The hybrid maximum principle for optimal control problems with spatially heterogeneous dynamics is a consequence of a Pontryagin maximum principle for L1 square'' local solutions - Archive ouverte HAL Access content directly
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## The hybrid maximum principle for optimal control problems with spatially heterogeneous dynamics is a consequence of a Pontryagin maximum principle for L1 square'' local solutions

Terence Bayen
• Function : Author
Anas Bouali
• Function : Author
Loic Bourdin
• Function : Author

#### Abstract

The title of the present work is a nod to the paper The hybrid maximum principle is a consequence of Pontryagin maximum principle by Dmitruk and Kaganovich (Systems and Control Letters, 2008). Here we investigate a similar but different framework of hybrid optimal control problems. Precisely we consider a general control system that is described by a differential equation involving a spatially heterogeneous dynamics. In that context the sequence of dynamics followed by the trajectory and the corresponding switching times are fully constrained by the state position. We prove with an explicit counterexample that the augmentation technique proposed by Dmitruk and Kaganovich cannot be fully applied to our setting, but we show that it can be adapted by introducing a new notion of local solution to classical optimal control problems and by establishing a corresponding Pontryagin maximum principle. Thanks to this method we derive a hybrid maximum principle adapted to our setting, with a simple proof that does not require any technical tool (such as implicit function arguments) to handle the dynamical discontinuities.

### Dates and versions

hal-03985420 , version 1 (13-02-2023)

### Identifiers

• HAL Id : hal-03985420 , version 1

### Cite

Terence Bayen, Anas Bouali, Loic Bourdin. The hybrid maximum principle for optimal control problems with spatially heterogeneous dynamics is a consequence of a Pontryagin maximum principle for L1 square'' local solutions. 2023. ⟨hal-03985420⟩

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