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Necessary optimality conditions for quasi-singular controls for systems with Caputo fractional derivatives

Shakir Sh. Yusubov Elimhan N. MahmudoV
Archives of Control Sciences | 2023 | vol. 33 | No 3 | 463–496 | DOI: 10.24425/acs.2023.146955
Keywords fractional derivative fractional optimal control necessary optimality condition
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Abstract

In this paper, we consider an optimal control problem in which a dynamical system is controlled by a nonlinear Caputo fractional state equation. First we get the linearized maximum principle. Further, the concept of a quasi-singular control is introduced and, on this basis, an analogue of the Legendre-Clebsch conditions is obtained. When the analogue of Legendre- Clebsch condition degenerates, a necessary high-order optimality condition is derived. An illustrative example is considered.
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Authors and Affiliations

Shakir Sh. Yusubov
1
Elimhan N. MahmudoV
2 3
ORCID: ORCID
  1. Department of Mechanics and Mathematics, Baku State University, Baku, Azerbaijan
  2. Department of Mathematics, Istanbul Technical University, Istanbul, Turkey
  3. Azerbaijan National Aviation Academy, Baku, Azerbaijan

A comparative study of the sensitivity analysis for systems with viscoelastic elements

Magdalena Łasecka-Plura
Archive of Mechanical Engineering | 2023 | vol. 70 | No 1 | 5-25 | DOI: 10.24425/ame.2022.144077
Keywords sensitivity analysis viscoelastic damping elements dynamic characteristics fractional derivatives
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Abstract

This paper discusses the different methods used for calculating first- and second-order sensitivity: the direct differentiation method, the adjoint variables method, and the hybrid method. The solutions obtained allow determining the sensitivity of dynamic characteristics such as eigenvalues and eigenvectors, natural frequencies, and nondimensional damping ratios. The methods were applied for analyzing systems with viscoelastic damping elements, whose behavior can be described by classical and fractional rheological models. However, the derived formulas are general and can also be applied to systems with damping elements described by other models. Their advantage is a compact and easy to code form. The paper also presents a comparison of the computational costs of the discussed methods. The correctness of all the proposed methods has been illustrated with numerical examples.
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Bibliography

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[27] R. Lewandowski. Sensitivity analysis of structures with viscoelastic dampers using the adjoint variable method. Civil-Comp Proceedings, 106, 2014.
[28] J.S. Arora and J.B. Cardoso. Variational principle for shape design sensitivity analysis. AIAA Journal, 30(2):538–547, 1992. doi: 10.2514/3.10949.
[29] Z. Pawlak and R. Lewandowski. The continuation method for the eigenvalue problem of structures with viscoelastic dampers. Computers and Structures, 125:53–61, 2013. doi: 10.1016/j.compstruc.2013.04.021.
[30] R. Lewandowski and M. Baum. Dynamic characteristics of multilayered beams with viscoelastic layers described by the fractional Zener model. Archive of Applied Mechanics, 85(12):1793–1814, 2015. doi: 10.1007/s00419-015-1019-2.
[31] R. Lewandowski, P. Litewka and P. Wielentejczyk. Free vibrations of laminate plates with viscoelastic layers using the refined zig-zag theory – Part 1: Theoretical background. Composite Structures, 278:114547, 2021. doi: 10.1016/j.compstruct.2021.114547.
[32] M. Kamiński, A. Lenartowicz, M. Guminiak, and M. Przychodzki. Selected problems of random free vibrations of rectangular thin plates with viscoelastic dampers. Materials, 15(19): 6811, 2022. doi: 10.3390/ma15196811.
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Authors and Affiliations

Magdalena Łasecka-Plura
1
ORCID: ORCID
  1. Poznan University of Technology, Institute of Structural Analysis, Poznan, Poland

On optimal control problem subject to fractional order discrete time singular systems

Tirumalasetty Chiranjeevi Raj Kumar Biswas Ramesh Devarapalli Naladi Ram Babu Fausto Pedro García Márquez
Archives of Control Sciences | 2021 | vol. 31 | No 4 | 849-863 | DOI: 10.24425/acs.2021.139733
Keywords fractional optimal control problem discrete time singular system fractional derivative Hamiltonian technique
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Abstract

In this work, we present optimal control formulation and numerical algorithm for fractional order discrete time singular system (DTSS) for fixed terminal state and fixed terminal time endpoint condition. The performance index (PI) is in quadratic form, and the system dynamics is in the sense of Riemann-Liouville fractional derivative (RLFD). A coordinate transformation is used to convert the fractional-order DTSS into its equivalent non-singular form, and then the optimal control problem (OCP) is formulated. The Hamiltonian technique is used to derive the necessary conditions. A solution algorithm is presented for solving the OCP. To validate the formulation and the solution algorithm, an example for fixed terminal state and fixed terminal time case is presented.
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Authors and Affiliations

