Dorota Mozyrska

Modified optimal energy and initial memory of fractional continuous-time linear systems

Fractional systems with Riemann-Liouville derivatives are considered. The initial memory value problem is posed and studied. We obtain explicit steering laws with respect to the values of the fractional integrals of the state variables. The Gramian is generalized and steering functions between memory values are characterized.

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.

Overview of fractional h-difference operators

Fractional difference operators and their properties are discussed. We give a characterization of three operators that we call Grunwald-Letnikov, Riemann-Liouville and Caputo like difference operators. We show relations among them. In the paper, linear fractional h-difference equations are described. We give formulas of solutions to initial value problems. Crucial formulas are gathered in the tables presented in the last section of the paper.

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.

Multiparameter Fractional Difference Linear Control Systems

The Riemann-Liouville-, Caputo-, and Grunwald-Letnikov-type fractional order difference operators are discussed and used to state and solve the controllability and observability problems of linear fractional order discrete-time control systems with multiorder and multistep. It is shown that the obtained results do not depend on the type of fractional operators and steps. The comparison of systems is made under the number of steps needed, firstly to achieve a final point, and secondly to distinguish initial conditions for particular operator.

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.

Fractional discrete-time linear control systems with initialisation

Linear control systems with fractional differences and finite memory are considered. The evaluation of trajectories of such systems is studied. The extension of the classical notion of observability and controllability is given and developed. The necessary and sufficient conditions for observability and controllability are proved.

Constrained Controllability of h-Difference Linear Systems with Two Fractional Orders

The problem of controllability in finite number of steps with control constrains of h-difference linear control systems with two fractional orders is studied. There are considered systems with the Caputo type h-difference operators and with controls which values are from a given convex and bounded subset of the control space. Necessary and sufficient conditions for constrained controllability in finite number of steps are given.

Fractional discrete-time of HegselmannKrause's type consensus model with numerical simulations

The leader-following consensus problem of fractional-order multi-agent discrete-time systems is considered. In the system, interactions between opinions are defined like in the classical HegselmannKrause models but the memory is included by taking the fractional-order discrete-time operator on the left-hand side of the nonlinear system. In the paper we investigate various models for the dynamics of discrete-time fractional order opinions by analytical methods as well as by computer simulations.

Stability of discrete fractional-order nonlinear systems with the nabla Caputo difference

Abstract In this paper, using the Lyapunovs direct method, the stability of discrete non-autonomous systems with the nabla Caputo fractional difference is studied. The conditions for different kind of stability are presented.