On semi-Classical Questions Related to Signal Analysis

This study explores the reconstruction of a signal using spectral quantities associated with some self-adjoint realization of an h-dependent Schr\"odinger operator when the parameter h tends to 0. Theoretical results in semi-classical analysis are proved. Some numerical results are also presented. We first consider as a toy model the sech^2 function. Then we study a real signal given by arterial blood pressure measurements. This approach seems to be very promising in signal analysis. Indeed it provides new spectral quantities that can give relevant information on some signals as it is the case for arterial blood pressure signal

Mean-Field Stochastic Differential Game for Fine Alignment Control of Cooperative Optical Beam Systems

The deployment of autonomous optical link communication platforms that benefit from mobility and optical data rates is essential in public safety communications. However, maintaining an accurate line-of-sight and perfect tracking between mobile platforms or unmanned aerial vehicles (UAVs) in free-space remains challenging for cooperative optical communication due to the underlying mechanical vibration and accidental shocks. Indeed, a misalignment can result in optical channel disconnection, leading to connectivity loss. To address this challenge, we propose a two-way optical link that coordinates mobile UAVs' closed-loop fine beam tracking operation in a swarm architecture to enhance terrestrial public safety communication systems. We study a dynamic of the optical beam tracking games in which each agent's dynamic and cost function are coupled with the other optical beam transceiver agents' states via a mean-field term. We describe a line-of-sight stochastic cooperative beam tracking communication through a mean field game paradigm that can provide reliable network structure and persistent distributed connectivity and communicability. We derive two optimal mean-field beam tracking control frameworks through decentralized and centralized strategies. The solutions of these strategies are derived from forward-backward ordinary differential equations and rely on the linearity Hamilton-Jacobi-Bellman Fokker-Planck (HJB-FP) equations and stochastic maximum principle. Furthermore, we numerically compute the solution pair to the two joint equations using Newton and fixed point iterations methods to verify the existence and uniqueness of the equilibrium that drives the control to a Nash equilibrium for both differential games

Reference Tracking AND Observer Design for Space-Fractional Partial Differential Equation Modeling Gas Pressures in Fractured Media

This paper considers a class of space fractional partial differential equations (FPDEs) that describe gas pressures in fractured media. First, the well-posedness, uniqueness, and the stability in $L_(\infty{R})$of the considered FPDEs are investigated. Then, the reference tracking problem is studied to track the pressure gradient at a downstream location of a channel. This requires manipulation of gas pressure at the downstream location and the use of pressure measurements at an upstream location. To achiever this, the backstepping approach is adapted to the space FPDEs. The key challenge in this adaptation is the non-applicability of the Lyapunov theory which is typically used to prove the stability of the target system as, the obtained target system is fractional in space. In addition, a backstepping adaptive observer is designed to jointly estimate both the system's state and the disturbance. The stability of the closed loop (reference tracking controller/observer) is also investigated. Finally, numerical simulations are given to evaluate the efficiency of the proposed method.Comment: 37 pages, 9 figure