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Feedback Maximum Principle for a Class of Linear Continuity Equations Inspired by Optimal Impulsive Control

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Mathematical Optimization Theory and Operations Research (MOTOR 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12755))

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Abstract

The paper deals with an optimal control problem for the simplest version of a “reduced” linear impulsive continuity equation. The latter is introduced in our recent papers as a model of dynamical ensembles enduring jumps, or impulsive ODEs having a probabilistic uncertainty in the initial data. The model, addressed in the manuscript, is, in fact, equivalent to the mentioned impulsive one, while it is stated within the usual, continuous setup. The price for this reduction is the appearance of an integral constraint on control, which makes the problem non-standard.

The main focus of our present study is on the theory of so-called feedback necessary optimality conditions, which are one of the recent achievements in the optimal control theory of ODEs. The paradigmatic version of such a condition, called the feedback maximum (or minimum) principle, is formulated with the use of standard constructions of the classical Pontryagin’s Maximum Principle but is shown to strengthen the latter (in particular, for different pathological cases). One of the main advantages of the feedback maximum principle is due to its natural algorithmic property, which enables using it as an iterative numeric algorithm.

The paper presents a version of the feedback maximum principle for linear transport equations with integrally bounded controls.

The second author is supported by the Russian science foundation (project No. 17-11-01093).

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Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory of the Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia

    Maxim Staritsyn, Nikolay Pogodaev & Elena Goncharova

  2. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

    Nikolay Pogodaev

Authors
  1. Maxim Staritsyn
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  2. Nikolay Pogodaev
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  3. Elena Goncharova
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Editor information

Editors and Affiliations

  1. University of Florida, Gainesville, FL, USA

    Panos Pardalos

  2. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  3. Matrosov Institute for System Dynamics and Control Theory, Irkutsk, Russia

    Alexander Kazakov

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Staritsyn, M., Pogodaev, N., Goncharova, E. (2021). Feedback Maximum Principle for a Class of Linear Continuity Equations Inspired by Optimal Impulsive Control. In: Pardalos, P., Khachay, M., Kazakov, A. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2021. Lecture Notes in Computer Science(), vol 12755. Springer, Cham. https://doi.org/10.1007/978-3-030-77876-7_24

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  • DOI: https://doi.org/10.1007/978-3-030-77876-7_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-77875-0

  • Online ISBN: 978-3-030-77876-7

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