Małgorzata Wyrwas

The -Transform Method and Delta Type Fractional Difference Operators

The Caputo-, Riemann-Liouville-, and Grunwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classical -transform method. We stress the formula for the image of the discrete Mittag-Leffler matrix function in the -transform. We also prove forms of images in the -transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.

Transfer Equivalence and Realization of Nonlinear Input-Output Delta-Differential Equations on Homogeneous Time Scales

Nonlinear control systems on homogeneous time scales are studied. First the concepts of reduction and irreducibility are extended to higher order delta-differential input-output equations. Subsequently, a definition of system equivalence is introduced which generalizes the notion of transfer equivalence in the linear case. Finally, the necessary and sufficient conditions are given for the existence of a state-space realization of a nonlinear input-output delta-differential equation.

Control systems on regular time scales and their differential rings

The paper describes an algebraic construction of the inversive differential ring, associated with a nonlinear control system, defined on a nonhomogeneous but regular time scale. The ring of meromorphic functions in system variables is constructed under the assumption that the system is submersive, and equipped with three operators (delta- and nabla-derivatives, and the forward shift operator) whose properties are studied. The formalism developed unifies the existing theories for continuous- and discrete-time nonlinear systems, and accommodates also the case of non-uniformly sampled systems. Compared with the homogeneous case the main difficulties are noncommutativity of delta (nabla) derivative and shift operators and the fact that the additional time variable t appears in the definition of the differential ring. The latter yields that the new variables of the inversive closure, depending on t, have to be chosen to be smooth at each dense point t of the time scale.

Comparison of h-Difference Fractional Operators

We compare three different types of h-difference fractional operators: Grunwald-Letnikov, Caputo, Riemann-Liouville types of operators. There is introduced the formula for fundamental matrix of solutions for linear systems of h-difference fractional equations with Grunwald-Letnikov type operator while the one with Caputo type or Riemann-Liouville type is well known. We present new formulas for linear control systems with the mentioned operators.

Fractional nonlinear systems with sequential operators

In the paper possible approximation of solutions to initial value problems stated for fractional nonlinear equations with sequential derivatives of Caputo type is presented. We proved that values of Caputo derivatives in continuous case can be approximated by corresponding values of h-difference operators with h being small enough. Numerical examples are presented.

Stability of nonlinear

In the paper we study the subject of stability of systems with

h

h

-difference systems with

-differences of Caputo-, Riemann-Liouville- and Grunwald-Letnikov-type with

n

n

fractional orders

fractional orders. The equivalent descriptions of fractional