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37

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## Publications

Publications (37)

The paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville derivatives. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are su...

This work was inspired by an example of a pursuit strategy whereby a pursuer approaches an evader while appearing stationary to the latter. This effect is achieved due to the fact that the pursuer P stays on the line connecting some fixed reference point R and current position of the evader E. According to recent researches of biologists (Mizutani...

Matrix Mittag-Leffler functions play a key role in numerous applications related to systems with fractional dynamics. That is why the methods for computing the matrix Mittag-Leffler function are so important. The matrix Mittag-Leffler function is a generalization of matrix exponential function. This implies that some of numerous existing methods fo...

Problem of time-optimal control of linear systems with fractional Caputo derivatives is examined using technique of attainability sets and their support functions.
A method to construct a control function that brings trajectory of the system to a strictly convex terminal set in the shortest time is elaborated. The proposed method uses technique of...

The collection of conference materials covers a wide range of theoretical and applied problems of socio-economic, technical-technological, information-analytical, socio-philosophical and educational support of Ukraine's transition to sustainable development, taking into account modern transformation processes. Particular attention is paid to the pr...

Linear systems described by fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives are examined in the paper. For these systems, a solution of the initial-value problem is derived in terms of the generalized Peano–Baker series and a time-optimal control problem is formulated. The optim...

This paper deals with the initial value problem for linear systems of fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives. Some basic properties of fractional derivatives and antiderivatives, including their non-symmetry w.r.t. each other, are discussed. The technique of the general...

Optimal control problem for linear systems of arbitrary fractional order in the sense of Riemann–Liouville is treated in the paper. The technique of attainability sets and their support functions is used to obtain sufficient conditions for time-optimal control similar to that of Pontryagin’s maximum principle. Theoretical results are supported by e...

Fractional optimal control problem is treated from convex-analytical viewpoint. Sufficient conditions for time-optimal control similar to that of Pontryagin's maximum principle are obtained for fractional-order systems in the sense of Riemann-Liouville and Caputo. Theoretical results are supported by examples.

A method for computation of the matrix Mittag-Leffler function is presented. The method is based on Jordan canonical form and implemented as a Matlab routine.

In 1993, Samko and Ross introduced the study of fractional integration and differentiation when the order is not a constant but a function. This suggestion gave rise to a number of further ideas and results. In particular, this implies a possibility of integration with respect to derivative's order. Here an identity is presented, in which an expres...

Problem of optimal operation speed for linear systems with fractional derivatives in the Riemann-Liouville sense is under consideration. By means of the methods of convex analysis and theory of multivalued mappings we constructed optimal controls and obtained sufficient conditions of optimality. Theoretical results are shown in practice.

The problem of time-optimal control of linear systems with fractional dynamics is treated in the paper from the convex-analytic standpoint. A linear system of fractional differential equations involving Riemann- Liouville derivatives is considered. A method to construct a control function that brings trajectory of the system to the terminal state i...

The paper deals with a game problem of approaching cylindrical terminal set for a dynamic system described by fractional differential equations subject to impulse effect at fixed time instants. By resolving functions method there are obtained the sufficient conditions for a terminal set approaching over a finite time. Theoretical results are illust...

Problem of time-optimal control of linear systems with fractional dynamics is
treated in the paper from the convex-analytic standpoint. A linear system of
fractional differential equations involving Riemann--Liouville derivatives is
considered. A method to construct a control function that brings trajectory of
the system to the terminal state in th...

Here we investigate a problem of approaching terminal (target) set by a system of impulse differential equations of fractional order in the sense of Caputo. The system is under control of two players pursuing opposite goals. The first player tries to bring the trajectory of the system to the terminal set in the shortest time, whereas the second pla...

There exists a number of definitions of the fractional order derivative. The classical one is the definition by Riemann–Liouville
[26, 19, 23, 21]. The Riemann–Liouville fractional derivatives have a singularity at zero. That is why differential equations
involving these derivatives require initial conditions of special form lacking clear physical...

The paper concerns game problems for controlled systems with arbitrary Riemann–Liouville fractional derivatives, regularized Dzhrbashyan-Caputo derivatives, and sequential Miller-Ross derivatives. Under fixed controls of players, solution to such systems is presented in the form of a Cauchy formula analog. On the basis of the method of resolving fu...