Tirumalasetty Chiranjeevi
1
Raj Kumar Biswas
2
Ramesh Devarapalli
3
ORCID: ORCID
Naladi Ram Babu
2
Fausto Pedro García Márquez
4
  1. Department of Electrical Engineering, Rajkiya Engineering College Sonbhadra, U. P., India
  2. Department of Electrical Engineering, National Institute of Technology, Silchar, India
  3. Department of Electrical Engineering, BIT Sindri, Dhanbad 828123, Jharkhand, India
  4. Ingenium Research Group, University of Castilla-La Mancha, Spain

Optimal control problems with a fixed terminal time in linear fractional-order systems

Mikhail I. Gomoyunov
Archives of Control Sciences | 2020 | vol. 30 | No 4 | 721-744 | DOI: 10.24425/acs.2020.135849
Keywords optimal control fractional derivatives linear systems open-loop control feed-back control reduction
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Abstract

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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Authors and Affiliations

Mikhail I. Gomoyunov

On LQ optimization problem subject to fractional order irregular singular systems

Muhafzan Admi Nazra Lyra Yulianti Zulakmal Refi Revina
Archives of Control Sciences | 2020 | vol. 30 | No 4 | 745-756 | DOI: 10.24425/acs.2020.135850
Keywords linear quadratic optimization fractional order irregular singular system Caputo fractional derivative Mittag-Leffler function
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Abstract

In this paper we discuss the linear quadratic (LQ) optimization problem subject to fractional order irregular singular systems. The aim of this paper is to find the control-state pairs satisfying the dynamic constraint of the form a fractional order irregular singular systems such that the LQ objective functional is minimized. The method of solving is to convert such LQ optimization into the standard fractional LQ optimization problem. Under some particularly conditions we find the solution of the problem under consideration.

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Authors and Affiliations

Muhafzan
Admi Nazra
Lyra Yulianti
Zulakmal
Refi Revina

Fixed terminal time fractional optimal control problem for discrete time singular system

Tirumalasetty Chiranjeevi Ramesh Devarapalli Naladi Ram Babu Kiran Babu Vakkapatla R. Gowri Sankara Rao Fausto Pedro Garcìa Màrquez
Archives of Control Sciences | 2022 | vol. 32 | No 3 | 489-506 | DOI: 10.24425/acs.2022.142846
Keywords fractional order differential equation discrete time singular system fractional derivative linear quadratic optimal control problem
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Abstract

This paper presents the formulation and numerical simulation for linear quadratic optimal control problem (LQOCP) of free terminal state and fixed terminal time fractional order discrete time singular system (FODSS). System dynamics is expressed in terms of Riemann-Liouville fractional derivative (RLFD), and performance index (PI) in terms of state and costate. Because of its complexity, finding analytical and numerical solutions to singular system (SS) is difficult. As a result, we use coordinate transformation to convert FODSS to its corresponding fractional order discrete time nonsingular system (FODNSS). After that, we obtain the necessary conditions by employing a Hamiltonian approach. The relevant conditions are solved using the general solution approach. For the analysis of formulation and solution algorithm, a numerical example is illustrated. Results are obtained for various �� values. According to state of the art, this is the first time that a formulation and numerical simulation of free terminal state and fixed terminal time optimal control problem (OCP) of FODSS is presented.
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Authors and Affiliations

Tirumalasetty Chiranjeevi
1
Ramesh Devarapalli
2
ORCID: ORCID
Naladi Ram Babu
3
Kiran Babu Vakkapatla
4
R. Gowri Sankara Rao
5
Fausto Pedro Garcìa Màrquez
6
  1. Department of Electrical Engineering, Rajkiya Engineering College Sonbhadra, U.P., India
  2. Department of EEE, Lendi Institute of Engineering and Technology, Vizianagaram-535005, India
  3. Department of EEE, Aditya Engineering College, Surampalem, Andhra Pradesh, India
  4. Lingayas Institute of Management and Technology Madalavarigudem, A.P., India
  5. Department of EEE, MVGR College of Engineering Vizianagaram, A.P., India
  6. Ingenium Research Group, University of Castilla-La Mancha, Spain

Numerical investigation of rotor-bearing systems with fractional derivative material damping models

Gregor Überwimmer Georg Quinz Michael Klanner Katrin Ellermann
Bulletin of the Polish Academy of Sciences Technical Sciences | 2023 | 71 | 6 | e148610 | DOI: 10.24425/bpasts.2023.148610
Keywords Numerical Assembly Technique rotor-bearing system steady-state harmonic vibration unbalance response fractional derivative damping model
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Abstract

The increasing demand for high-speed rotor-bearing systems results in the application of complex materials, which allow for a better control of the vibrational characteristics. This paper presents a model of a rotor including viscoelastic materials and valid up to high spin speeds. Regarding the destabilization of rotor-bearing systems, two main effects have to be investigated, which are strongly related to the associated internal and external damping of the rotor. For this reason, the internal material damping is modeled using fractional time derivatives, which can represent a large class of viscoelastic materials over a wide frequency range. In this paper, the Numerical Assembly Technique (NAT) is extended for the rotating viscoelastic Timoshenko beam with fractional derivative damping. An efficient and accurate simulation of the proposed rotor-bearing model is achieved. Several numerical examples are presented and the influence of internal damping on the rotor-bearing system is investigated and compared to classical damping models.
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Authors and Affiliations

