Photonic Structures Optimization Using Highly Data-Efficient Deep Learning: Application To Nanofin And Annular Groove Phase Masks

Metasurfaces offer a flexible framework for the manipulation of light properties in the realm of thin film optics. Specifically, the polarization of light can be effectively controlled through the use of thin phase plates. This study aims to introduce a surrogate optimization framework for these devices. The framework is applied to develop two kinds of vortex phase masks (VPMs) tailored for application in astronomical high-contrast imaging. Computational intelligence techniques are exploited to optimize the geometric features of these devices. The large design space and computational limitations necessitate the use of surrogate models like partial least squares Kriging, radial basis functions, or neural networks. However, we demonstrate the inadequacy of these methods in modeling the performance of VPMs. To address the shortcomings of these methods, a data-efficient evolutionary optimization setup using a deep neural network as a highly accurate and efficient surrogate model is proposed. The optimization process in this study employs a robust particle swarm evolutionary optimization scheme, which operates on explicit geometric parameters of the photonic device. Through this approach, optimal designs are developed for two design candidates. In the most complex case, evolutionary optimization enables optimization of the design that would otherwise be impractical (requiring too much simulations). In both cases, the surrogate model improves the reliability and efficiency of the procedure, effectively reducing the required number of simulations by up to 75% compared to conventional optimization techniques

The LQ-Optimal Control Problem for Invariant Linear Systems

This work is concerned with the study of the linear quadratic (LQ) optimal control problem for linear systems with affine inequality constraints on the state and/or the input tra- jectories, and in particular for input/state-invariant linear systems. The study of such systems is motivated notably by the coexistence problem in a chemostat model where, for biologi- cal reasons, it is meaningful to aim at forcing the state and the input trajectories to remain in a cone. Necessary and sufficient optimality conditions are established for the input/state- invariant LQ problem by using the maximum principle with state and input constraints and by using the admissibility of the solution of the standard LQ problem. Similar and specific results are obtained for the particular LQ problem for positive systems, which are character- ized by the invariance of the nonnegative orthant of the state space. The methods developed in this thesis are applied to the chemostat model via the study of locally positively input/state- invariant nonlinear systems. The main results of this work are illustrated by some numerical examples.Ce travail a pour objet l’étude du problème de commande optimale au sens linéaire quadratique (LQ) pour des systèmes linéaires avec contraintes d’inégalité affines sur les trajectoires d’état et/ou d’entrée, et en particulier pour des systèmes linéaires entrée/état- invariants. L’étude de ces systèmes est motivée notamment par le problème de coexistence dans un modèle de chémostat où, pour des raisons biologiques, il est important de chercher à forcer les trajectoires d’état et d’entrée de rester dans un cône. Des conditions nécessaires et suffisantes d’optimalité sont établies pour le problème LQ invariant entrée/état en utilisant le principe du maximum avec contraintes sur l’état et l’entrée et à l’aide de l’admissibilité de la solution du problème LQ standard. Des résultats similaires et spécifiques sont obtenus pour le problème LQ appliqué aux systèmes positifs, qui sont caractérisés par l’invariance de l’orthant non négatif de l’espace d’état. Les méthodes développées dans cette thèse sont appliquées au modèle de chémostat via l’étude des systèmes non linéaires localement entrée/état-invariants. Les principaux résultats de ce travail sont illustrés par des exemples numériques.(DOCSC00) -- FUNDP, 201