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Positive Switching Systems and the Wealth Distribution Problem

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Abstract

The paper shows how a dynamic model, belonging to the class of positive switching systems, can provide an interesting point of view to the study of the generation and the distribution of wealth in a society. In this model, the evolution of the overall wealth is affected by the way in which individuals share their resources in order to maximize the total income. Different formulations of the corresponding optimal control problem are considered and some general properties are pointed out. Approximated numerical solutions, as well as upper and lower bounds are also investigated. A final section dedicated to an extended numerical analysis yields some qualitative conclusion.

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Data Availability

The data supporting the findings of this study are available within the paper or from the corresponding author upon reasonable request.

Notes

  1. In Eq. (27) the index “lm” stays for “Lyapunov–Metzler”.

  2. Recall that a function \(V:\mathbb {R}^n\rightarrow \mathbb {R}\) is said to be co-positive if \(V(x)>0\) for every \(x>0\) and \(V(0)=0\).

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Acknowledgements

Franco Blanchini, Daniele Casagrande, and Patrizio Colaneri have been supported by the Italian Ministry for Research in the framework of project “Proliferation, Resistance and Infection Dynamics” (PRIDE), PRIN 2022 by the European Union - NextGenerationEU grant no. 2022LP77J4.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università degli Studi di Udine, Via delle Scienze 208, 33100, Udine, Italy

    Franco Blanchini

  2. Dipartimento Politecnico di Ingegneria e Architettura, Università degli Studi di Udine, Via delle Scienze 208, 33100, Udine, Italy

    Daniele Casagrande

  3. Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Via Ponzio 34/5, 20133, Milano, Italy

    Patrizio Colaneri

  4. IEIIT-CNR, Turin, Italy

    Patrizio Colaneri

  5. Jan C. Willems Center for Systems and Control, ENTEG, Faculty of Science and Engineering, University of Groningen, 9747 AG, Groningen, The Netherlands

    Stefanny Ramirez & Dario Bauso

  6. Dipartimento di Ingegneria, Università di Palermo, 90128, Palermo, Italy

    Dario Bauso

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  1. Franco Blanchini
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  2. Daniele Casagrande
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  3. Patrizio Colaneri
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  4. Stefanny Ramirez
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Correspondence to Daniele Casagrande.

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Communicated by Nikolai Osmolovskii.

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Blanchini, F., Casagrande, D., Colaneri, P. et al. Positive Switching Systems and the Wealth Distribution Problem. J Optim Theory Appl 204, 29 (2025). https://doi.org/10.1007/s10957-024-02579-z

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