The paper is concerned with studying approach game problems for linear conflict-controlled processes with fractional derivatives
of arbitrary order. Namely, the classical Riemann-Liouville fractional derivatives, Dzhrbashyan-Nersesyan or Caputo regularized
derivatives, and Miller-Ross sequential derivatives are considered. Under fixed controls of t...

Game approach problems for linear conflict controlled processes with fractional derivatives of arbitrary order are studied. The classical fractional Riemann-Liouville derivatives, regularized Dzhrbashyan-Nersesyan or Caputo derivatives as well as sequential Miller-Ross derivatives are considered. For the fixed controls of players, the solutions in...

Consideration is given to inhomogeneous linear systems of differential equations with classical Riemann-Liouville fractional derivatives as well as the regularized Caputo derivative and sequential Miller-Ross fractional derivatives. Using Laplace transform the solutions of such systems are presented in terms of matrix Mittag-Leffler functions as an...

This chapter deals with the pursuit games in which players (pursuer, evader, or both) employ impulse control.We consider continuous-time
dynamical systems modeled by ordinary differential equations that are affected by jumps in state at discrete instants. The
moments of jump comply with the condition for a finite number of jumps in finite time. In...

Summary: Non-homogeneous linear systems of differential equations with classical Riemann-Liouville fractional derivatives, as well as regularized Caputo's fractional derivatives, are considered. Using the Laplace transform, the solutions to such systems are represented in the form of analogues of the Cauchy formula for arbitrary measurable and boun...

We consider pursuit games in which players (the pursuer, evader or both players) use impulse controls, which is expressed in terms of the Dirac delta-function. We study linear dynamical system described by ordinary differential equations whose trajectories have discontinuities at discrete time instants. Such systems are a kind of hybrid ones. The r...

Motion camouflage is a pursuit strategy whereby a pursuer approaches an evader while appearing stationary to the evader. Such strategy has been observed in insects by biological researchers. Here, a method is proposed, providing rigorous mathematical substantiation for the motion camouflage strategy. This method is based on some ideas of the method...

Linear differential games with pulse control of players, which is represented by the Dirac delta-function, are analyzed. For this class of games, the sufficient conditions for the solvability of the approach problem are derived. The problems are analyzed based on the method of resolving functions. The result is exemplified by a pursuit game with a...

Summary: Differential games with impulse control for the evader that is represented by the Dirac measure are treated. For this class of games, sufficient conditions for the solvability of the approach problem are derived. The basis for the research of these problems is the method of resolving functions. The result is supported by a model example wi...

The pursuit problem for differential game with simple motion is considered under different assumptions about information distribution of players. It is shown that classical methods of parallel and pure pursuit can be obtained with application of the unified approach within the framework of the method of resolving functions.

Pursuit problem for differential game with simple motion is considered under different assumptions regarding information available to the players. It is demonstrated that such classical pursuit methods as parallel and pure pursuit method can be derived using a single approach, in the framework of resolving function method. It is shown that for the...

We consider quasi-linear differential variable structure games with a finite number of s witching. Using the method of resolving functions, we found sufficient conditions for solution of the approach problem. A model example illustrates the result.

A dynamic process with a variable structure is considered under game conditions. The sufficient conditions of resolvability of the approach problem are derived based on the method of resolving functions. The results are illustrated by a model example.

## Questions

Question (1)

The author obtains interesting results that seem to be groundbreaking. However they are based on the identity $E_\alpha(x^\alpha)E_\alpha(y^\alpha)=E_\alpha((x+y)^\alpha)$ (Equation(2.21)), which is similar to the basic exponentiation identity $e^xe^y=e^(x+y)$. Some other authors (Guy Jumarie) also use this identity and prove it using the fractional differentiation formulas similar to the product rule and chain rule, which are generally speaking not true. On the other hand, it can be easily verified using the Matlab routine for calculating Mittag-Leffler function (http://www.mathworks.com/matlabcentral/fileexchange/8738) that the identity in question does not hold for arbitrary alpha. Can anyone provide feasible proof of the identity or otherwise clarify the issue? Thank you.

## Projects

Project (1)

We invite you to contribute to the Special Issue “Fractional Differential Equations and Control Problems” of Mathematics (ISSN 2227-7390, IF 1.747). The journal is indexed in the Science Citation Indexed Expanded - SCIE (Web of Science) and Scopus. This Special Issue is aimed at presenting the recent developments in the theory and applications of any types of fractional differential equations and inclusions, with a special emphasis on control problems for fractional ordinary and partial differential equations.