Uniqueness and long time asymptotics for the parabolic-parabolic Keller-Segel equation

The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane inthe general framework of weak (or "free energy") solutions associated to an initial datum with finite mass M\textless{} 8\pi, finite second log-moment and finite entropy. The aim of the paper is twofold:(1) We prove the uniqueness of the "free energy" solution. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the "optimal regularity" as well as an estimate of the difference of two possible solutions in the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$ vorticity Navier-Stokes equation. (2) We prove a radially symmetric and polynomial weighted $L^2$ exponential stability of the self-similar profile in the quasi parabolic-elliptic regime. The proof is based on a perturbation argument which takes advantage of the exponential stability of the self-similar profile for the parabolic-elliptic Keller-Segel equation established by Campos-Dolbeault and Egana-Mischler

State Estimation for Distributed and Hybrid Systems

This thesis deals with two aspects of recursive state estimation: distributed estimation and estimation for hybrid systems. In the first part, an approximate distributed Kalman filter is developed. Nodes update their state estimates by linearly combining local measurements and estimates from their neighbors. This scheme allows nodes to save energy, thus prolonging their lifetime, compared to centralized information processing. The algorithm is evaluated experimentally as part of an ultrasound based positioning system. The first part also contains an example of a sensor-actuator network, where a mobile robot navigates using both local sensors and information from a sensor network. This system was implemented using a component-based framework. The second part develops, a recursive joint maximum a posteriori state estimation scheme for Markov jump linear systems. The estimation problem is reformulated as dynamic programming and then approximated using so called relaxed dynamic programming. This allows the otherwise exponential complexity to be kept at manageable levels. Approximate dynamic programming is also used to develop a sensor scheduling algorithm for linear systems. The algorithm produces an offline schedule that when used together with a Kalman filter minimizes the estimation error covariance