Zapraszamy na seminarium „Informatyka Stosowana i Modelowanie Matematyczne", które odbędzie się w czwartek 9 czerwca 2022 r. o godz. 13:15 w sali E1/10.
Prelegent: Ivan Matychyn
Tytuł referatu: Game Problems for Fractional-Order Nonstationary Systems
Fractional differential equations (FDEs) provide a powerful tool to describe memory effect and hereditary properties of various materials and processes. While linear systems of FDEs represent a fairly well investigated field of research, relatively few papers deal with nonstationary fractional-order systems described by linear FDEs with variable coefficients. Meanwhile, a number of real-life systems and processes can be described by linear FDEs with variable coefficients, e.g. linearized aircraft models, linearized models of population restricted growth, models related to the distribution of parameters in the charge transfer and the diffusion of the batteries etc.
Linear differential equations with variable coefficients arise in a natural way when modeling RLC-circuits with variable capacitance or inductance. With the advent of electronic components like super-capacitors (also called ultracapacitors) and fractances, one should employ fractional differential equations for circuit models. This provides motivation for research on FDEs with variable coefficients as well as related control and game problems.
Explicit solutions to linear systems of differential equations provide basis to perform stability analysis and to solve control and game-theoretical problems. Explicit solutions to linear systems of differential equations are usually expressed in terms of state transition matrix. In the case of FDEs with constant coefficients the state transition matrix can be represented using the matrix Mittag-Leffler function. Recently explicit solutions for the linear systems of FDEs with variable coefficients were obtained in terms of generalized Peano--Baker series.
Differential games represent a special topic of the control theory. Differential games described by the systems of linear FDEs with variable coefficients involving Riemann--Liouville and Caputo derivatives are treated from convex-analytical viewpoint using the method of resolving functions. On the basis of the resolving functions method sufficient conditions for the finite-time game termination from given initial states are derived. Theoretical results are supported by an illustrative example.
Mariusz Bodzioch, Aleksander Denisiuk, Mikhail Kolev, Ivan Matychyn
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