A comprehensive study of a feedback control problem with a state-dependent implicit pantograph equation of Chandrasekhar type

Ahmed M. A. El-Sayed; Eman M. Al-Barg; Hanaa R. Ebead; Ahmed M. A. El-Sayed; Eman M. Al-Barg; Hanaa R. Ebead

doi:10.3934/math.2025045

AIMS Mathematics

2025, Volume 10, Issue 1: 951-971. doi: 10.3934/math.2025045

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Research article

A comprehensive study of a feedback control problem with a state-dependent implicit pantograph equation of Chandrasekhar type

1.
Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, 21521, Egypt
2.
Department of Mathematics, Faculty of Science, Sirt University, Sirt, 53950, Libya

Received: 18 October 2024 Revised: 31 December 2024 Accepted: 07 January 2025 Published: 16 January 2025
MSC : 34A12, 34C08, 34K43, 47H09, 47H10

In this research, we investigate the existence of at least one continuous solution of a problem with feedback control involving implicit pantograph equations of the Chandrasekhar type with state-dependent delay. In addition, we examine the possibility of the uniqueness of the solution under suitable assumptions. Furthermore, we analyze the problem's Hyers-Ulam stability and the continuous dependency of the unique solution on the original data and the parameter. Moreover, we look into this problem in the absence of feedback control. We provided a few instances to indicate our findings.
- Schauder fixed point theorem,
- state-dependant,
- feedback control,
- pantograph equation,
- existence of solution,
- continuous dependence,
- Hyres-Ulam stability
Citation: Ahmed M. A. El-Sayed, Eman M. Al-Barg, Hanaa R. Ebead. A comprehensive study of a feedback control problem with a state-dependent implicit pantograph equation of Chandrasekhar type[J]. AIMS Mathematics, 2025, 10(1): 951-971. doi: 10.3934/math.2025045

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Abstract

In this research, we investigate the existence of at least one continuous solution of a problem with feedback control involving implicit pantograph equations of the Chandrasekhar type with state-dependent delay. In addition, we examine the possibility of the uniqueness of the solution under suitable assumptions. Furthermore, we analyze the problem's Hyers-Ulam stability and the continuous dependency of the unique solution on the original data and the parameter. Moreover, we look into this problem in the absence of feedback control. We provided a few instances to indicate our findings.

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