Controllability of Fractional-Order Directed Complex Networks, with Self Loop and Double Edge Structure

Abstract

For that the conclusion of maximum matching is an important basic theory for controllability of complex networks, we¯rst study the validity of maximum matching for fractional-order directed complex networks. We also develop a new analytical tool to study the controllability of an arbitrary fractional-order directed complex directed network with self loop by identifying the set of driver nodes with time-dependent control that can guide the system's entire dynamics. Through analyzing a mass of typical examples, we propose a new theory named \variant maximum matching" which is superior to the old one. Finally, we present some typical examples to prove the correctness of our conclusions.

Controllability of Fractional Complex Networks

“…The nonlinear network (27) can be recast as follows: 15) is controllable, then network (27) with the Laplacian matrix can be controlled over interval I.…”

“…2024, 1, 0 9 of 16 Theorem 3. If A is replaced by −L, conditions (A 1 ) − (A 3 ) are true and system (15) is controllable, then network (27) with the Laplacian matrix can be controlled over interval I.…”

“…Some papers have addressed the controllability of fractional systems [21][22][23][24][25], but only a few papers have delved into the controllability of fractional complex networks because of their inherent complexity and long memory characteristics. Zhang et al [26] examined the controllability of linear fractional directed complex networks, and subsequently, they [27] extended this work to explore the controllability of linear fractional dynamical networks with specific topological structures. However, given the ubiquitous nature of nonlinearity, it is imperative to consider the control of fractional complex networks with nonlinear dynamics.…”

Controllability of Flow‐Conservation Transportation Networks with Fractional‐Order Dynamics

“…e problem of stability with fractional-order system, for example, has been studied in [11,13]. In [18,19], the controllability of complex networks is extended from integer order to fractional order, which shows that the control theory is applicable to fractional-order complex networks. In reality, many dynamic systems cannot be accurately described by a linear system, so the study of nonlinear systems is indispensable.…”

“…e numerical approximation for the expansion of Caputo fractional-order nonlinear systems was studied in [16,17]. Controllability is one of the important issues in the study of fractional-order systems; the controllability of Caputo fractional-order complex networks is discussed in [18][19][20]. Transportation networks are typical complex networks, so we introduce the Caputo fractional derivative to describe the dynamics of the fractional-order transportation networks.…”