Controllability of Hilfer Fractional Langevin Dynamical System with Impulse in an Abstract Weighted Space

Radhakrishnan, B.; Sathya, Thangavel

doi:10.1007/s10957-022-02081-4

Cited by 8 publications

(5 citation statements)

References 15 publications

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“…which means that RL D α 1 a + x(t) exists and belongs to C 1−α 1 . Hence, (17). The proof is complete.…”

Section: Equivalence With An Integral Equationmentioning

confidence: 73%

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Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

Li,

Yang,

Wang

2024

Axioms

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In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results.

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“…which means that RL D α 1 a + x(t) exists and belongs to C 1−α 1 . Hence, (17). The proof is complete.…”

Section: Equivalence With An Integral Equationmentioning

confidence: 73%

“…Moreover, we refer to some recent works [17,18] that deal with a qualitative analysis of FLEs with Hilfer fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

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Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

Li,

Yang,

Wang

2024

Axioms

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“…Zhao, in [3][4][5], discussed the stability of several types of nonlinear fractional Langevin equations with delays and controls. In [6][7][8], the authors explored the controllability problem of fractional Langevin equations. Other papers are [9][10][11] concerned with the dynamics of stochastic Langevin equations.…”

Section: Introductionmentioning

confidence: 99%

A Unified Approach to Solvability and Stability of Multipoint BVPs for Langevin and Sturm–Liouville Equations with CH–Fractional Derivatives and Impulses via Coincidence Theory

Zhao,

Liu,

2024

Fractal Fract

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The Langevin equation is a model for describing Brownian motion, while the Sturm–Liouville equation is an important mechanical model. This paper focuses on the solvability and stability of nonlinear impulsive Langevin and Sturm–Liouville equations with Caputo–Hadamard (CH) fractional derivatives and multipoint boundary value conditions. To unify the two types of equations, we investigate a general nonlinear impulsive coupled implicit system. By cleverly constructing relevant operators involving impulsive terms, we establish the coincidence degree theory and obtain the solvability. We explore the stability of solutions using nonlinear analysis and inequality techniques. As the most direct application, we naturally obtained the solvability and stability of the Langevin and Sturm–Liouville equations mentioned above. Finally, an example is provided to demonstrate the validity and availability of our major findings.

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“…Fractional differential equations with the Hilfer derivative have been studied by many authors, see for example [8,10,13,15,24,45,47]. However, only a few papers have been published on the controllability of differential systems involving the Hilfer fractional derivative [21,25,31,41].…”

Section: Introductionmentioning

confidence: 99%

On the constrained and unconstrained controllability of semilinear Hilfer fractional systems

2023

Archives of Control Sciences

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In the paper finite-dimensional semilinear dynamical control systems described by fractional-order state equations with the Hilfer fractional derivative are discussed. The formula for a solution of the considered systems is presented and derived using the Laplace transform. Bounded nonlinear function 𝑓 depending on a state and controls is used. New sufficient conditions for controllability without constraints are formulated and proved using Rothe's fixed point theorem and the generalized Darbo fixed point theorem. Moreover, the stability property is used to formulate constrained controllability criteria. An illustrative example is presented to give the reader an idea of the theoretical results obtained. A transient process in an electrical circuit described by a system of Hilfer type fractional differential equations is proposed as a possible application of the study.

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Controllability of Hilfer Fractional Langevin Dynamical System with Impulse in an Abstract Weighted Space

Cited by 8 publications

References 15 publications

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

A Unified Approach to Solvability and Stability of Multipoint BVPs for Langevin and Sturm–Liouville Equations with CH–Fractional Derivatives and Impulses via Coincidence Theory

On the constrained and unconstrained controllability of semilinear Hilfer fractional systems

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