Gregor Überwimmer
1
ORCID: ORCID
Georg Quinz
1
Michael Klanner
1
ORCID: ORCID
Katrin Ellermann
1
  1. Graz University of Technology, Institute of Mechanics, Kopernikusgasse 24/IV, 8010 Graz, Austria

The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term

Marek Błasik
Bulletin of the Polish Academy of Sciences Technical Sciences | 2021 | 69 | 6 | e138240 | DOI: 10.24425/bpasts.2021.138240
Keywords fractional derivatives and integrals integro-differential equations numerical methods finite difference methods
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Abstract

In the paper, the numerical method of solving the one-dimensional subdiffusion equation with the source term is presented. In the approach used, the key role is played by transforming of the partial differential equation into an equivalent integro-differential equation. As a result of the discretization of the integro-differential equation obtained an implicit numerical scheme which is the generalized Crank-Nicolson method. The implicit numerical schemes based on the finite difference method, such as the Carnk-Nicolson method or the Laasonen method, as a rule are unconditionally stable, which is their undoubted advantage. The discretization of the integro-differential equation is performed in two stages. First, the left-sided Riemann-Liouville integrals are approximated in such a way that the integrands are linear functions between successive grid nodes with respect to the time variable. This allows us to find the discrete values of the integral kernel of the left-sided Riemann-Liouville integral and assign them to the appropriate nodes. In the second step, second order derivative with respect to the spatial variable is approximated by the difference quotient. The obtained numerical scheme is verified on three examples for which closed analytical solutions are known.
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Authors and Affiliations

Marek Błasik
1
  1. Institute of Mathematics, Czestochowa University of Technology, al. Armii Krajowej 21, 42-201 Czestochowa, Poland

Some applications of the generalized Laplace transform and the representation of a solution to Sobolev-type evolution equations with the generalized Caputo derivative

Mustafa Aydin Nazim Mahmudov
Bulletin of the Polish Academy of Sciences Technical Sciences | 2024 | 72 | 2 | e149170 | DOI: 10.24425/bpasts.2024.149170
Keywords degenerate evolution equation Sobolev generalized Laplace transform fractional derivative with respect to another function Mittag-Leffler type function
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Abstract

We introduce the Sobolev-type multi-term μ-fractional evolution with generalized fractional orders with respect to another function. We make some applications of the generalized Laplace transform. In the sequel, we propose a novel type of Mittag-Leffler function generated by noncommutative linear bounded operators with respect to the given function and give a few of its properties. We look for the mild solution formula of the Sobolev-type evolution equation by building on the aforementioned Mittag-Leffler-type function with the aid of two different approaches. We share new special cases of the obtained findings.
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Authors and Affiliations

Mustafa Aydin
1
ORCID: ORCID
Nazim Mahmudov
2 3
ORCID: ORCID
  1. Department of Medical Services and Techniques, Muradiye Vocational School, Van Yuzuncu Yil University, Tu¸sba 65080 Van, Turkey
  2. Department of Mathematics, Eastern Mediterranean University, Famagusta 99628 T.R. North Cyprus, Turkey
  3. Research Center of Econophysics, Azerbaijan State University of Economics (UNEC), Istiqlaliyyat Str. 6, Baku 1001, Azerbaijan

On the constrained and unconstrained controllability of semilinear Hilfer fractional systems

Beata Sikora
Archives of Control Sciences | 2023 | vol. 33 | No 1 | 155-178 | DOI: 10.24425/acs.2023.145118
Keywords Hilfer fractional derivative fractional systems semilinear control systems nonautonomous systems Rothe’s fixed point theorem generalized Darbo fixed point theorem
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Abstract

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 �� 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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Authors and Affiliations

Beata Sikora
1
ORCID: ORCID
  1. Department of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland

On theoretical and practical aspects of Duhamel’s integral

Michał Różański Beata Sikora Adrian Smuda Roman Wituła
Archives of Control Sciences | 2021 | vol. 31 | No 4 | 815-847 | DOI: 10.24425/acs.2021.139732
Keywords Duhamel’s integral Duhamel’s principle Duhamel’s formula Laplace transformation semigroup of operators Leibniz integral rule Volterra integral equation Caputo fractional derivative
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Abstract

The paper is a newapproach to the Duhamel integral. It contains an overviewof formulas and applications of Duhamel’s integral as well as a number of new results on the Duhamel integral and principle. Basic definitions are recalled and formulas for Duhamel’s integral are derived via Laplace transformation and Leibniz integral rule. Applications of Duhamel’s integral for solving certain types of differential and integral equations are presented. Moreover, an interpretation of Duhamel’s formula in the theory of operator semigroups is given. Some applications of Duhamel’s formula in control systems analysis are discussed. The work is also devoted to the usage of Duhamel’s integral for differential equations with fractional order derivative.
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Authors and Affiliations

Michał Różański
1
Beata Sikora
1
ORCID: ORCID
Adrian Smuda
1
Roman Wituła
1
  1. Department of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland

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