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1

Behaviour of fractional discrete-time consensus models with delays for summator dynamics

Girejko E., Mozyrska D., Wyrwas M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 4

403--410

EN

The leader-following consensus problem of fractional-order multi-agent discrete-time systems with delays is considered. In the systems, interactions between agents are defined like in Krause and Cucker-Smale models, but the memory is included by taking both the fractional-order discrete-time operator on the left hand side of the nonlinear systems and the delays. Since in practical problems only bounded number of delays can be considered, we study the fractional order discrete-time models with a finite number of delays. The models of opinions under consideration are investigated for single- and double-summator dynamics of discrete-time by means of analytical methods as well as computer simulations.

2

Cone solutions of multi-order fractional difference systems

Girejko E., Mozyrska D.

Control and Cybernetics

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2013

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Vol. 42, no. 2

419--429

EN

The fractional difference system of equations with different fractional orders is considered. We obtain the existence and uniqueness results for the initial value problem. Cone solutions are presented. An example is given to illustrate the results.

3

Viability and generalized differential quotients

Girejko E., Bartosiewicz Z.

Control and Cybernetics

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2006

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Vol. 35, no 4

815-829

EN

Necessary and sufficient conditions for a set-valued map K : R → Rn to be GDQ-differentiable are given. It is shown that K is GDQ differentiate at to if and only if it has a local multiselection that is Cellina continuously approximable and Lipschitz at to. It is also shown that any minimal GDQ of K at (to,yo) is a subset of the contingent derivative of K at (to,yo), evaluated at 1. Then this fact is used to prove a viability theorem that asserts existence of a solution to the initial value problem y(t) ∈ F(t, y(t)), with y(to) =yo, where F : Gr(K) → Rn is an orientor field (i.e. multivalued vector field) defined only on the graph of K and K : T → Rn is a time-varying constraint multifunction. One of the assumptions is GDQ differentiability of K.

4

On multidifferentials of multifunctions

Girejko E.

Zeszyty Naukowe Politechniki Białostockiej. Matematyka, Fizyka, Chemia

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2001

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Z. 20

23--35

EN

In this article some properties of multidiffereiitials of set-valued maps (multifunctions) are studied. The functions considered here are mostly those that are not differentiable in a classical sense. Existence of minimal multidifferentials has been proved.

PL

W artykule tym badane są pewne własności multiróżniczek funkcji wielo wartościowych (multifunkcji). Rozpatrywane funkcje nie są, zwykle różniczkowalne w klasycznym sensie. Pokazano istnienie minimalnych multiróżniczek.