A review of the Lurie problem and its applications in the medical and biological fields

Rafael F. Pinheiro; Rui Fonseca-Pinto; Diego Colón; Rafael F. Pinheiro; Rui Fonseca-Pinto; Diego Colón

doi:10.3934/math.20241577

AIMS Mathematics

2024, Volume 9, Issue 11: 32962-32999. doi: 10.3934/math.20241577

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Review

A review of the Lurie problem and its applications in the medical and biological fields

1.
Center for Innovative Care and Health Technology (ciTechCare), School of Health Sciences (ESSLei), Polytechnic University of Leiria, R. das Olhalvas, 2414-016, Leiria, Portugal
2.
Laboratory of Automation and Control (LAC), Telecommunications and Control Department, Polytechnic School, University of São Paulo, Av. Prof. Luciano Gualberto, Travessa 03,158, São Paulo, 05508-900, São Paulo, Brazil

Received: 29 July 2024 Revised: 14 October 2024 Accepted: 05 November 2024 Published: 21 November 2024
MSC : 34D23, 68T07, 92-10, 93-10, 93B36, 93C10, 93C35, 93D09

This paper provided a review of the Lurie problem and its applications to control as well as modeling problems in the medical and biological fields, highlighting its connection with robust control theory, more specifically the works of Doyle, Skogestad, and Zhou. The Lurie problem involved the study of control systems with nonlinearities incorporated into the feedback loop. Providing a simpler and broader approach, this review returned to the Lurie problem, covering basic stability concepts and Aizerman's conjecture, establishing it as a special instance of the Lurie problem. The paper also explained the connection between the Lurie problem and robust control theory, which resulted in the establishment of new conditions for the Lurie problem. The principal contribution of this paper was a comprehensive review, utilizing the preferred reporting items for systematic reviews and meta-analyses (PRISMA) methodology of the applications of the Lurie problem in the medical and biological fields, demonstrating its significance in various domains such as medical device controllers, mechanical ventilation systems, patient-robot-therapist collaboration, tele-surgery, fluid resuscitation control, nanobiomedicine actuators, anesthesia systems, cardiac mechanics models, oncology cell dynamics, epidemiological models, diabetes modeling, population dynamics and neuroscience, including artificial neural networks (ANN). This article seeked to present the latest advancements in the Lurie problem, offering an update for researchers in the area and a valuable starting point for new researchers with several suggestions for future work, showcasing the importance of Lurie-type systems theory in advancing medical research and applications.
- problem of Lur'e,
- absolute stability,
- disease modeling,
- Aizerman conjecture,
- nonlinear systems,
- robust control
Citation: Rafael F. Pinheiro, Rui Fonseca-Pinto, Diego Colón. A review of the Lurie problem and its applications in the medical and biological fields[J]. AIMS Mathematics, 2024, 9(11): 32962-32999. doi: 10.3934/math.20241577

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Abstract

This paper provided a review of the Lurie problem and its applications to control as well as modeling problems in the medical and biological fields, highlighting its connection with robust control theory, more specifically the works of Doyle, Skogestad, and Zhou. The Lurie problem involved the study of control systems with nonlinearities incorporated into the feedback loop. Providing a simpler and broader approach, this review returned to the Lurie problem, covering basic stability concepts and Aizerman's conjecture, establishing it as a special instance of the Lurie problem. The paper also explained the connection between the Lurie problem and robust control theory, which resulted in the establishment of new conditions for the Lurie problem. The principal contribution of this paper was a comprehensive review, utilizing the preferred reporting items for systematic reviews and meta-analyses (PRISMA) methodology of the applications of the Lurie problem in the medical and biological fields, demonstrating its significance in various domains such as medical device controllers, mechanical ventilation systems, patient-robot-therapist collaboration, tele-surgery, fluid resuscitation control, nanobiomedicine actuators, anesthesia systems, cardiac mechanics models, oncology cell dynamics, epidemiological models, diabetes modeling, population dynamics and neuroscience, including artificial neural networks (ANN). This article seeked to present the latest advancements in the Lurie problem, offering an update for researchers in the area and a valuable starting point for new researchers with several suggestions for future work, showcasing the importance of Lurie-type systems theory in advancing medical research and applications.

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