Pierre Cardaliaguet

Original Contribution: Approximation of a function and its derivative with a neural network

This paper deals with the approximation of both a function and its derivative by feedforward neural networks. We propose an explicit formula of approximation which is noise resistant and can be easily modified with the patterns. We apply these results to approach a function defined implicitly, which is useful in control theory.

Set-Valued Numerical Analysis for Optimal Control and Differential Games

This chapter deals with theoretical and numerical results for solving qualitative and quantitative control and differential game problems. These questions are treated in the framework of set-valued analysis and viability theory. In a way, this approach is rather well adapted to look at these several problems with a unified point of view. The idea is to characterize the value function as a viability kernel instead of solving a Hamilton—Jacobi—Bellmann equation. This allows us to easily take into account state constraints without any controllability assumptions on the dynamic, neither at the boundary of targets, nor at the boundary of the constraint set. In the case of two-player differential games, the value function is characterized as a discriminating kernel. This allows dealing with a large class of systems with minimal regularity and convexity assumptions. Rigorous proofs of the convergence, including irregular cases, and completely explicit algorithms are provided.

Optimal times for constrained nonlinear control problems without local controllability

We study optimal times to reach a given closed target for controlled systems with a state constraint. Our goal is to characterize these optimal time functions in such a way that it is possible to compute them numerically and we do not need to compute trajectories of the controlled system. In this paper we provide new results using viability theory. This allows us to study optimal time functions free from the controllability assumptions classically made in the partial differential equations approach.

Long time average of mean field games

We consider a model of mean field games system defined on a time interval

Nash Equilibrium Payoffs for Nonzero-Sum Stochastic Differential Games

[0,T]

Existence and uniqueness for dislocation dynamics with nonnegative velocity

and investigate its asymptotic behavior as the horizon

Pursuit Differential Games with State Constraints

T

Weak Solutions for First Order Mean Field Games with Local Coupling

tends to infinity. We show that the system, rescaled in a suitable way, converges to a stationary ergodic mean field game. The convergence holds with exponential rate and relies on energy estimates and the Hamiltonian structure of the system.

DETERMINISTIC DIFFERENTIAL GAMES UNDER PROBABILITY KNOWLEDGE OF INITIAL CONDITION

Existence and characterization of Nash equilibrium payoffs are proved for stochastic nonzero-sum differential games.

A Representation Formula for the Mean Curvature Motion

We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and nonlocal eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows us to prove that the energy associated to the dislocation dynamics is nonincreasing.