Controllability and Stability of Semilinear Fractional Order Systems

Abstract

In the chapter two the most important properties of fractional order dynamical systems, namely, controllability and stability are presented. At the beginning the basic notations and the fundamental definitions are recalled. The first part of the chapter is devoted to controllability and contains the formulation of the problem, main hypotheses and theorems about controllability of semilinear fractional order systems with distributed and point multiplicities constant or variable delays in the state variables and controls. Next, using fixed point theorems approximate controllability problem in infinite dimensional spaces, in particular Banach or Hilbert space, are discussed. Second part of the chapter is devoted to the stability problem of fractional order systems. The problem of stability and the problem of the existence of solutions for linear and nonlinear fractional order systems are also presented.

Mittag-Leffler stability for a Timoshenko problem

“…We refer also to Anderson et al (2018). In the survey paper by Klamka et al (2020), the authors presented the most important results on the controllability and stability of some fractional differential equations of an order between zero and one (see also the work of Kaczorek (2020) and the references therein). The importance of these questions and the difficulties encountered in their treatments are highlighted.…”