Convexity and duality; Proximal concepts; Nonsmooth calculus; Favorable classes of nonsmooth objects; Derivation of necessary conditions in nonlinear programming

Optimal control problems; Dierential inclusions; Filippov's lemma; Relaxation; Pontryagin maximum principle and nonsmooth variants; Value functions and Hamilton-Jacobi equations; Flow invariance.

Bressan and Hong recently introduced an optimal control model (called stratified systems) whose dynamics have structured discontinuities in the state variable. We shall discuss this model in detail and add more assumptions to modify their attempt to characterize the value function as a generalized solution to a Hamilton-Jacobi equation.

In this presentation electricity markets are considered including a transmission network, producers generating electricity and an agent doing coordination (Independent System Operator, ISO). Production is organized by means of an auction. Once producers simultaneously bid cost functions, the ISO decides the quantity each generator produces and the flows through the network lines. Producers play strategically with the ISO. When bidding, each firm tries to obtain revenues as high as possible. We prove existence of equilibrium for this discontinuous game and then by using optimal mechanism design, we derive an optimal regulation mechanism for pricing and compare its performance with the bayesian version of the usual price equal to Lagrange multiplier. Finally, we develop the sensitivity analysis with respect to probabilities distributions involved.

L'exposé sera divisé en deux parties. Dans une premier temps, on présentera une extension de la méthode des bases réduites qui a été introduite pour résoudre des EDPs paramétrées à des problèmes de calcul de moyennes par Monte Carlo. Dans un deuxième temps, on s'intéressera à un algorithme qui a été récemment proposé pour résoudre des EDPs en grande dimension, et qui s'applique en particulier à des problèmes de propagation des incertitudes. Des résultats de convergence de cette méthode seront présentés.

We will give a short introduction into sparse grids how they are obtained from an univariate multi scale basis by a sparse tensor product construction and how they allow to overcome the curse of dimensionality to a certain extent. Furthermore, the sparse grid combination technique will be presented, where an approximate solution on a sparse grid for a particular problem is constructed using a linear combination of approximations on a certain sequance of small anisotropic partial grids. In the main part of the talk we consider an adaptive sparse grid approach for the numerical solution of Hamilton-Jacobi-Bellman equations in higher dimensions. We concentrate on equations of the form that describe problems in continuous time and state space with finite time horizon. We use spatial adaptive sparse grids for the discretization of the continuous state spaces and apply a Semi-Lagrangian Scheme based on the dynamic programming approach in order to approximate the HJB-solution. We show numerical results where sparse grids enable us to treat problems in up to 6 dimensions. This is joint work with O. Bokanowski, M. Griebel, and I. Klompmaker.

On caracterise les propriétés de convexité du domaine d'injectivité sur un ellipsoïde de révolution oblat, en lien avec des résultats de régularité en transport optimal de Figalli et al.