Abstract
In this paper, we investigate a class of fractional-order time-varying delays gene regulatory networks with structured uncertainties and controllers (DFGRNs). Our contributions lie in three aspects: first, a necessary and sufficient condition on the existence of the solution for the DFGRNs is given by using the properties of the Riemann–Liouville fractional derivative and Caputo’s fractional derivative; second, the unique solution of the DFGRNs is proved under given initial function and certain condition; third, some novel sufficient conditions on finite-time stability of the DFGRNs are established by using a generalized Gronwall inequality and norm technique, and some conclusions on the finite-time stability of the DFGRNs with memory state-feedback controllers are reached, and those conditions and conclusions depend on the fractional order of the DFGRNs. One of the most interesting findings is that the “estimated time” of the finite-time stability is indeed related to the structured uncertainties, state-feedback controllers, time delays, and the fractional order of the system.
Highlights
As time delays often change with time and their precise measurement is difficult in real Genetic regulatory networks (GRNs), the dynamics of fractional-order linear and nonlinear systems with time-varying delays has attracted increasing interest, and the results show that it is naturally of better practical significance than those with constant delays [21,22,23,24,25]
In order to investigate the effects of structured uncertainties, controllers and time delays on the stability of the DFGRNs, we calculate the “estimated time” Te of finite-time stability for above four examples and the corresponding fractional-order gene regulatory networks without time delays (FGRNs) with different fractional-order q; the results are shown in Tables 1 and 2, respectively
It should be pointed out that the conditions of finitetime stability in the present paper are dependent on the fractional-order q, which is more different from the previous stability results for the case of integer order, i.e., the finite-time stability is independent of the integer order
Summary
Genetic regulatory networks (GRNs), which describe the interaction functions in gene expressions between DNAs, RNAs, proteins, and small molecules in an organism, are fundamental and important biological networks. e analysis and control of GRNs involve two aspects: first, understanding the widespread phenomena in living organisms and providing potential routes to prolong life span, cure cancer and diabetes, and so on; second, potential application of GRNs in the development of related disciplines, such as synthetic biology, network medicine, and personalized medicine [1,2,3,4,5]. In [28], the authors studied the robust stabilization and state-feedback controller design for a class of integer-order GRNs with time-varying delays (DGRNs) and structured uncertainties and established some delay-dependent stability results by using some matrix techniques. Phat and anh [23] established some new sufficient conditions of robust finite-time stability for a class of nonlinear fractional-order differential systems with time-varying delays. The discussions on the existence and uniqueness of the solutions and the finite-time stability results for the fractional-order uncertain GRNs with timevarying delays and controllers seem rare. We focus on the existence and uniqueness of the solution and the finite-time stability for a class of DFGRNs with structured uncertainties and controllers.
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