On Max-Stable Processes and the Functional D-Norm

We introduce a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. The distribution function G of a continuous max-stable process on [0,1] is introduced and it is shown that G can be represented via a norm on functional space, called D-norm. This is in complete accordance with the multivariate case and leads to the definition of functional generalized Pareto distributions (GPD) W. These satisfy W=1+log(G) in their upper tails, again in complete accordance with the uni- or multivariate case. Applying this framework to copula processes we derive characterizations of the domain of attraction condition for copula processes in terms of tail equivalence with a functional GPD. \delta-neighborhoods of a functional GPD are introduced and it is shown that these are characterized by a polynomial rate of convergence of functional extremes, which is well-known in the multivariate case.Comment: 22 page

The Linear Constrained Control Problem for Discrete-Time Systems: Regulation on the Boundaries

International audienceThe chapter deals with the problem of regulation of linear systems around an equilibrium lying on the boundary of a polyhedral domain where linear constraints on the control and/or the state vectors are satisfied. In the first part of the chapter, the fundamental limitations for constrained control with active constraints at equilibrium are exposed. Next, based on the invariance properties of polyhe-dral and semi-ellipsoidal sets, design methods for guaranteeing convergence to the equilibrium while respecting linear control constraints are proposed. To this end, Lyapunov-like polyhedral functions, LMI methods and eigenstructure assignment techniques are applied