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1

On the one-dimensional mean-field games with congestion

Sedjro Marc

Mathematica Applicanda

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2022

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Vol. 50, no. 1

139--151

EN

In this work, we consider a one dimensional forward-forward model of Mean-field Games with congestion. We establish a connection between such models and conservation laws. Next, we show the existence of a non trivial convex entropies. Finally, we investigate the existence of solutions in the parabolic case and derived some estimates thanks to the existence of such convex entropies.

PL

Praca poświęcona jest jednowymiarowym modelom gier z uśrednionym oddziaływaniem w zarządzaniu. Takie badania mają na celu analizę podejmowania strategicznych decyzji przez czynniki mało oddziaływające w bardzo dużych populacjach. Ustalany jest związek między takimi modelami a prawami zachowania. W wyniku tych badań pokazano istnienie nietrywialnych entropii wypukłych. W końcowej części badane jest istnienie rozwiązań w przypadku parabolicznym i wyprowadzane są pewne oszacowania z istnienia takich wypukłych entropii.

2

On the numerical approximation of first-order Hamilton-Jacobi equations

Abgrall R., Perrier V.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 3

403-412

EN

Some methods for the numerical approximation of time-dependent and steady first-order Hamilton-Jacobi equations are reviewed. Most of the discussion focuses on conformal triangular-type meshes, but we show how to extend this to the most general meshes. We review some first-order monotone schemes and also high-order ones specially dedicated to steady problems.

3

A Method for Constructing ε-value Functions for The Bolza Problem of Optimal Control

Pustelnik J.

International Journal of Applied Mathematics and Computer Science

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2005

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Vol. 15, no 2

177-186

EN

The problem considered is that of approximate minimisation of the Bolza problem of optimal control. Starting from Bellman's method of dynamic programming, we define the ε-value function to be an approximation to the value function being a solution to the Hamilton-Jacobi equation. The paper shows an approach that can be used to construct an algorithm for calculating the values of an ε-value function at given points, thus approximating the respective values of the value function.

4

Optimal synthesis via superdifferentials of value function

Frankowska H.

Control and Cybernetics

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2005

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Vol. 34, no 3

787-803

EN

We derive a differential inclusion governing the evolution of optimal trajectories to the Mayor problem. The value function is allowed to be discontinuous. This inclusion has convex compact right-hand sides.

5

An Algorithm for Construction of varepsilon-Value Functions for the Bolza Control Problem

Jacewicz E.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 2

391-428

EN

The problem considered is that of approximate numerical minimisation of the non-linear control problem of Bolza. Starting from the classical dynamic programming method of Bellman, an varepsilon-value function is defined as an approximation for the value function being a solution to the Hamilton-Jacobi equation. The paper shows how an varepsilon-value function which maintains suitable properties analogous to the original Hamilton-Jacobi value function can be constructed using a stable numerical algorithm. The paper shows the numerical closeness of the approximate minimum to the infimum of the Bolza functional.