Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces

Bandaliyev, Rovshan A.; Mamedov, Ilgar G.; Mardanov, Mısır J.; Melikov, T. K.

doi:10.1007/s11590-019-01518-6

Cited by 11 publications

(10 citation statements)

References 19 publications

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“…and, together with u(•), satisfies the differential equation in (1) for almost every t ∈ [t 0 , ϑ]. Due to conditions ( A.1) and ( A.2), such a motion x(•) exists and is unique (see, e.g., [5,Theorem 3.1]), and we denote it by x(• | t 0 , x 0 , ϑ, u(•)).…”

Section: Problem Statementmentioning

confidence: 99%

Archives of Control Sciences

Gomoyunov¹

2020

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The paper deals with an optimal control problem in a dynamical system described by a linear differential equation with the Caputo fractional derivative. The goal of control is to minimize a Bolza-type cost functional, which consists of two terms: the first one evaluates the state of the system at a fixed terminal time, and the second one is an integral evaluation of the control on the whole time interval. In order to solve this problem, we propose to reduce it to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation. The reduction is based on the representation formula for solutions to linear fractional differential equations and is performed by some linear transformation, which is called the informational image of a position of the original system and can be treated as a special prediction of a motion of this system at the terminal time. A connection between the original and auxiliary problems is established for both open-loop and feedback (closed-loop) controls. The results obtained in the paper are illustrated by examples.

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Section: Problem Statementmentioning

confidence: 99%

Archives of Control Sciences

Gomoyunov¹

2020

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“…Therefore, in recent decades, fractional calculus has gradually become a powerful tool for discussing and resolving the problems of modern production technology in relation to the rapid development of such technology and natural science. It is worth noting that, in the study of control theory, some researchers use fractional calculus to obtain better simulation results [5]. There are also some related findings that can be referred to in the articles [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

Niu

Zou

2023

Fractal Fract

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Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results.

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“…Main directions of research here are related to necessary optimality conditions (see, e.g., [4,30] and the references therein) and numerical methods for constructing optimal controls (see, e.g., [29,45,51] and the references therein). In addition, note that several problems for linear systems are considered and studied in detail in, e.g., [2,13,18,20,27,40]. The reader is also referred to [5] for an overview of works on various control problems for fractional-order systems.…”

Section: Introductionmentioning

confidence: 99%

Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives

Gomoyunov

2022

ESAIM: COCV

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We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order $\alpha \in (0, 1)$. Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives of the order $\alpha$. We propose a notion of a generalized in the minimax sense solution of the considered problem. We prove that a minimax solution exists, is unique, and is consistent with a classical solution of this problem. In particular, we give a special attention to the proof of a comparison principle, which requires construction of a suitable Lyapunov--Krasovskii functional.

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Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces

Cited by 11 publications

References 19 publications

Archives of Control Sciences

Archives of Control Sciences

A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

Minimax solutions of Hamilton–Jacobi equations with fractional coinvariant derivatives

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