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Hybrid optimal control: optimality conditions and applications

Authors
  • Bouali, Anas
Publication Date
Nov 06, 2023
Source
HAL-Descartes
Keywords
Language
English
License
Unknown
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Abstract

This thesis deals with the mathematical field of optimal control theory, specifically focusing on spatially hybrid optimal control problems. Here, the term spatially indicates that we consider a hybrid control system defined over a partition of the state space that is divided into disjoint regions. Furthermore, we assume that the control system depends on a regionally switching parameter, which remains constant within each region but can change its value when the state position crosses boundaries. This new framework allows us to address control systems that includes loss control regions, which presents our initial motivation. In such systems, given a partition of the state space, the control behavior varies depending on the position of the state. It can be modified at any time (referred to as permanent controls) when the state belongs to regions referred to as control region, or it can remain constant when the state belongs to regions referred to as loss control regions. In both frameworks, our goals are, first, to to derive a spatially hybrid maximum principle (in short, HMP) with regionally switching parameter, second, to derive a Pontryagin maximum principle with loss control regions, and third, to provide a numerical approach allowing to solve optimal control problems with loss control regions. To achieve these purposes, we introduce new tools and concepts that address certain challenges that can arise in a spatially hybrid setting. Specifically, based on careful investigation, we identify two main challenges: the nonadmissibility of needle-like perturbations and the inability to directly apply the well-known augmentation technique in a spatially hybrid setting.

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