Relative controllability of fractional dynamical systems with delays in control

Keywords: Relative controllability Time delays Distributed delays Fractional derivative Mittag–Leffler function a b s t r a c t This paper is concerned with the controllability of nonlinear fractional dynamical systems with time varying multiple delays and distributed delays in control defined in finite dimensional spaces. Sufficient conditions for controllability results are obtained using the Schau-der fixed point theorem and the controllability Grammian matrix which is defined by Mittag–Leffler matrix function. Examples are provided to illustrate the theory. Fractional order differentiation is the generalization of classical integer order differentiation. It is well known that the state of many systems (biological, electrochemical, viscoelastic, etc.) at a given time depends on their configuration at previous times. The fractional derivative takes into account this history in its definition as a convolution with a function whose amplitude decays at earlier times as a power-law. Thus, the fractional derivative is natural to use when modeling biological or adaptive systems. In Cole [16], the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by the differential equations of fractional order. A systematic presentation of the applications of fractional differential equations can be found in the book of Oldham and Spanier [29] and Sabatier et al. [31]. The subject of fractional differential equations is gaining much importance and attention. For more details, see the monographs of Kilbas et al. [22], Miller and Ross [27], Podlubny [30] and Samko et al. [32]. In consequence, there has also been a surge in the study of the theory of fractional differential equations [8–10,34]. Controllability for nonlinear dynamical systems is not so uniform and connected as in the case of linear dynamical systems. Most of the results obtained and of the controllability criteria have a local character or concern only a very narrow class of dynamical systems. The main difficulty arising in the investigation of controllability for nonlinear dynamical systems is the lack of general methods for solving nonlinear differential or functional differential equations. Fixed point technique is the most powerful method to obtain the controllability results for nonlinear dynamical systems (see, for instance [4,5]). It has also been employed successfully in controllability problems of dynamical systems with time-delay such as time varying 1007-5704/$-see front matter Ó 2012 Elsevier B.V. All rights reserved.