Hybrid optimal control : optimalityconditions and applications

Overview & Synopsis

This manuscript 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 regions, or it can remain constant when the state belongs to regions referred to as loss control regions. In both frameworks, our goals are: (i) to derive a spatially hybrid maximum principle (in short, HMP) with regionally switching parameter. (ii) to derive a Pontryagin maximum principle with loss control regions. (iii) 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. This manuscript is made up of 6 chapters: (1) Chapter 1 is devoted to notations and the basic framework needed to describe the hybrid optimal control problem with regionally switching parameters and optimal control problems with loss control regions encountered in the manuscript. (2) Chapter 2 is devoted to deriving a spatially HMP with regionally switching parameters for hybrid optimal control problems with fixed initial conditions. We also provide first-order necessary conditions for optimal control problems with loss control regions. (3) Chapter 3 is devoted to the study of a variant of the minimum time problem for the double integrator with a loss control region. The latter is based on a PMP adapted to a minimum time problem with a loss control region. (4) Chapter 4 focuses on the derivation of a spatially HMP based on a careful use of the augmentation technique. The latter is based on the introduction of a new concept of local minimum and the subsequentderivation of a corresponding PMP. (5) Chapter 5 is devoted to the derivation of a PMP with loss control regions, as well as a two-step numerical approach to solve this type of problem. To do so, we first provide a HMP with regionally switching parameters in a similar manner to Chapter 4. (6) In Chapter 6, we provide a general conclusion that briefly presents our research findings and provides someperspectives for future research work.