On Markov Chains with Uncertain Data

In this paper, a general method is described to determine uncertainty intervals for performance measures of Markov chains given an uncertainty region for the parameters of the Markov chains. We investigate the effects of uncertainties in the transition probabilities on the limiting distributions, on the state probabilities after n steps, on mean sojourn times in transient states, and on absorption probabilities for absorbing states. We show that the uncertainty effects can be calculated by solving linear programming problems in the case of interval uncertainty for the transition probabilities, and by second order cone optimization in the case of ellipsoidal uncertainty. Many examples are given, especially Markovian queueing examples, to illustrate the theory.Markov chain;Interval uncertainty;Ellipsoidal uncertainty;Linear Programming;Second Order Cone Optimization

Automates programmables et réseaux de terrain: conception des systèmes de régulation

Depuis le début des années 80, nous intégrons des automates programmables pour le contrôle des différents processus industriels. A l'origine, l'automate programmable était considéré comme une machine séquentielle, capable de suppléer des automatismes réalisés en logique traditionnelle, en apportant toutefois de profonds bouleversements dans la manière de concevoir et d'organiser le contrôle d'un processus. L'intégration de l'automate programmable renforce le degré de fiabilité de l'équipement et offre une très grande adaptabilité face aux évolutions de l'environnement. Aujourd'hui, l'automate programmable n'est plus seulement une machine séquentielle mais il est beaucoup plus considéré comme un calculateur de processus grâce aux énormes progrès quant à la structure de base, la qualité et la diversité des outils proposés. Son intégration sur Fieldbus (Profibus, WorldFip), sur Ethernet (Standard TCP-IP), accroît ses possibilités et constitue un passage obligé pour augmenter la performance des processus

Fuzzy Logic Application for Optimization of the Cooling Towers Control System

The control system for the SPS-BA6 cooling towers station is considered in order to introduce the concept of a multivariable process. Multivariable control means the maintenace of several controlled variables at independent set points. In a single-variable system, to keep the single process variables within their critical values is considered a rather simple operation. In a complex multivariable system, the determination of the optimal operation point results in a combination of all set values of the variables. Control of a multivariable system requires therefore a more complex analysis. As the solution based on a mathematical model of the process is far beyond acceptable complexity, most mathematical models involve extensive simplifications and linearizations to optimize the resulting controllers. In this report the author will demonstrate how fuzzy logic might provide elegant and efficient solutions in the design of multivariable control based on experimental results rather than on mathematical models

Games for eigenvalues of the Hessian and concave/convex envelopes

We study the PDE $\lambda_j(D^2 u) = 0$, in $\Omega$, with $u=g$, on $\partial \Omega$. Here $\lambda_1(D^2 u) \leq ... \leq \lambda_N (D^2 u)$ are the ordered eigenvalues of the Hessian $D^2 u$. First, we show a geometric interpretation of the viscosity solutions to the problem in terms of convex/concave envelopes over affine spaces of dimension $j$. In one of our main results, we give necessary and sufficient conditions on the domain so that the problem has a continuous solution for every continuous datum $g$. Next, we introduce a two-player zero-sum game whose values approximate solutions to this PDE problem. In addition, we show an asymptotic mean value characterization for the solution the the PDE

The evolution problem associated with eigenvalues of the Hessian

In this paper we study the evolution problem $$\left\lbrace\begin{array}{ll} u_t (x,t)- \lambda_j(D^2 u(x,t)) = 0, & \text{in } \Omega\times (0,+\infty), \\ u(x,t) = g(x,t), & \text{on } \partial \Omega \times (0,+\infty), \\ u(x,0) = u_0(x), & \text{in } \Omega, \end{array}\right.$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ (that verifies a suitable geometric condition on its boundary) and $\lambda_j(D^2 u)$ stands for the $j-$st eigenvalue of the Hessian matrix $D^2u$. We assume that $u_0$ and $g$ are continuous functions with the compatibility condition $u_0(x) = g(x,0)$, $x\in \partial \Omega$. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, $g(x,t) =g(x)$, we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as $t\to \infty$. For $j=1$ the limit profile is just the convex envelope inside $\Omega$ of the boundary datum $g$, while for $j=N$ it is the concave envelope. We obtain this result with two different techniques: with PDE tools and and with game theoretical arguments. Moreover, in some special cases (for affine boundary data) we can show that solutions coincide with the stationary solution in finite time (that depends only on $\Omega$ and not on the initial condition $u_0$)

Nonclassical rotational inertia for a supersolid under rotation

As proposed by Leggett [4], the supersolidity of a crystal is characterized by the Non Classical Rotational Inertia (NCRI) property. Using a model of quantum crystal introduced by Josserand, Pomeau and Rica [5], we prove that NCRI occurs. This is done by analyzing the ground state of the aforementioned model, which is related to a sphere packing problem, and then deriving a theoretical formula for the inertia momentum. We infer a lower estimate for the NCRI fraction, which is a landmark of supersolidity