existence of multiple Wardrop equilibria
Linear-Quadratic -person and Mean-Field Games with Ergodic Cost
We consider stochastic differential games with players, linear-Gaussian
dynamics in arbitrary state-space dimension, and long-time-average cost with
quadratic running cost. Admissible controls are feedbacks for which the system
is ergodic. We first study the existence of affine Nash equilibria by means of
an associated system of Hamilton-Jacobi-Bellman and
Kolmogorov-Fokker-Planck partial differential equations. We give necessary and
sufficient conditions for the existence and uniqueness of quadratic-Gaussian
solutions in terms of the solvability of suitable algebraic Riccati and
Sylvester equations. Under a symmetry condition on the running costs and for
nearly identical players we study the large population limit, tending to
infinity, and find a unique quadratic-Gaussian solution of the pair of Mean
Field Game HJB-KFP equations. Examples of explicit solutions are given, in
particular for consensus problems.Comment: 31 page
Nearly Optimal Patchy Feedbacks for Minimization Problems with Free Terminal Time
The paper is concerned with a general optimization problem for a nonlinear
control system, in the presence of a running cost and a terminal cost, with
free terminal time. We prove the existence of a patchy feedback whose
trajectories are all nearly optimal solutions, with pre-assigned accuracy.Comment: 13 pages, 3 figures. in v2: Fixed few misprint
Modeling rationality to control self-organization of crowds: An environmental approach
In this paper we propose a classification of crowd models in built
environments based on the assumed pedestrian ability to foresee the movements
of other walkers. At the same time, we introduce a new family of macroscopic
models, which make it possible to tune the degree of predictiveness (i.e.,
rationality) of the individuals. By means of these models we describe both the
natural behavior of pedestrians, i.e., their expected behavior according to
their real limited predictive ability, and a target behavior, i.e., a
particularly efficient behavior one would like them to assume (for, e.g.,
logistic or safety reasons). Then we tackle a challenging shape optimization
problem, which consists in controlling the environment in such a way that the
natural behavior is as close as possible to the target one, thereby inducing
pedestrians to behave more rationally than what they would naturally do. We
present numerical tests which elucidate the role of rational/predictive
abilities and show some promising results about the shape optimization problem
Well-posedness of 2D and 3D swimming models in incompressible fluids governed by Navier--Stokes equations
We introduce and investigate the wellposedness of two models describing the
self-propelled motion of a "small bio-mimetic swimmer" in the 2D and 3D
incompressible fluids modeled by the Navier-Stokes equations. It is assumed
that the swimmer's body consists of finitely many subsequently connected parts,
identified with the fluid they occupy, linked by the rotational and elastic
forces. The swimmer employs the change of its shape, inflicted by respective
explicit internal forces, as the means for self-propulsion in a surrounding
medium. Similar models were previously investigated in [15]-[19] where the
fluid was modeled by the liner nonstationary Stokes equations. Such models are
of interest in biological and engineering applications dealing with the study
and design of propulsion systems in fluids and air.Comment: 28 pages, 5 figures, Corresponding author: Alexandre Khapalov,
Department of Mathematics, Washington State University, USA (email:
[email protected]
Infinite Horizon Noncooperative Differential Games with Non-Smooth Costs
In the present paper, we consider a class of two players infinite horizon
differential games, with piecewise smooth costs exponentially discounted in
time. Through the analysis of the value functions, we study in which cases it
is possible to establish the existence Nash equilibrium solutions in feedback
form. We also provide examples of piecewise linear costs whose corresponding
games have either infinitely many Nash equilibria or no solutions at all.Comment: 17 pages, 5 figure
Infinite Horizon Noncooperative Differential Games
For a non-cooperative differential game, the value functions of the various
players satisfy a system of Hamilton-Jacobi equations. In the present paper, we
consider a class of infinite-horizon games with nonlinear costs exponentially
discounted in time. By the analysis of the value functions, we establish the
existence of Nash equilibrium solutions in feedback form and provide results
and counterexamples on their uniqueness and stability.Comment: 25 pages, 7 figure
Generalized Control Systems in the Space of Probability Measures
In this paper we formulate a time-optimal control problem in the space of probability measures. The main motivation is to face situations in finite-dimensional control systems evolving deterministically where the initial position of the controlled particle is not exactly known, but can be expressed by a probability measure on \u211d^d. We propose for this problem a generalized version of some concepts from classical control theory in finite dimensional systems (namely, target set, dynamic, minimum time function..) and formulate an Hamilton-Jacobi-Bellman equation in the space of probability measures solved by the generalized minimum time function, by extending a concept of approximate viscosity sub/superdifferential in the space of probability measures, originally introduced by Cardaliaguet-Quincampoix in Cardaliaguet and Quincampoix (Int. Game Theor. Rev. 10, 1\u201316, 2008). We prove also some representation results linking the classical concept to the corresponding generalized ones. The main tool used is a superposition principle, proved by Ambrosio, Gigli and Savar\ue9 in Ambrosio et al. [3], which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system