Necessary Optimality Conditions for Singular Controls of Caputo Fractional Systems with Delay in Control

Shakir Sh Yusubov; Elimhan N. Mahmudov

doi:10.1007/s12346-025-01225-x

Necessary Optimality Conditions for Singular Controls of Caputo Fractional Systems with Delay in Control

Shakir Sh Yusubov
, Elimhan N. Mahmudov^*

^*Corresponding author for this work

Baku State University
Azerbaijan University of Architecture and Construction
Azerbaijan National Academy of Aviation
Azerbaijan State University of Economics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In this paper, we consider the optimal control of nonlinear systems of fractional-order with control delay, where the fractional derivatives are expressed in the Caputo sense. For this problem, we first obtained the necessary first-order optimality condition in the form of the Pontryagin maximum principle. For a control that is singular in the sense of the Pontryagin maximum principle, a higher-order necessary optimality condition is obtained. Further, in one particular case, a multipoint necessary second-order optimality condition is obtained. The effectiveness of the results obtained is illustrated by specific examples.

Original language	English
Article number	61
Journal	Qualitative Theory of Dynamical Systems
Volume	24
Issue number	2
DOIs	https://doi.org/10.1007/s12346-025-01225-x
Publication status	Published - Apr 2025
Externally published	Yes

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

Keywords

Caputo fractional
Delay
Increment method
Necessary conditions
Singular controls

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10.1007/s12346-025-01225-x

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title = "Necessary Optimality Conditions for Singular Controls of Caputo Fractional Systems with Delay in Control",

abstract = "In this paper, we consider the optimal control of nonlinear systems of fractional-order with control delay, where the fractional derivatives are expressed in the Caputo sense. For this problem, we first obtained the necessary first-order optimality condition in the form of the Pontryagin maximum principle. For a control that is singular in the sense of the Pontryagin maximum principle, a higher-order necessary optimality condition is obtained. Further, in one particular case, a multipoint necessary second-order optimality condition is obtained. The effectiveness of the results obtained is illustrated by specific examples.",

keywords = "Caputo fractional, Delay, Increment method, Necessary conditions, Singular controls",

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AU - Yusubov, Shakir Sh

AU - Mahmudov, Elimhan N.

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

PY - 2025/4

Y1 - 2025/4

N2 - In this paper, we consider the optimal control of nonlinear systems of fractional-order with control delay, where the fractional derivatives are expressed in the Caputo sense. For this problem, we first obtained the necessary first-order optimality condition in the form of the Pontryagin maximum principle. For a control that is singular in the sense of the Pontryagin maximum principle, a higher-order necessary optimality condition is obtained. Further, in one particular case, a multipoint necessary second-order optimality condition is obtained. The effectiveness of the results obtained is illustrated by specific examples.

AB - In this paper, we consider the optimal control of nonlinear systems of fractional-order with control delay, where the fractional derivatives are expressed in the Caputo sense. For this problem, we first obtained the necessary first-order optimality condition in the form of the Pontryagin maximum principle. For a control that is singular in the sense of the Pontryagin maximum principle, a higher-order necessary optimality condition is obtained. Further, in one particular case, a multipoint necessary second-order optimality condition is obtained. The effectiveness of the results obtained is illustrated by specific examples.

KW - Caputo fractional

KW - Delay

KW - Increment method

KW - Necessary conditions

KW - Singular controls

UR - https://www.scopus.com/pages/publications/85217676008

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DO - 10.1007/s12346-025-01225-x

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SN - 1575-5460

VL - 24

JO - Qualitative Theory of Dynamical Systems

JF - Qualitative Theory of Dynamical Systems

IS - 2

M1 - 61

ER -

Necessary Optimality Conditions for Singular Controls of Caputo Fractional Systems with Delay in Control

Abstract

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Keywords

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