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Journal Articles SIAM Journal on Applied Dynamical Systems Year : 2023

Optimal Control of the Controlled Lotka-Volterra Equations with Applications - The Permanent Case

Abstract

In this article motivated by the control of complex microbiota in view to reduce the infection by a pathogenic agent, we introduce the theoretical frame from optimal control to analyze the problem. Two complementary approaches can be applied in the analysis: one is the so-called permanent case, where no digital constraints are concerning the control (taken as a measurable mapping) versus the sampled-data control case taking into account the logistic constraints, e.g. frequency of the medical interventions. The model is the n-dimensional Lotka-Volterra equation controlled using either probiotics or antibiotic agents or transplantation and bactericides. In the permanent case the Maximum principle is used to parameterize the geodesics and the optimal synthesis boils down to analyze mainly the singular trajectories and their concatenation with bang arcs. This analysis is preliminary and complementary to the study of the problem in the frame of optimal sampleddata control, where convergence of the MPC algorithms is related to singular trajectories and nonuniqueness of the optimal dynamics encountered in the permanent case.
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Dates and versions

hal-03757060 , version 1 (22-08-2022)
hal-03757060 , version 2 (04-07-2023)

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Bernard Bonnard, Jérémy Rouot. Optimal Control of the Controlled Lotka-Volterra Equations with Applications - The Permanent Case. SIAM Journal on Applied Dynamical Systems, 2023, 22 (4), pp.2761-2791. ⟨10.1137/22M151978X⟩. ⟨hal-03757060v2⟩
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