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1
Content available Remarks on the Caputo fractional derivative
Sikora Beata
MINUT Matematyka i Informatyka na Uczelniach Technicznych
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2023
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Nr 5
76-84
EN
The purpose of the paper is to familiarise the reader with the concept of the Caputo fractional derivative. The definition and basic properties of the Caputo derivative are given. Formulas for the derivatives of selected functions are derived. Examples of calculating the derivatives of basic functions are presented. The paper also contains a number of self-solving exercises, with answers.
2
Content available On the constrained and unconstrained controllability of semilinear Hilfer fractional systems
Sikora Beata
Archives of Control Sciences
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2023
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Vol. 33, No. 1
155--178
EN
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.
3
Content available On theoretical and practical aspects of Duhamel’s integral
Różański Michał, Sikora Beata, Smuda Adrian, Wituła Roman
Archives of Control Sciences
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2021
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Vol. 31, No. 4
815--847
EN
The paper is a new approach to the Duhamel integral. It contains an overview of 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.
4
Content available On controllability of fractional positive continuous-time linear systems with delay
Sikora Beata, Matlok Natalia
Archives of Control Sciences
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2021
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Vol. 31, No. 1
29--51
EN
In the paper positive fractional continuous-time linear systems are considered. Positive fractional systems without delays and positive fractional systems with a single delay in control are studied. New criteria for approximate and exact controllability of systems without delays as well as a relative controllability criterion of systems with delay are established and proved. Numerical examples are presented for different controllability criteria. A practical application is proposed.
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