Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays
Abstract
:1. Introduction
- (1)
- An impulsive control strategy is considered for a class of fractional-order GRNs with time-varying delays;
- (2)
- The almost periodicity notion is introduced to the model under consideration which initiates the development of the almost periodicity theory for impulsive fractional GRNs;
- (3)
- New existence and uniqueness results for the almost periodic states are established;
- (4)
- New criteria for global Mittag–Leffler stability of an almost periodic state of the impulsive model under consideration are also proved;
- (5)
- We apply an extended Lyapunov function approach which allows representing the obtained results in terms of the model’s parameters, and leads to a better exploration of the impulsive effect.
2. The Fractional-Order Impulsive Delayed GRN Model. Preliminaries
- P1.
- P2.
- .
- P3.
- Ref. [51]
- , , ;
- The parameters are the same as in (1);
- The functions of regulation , are the same as in (1) and satisfy
- The translation rates and basal rates are extended to functions and , respectively, , , where is the set of all the j which are repressors of the gene i;
- The functions are represented as:
- The time-varying delays and are different for different mRNA and protein molecules, respectively, and satisfy and ( = const), , ( = const), ;
- are the impulsive moments (impulsive control instants) and , where is the set of all sequences of the type
- and denote the i-th mRNA concentration and i-th protein concentration at time , respectively, and and denote the level of the i-th mRNA concentration and i-th protein concentration, respectively, at , i.e., after an impulsive short-term effect on them at ;
- The impulsive functions and denote the amounts of the abrupt variation in and , respectively, at the impulsive instants , i.e., and , , .
- A1.
- A2.
- The sequences of functions and , , are almost periodic in the sense of Bohr;
- A3.
- The initial functions and are almost periodic;
- A4.
- The set of sequences , , is UAP.
- (i)
- ;
- (ii)
- The inequality
3. Fractional Order Almost Periodicity Theorems
- (i)
- There exists a real number
- (ii)
- The functions , , , where , and are positive constants and the constants , , are such that
- (iii)
- A solution of the model (2) exists such that
- (a)
- ;
- (b)
- , where is the hull of the almost periodic solution .
4. A Numerical Example
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
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Stamova, I.; Stamov, G. Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays. Fractal Fract. 2021, 5, 268. https://doi.org/10.3390/fractalfract5040268
Stamova I, Stamov G. Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays. Fractal and Fractional. 2021; 5(4):268. https://doi.org/10.3390/fractalfract5040268
Chicago/Turabian StyleStamova, Ivanka, and Gani Stamov. 2021. "Lyapunov Approach for Almost Periodicity in Impulsive Gene Regulatory Networks of Fractional Order with Time-Varying Delays" Fractal and Fractional 5, no. 4: 268. https://doi.org/10.3390/fractalfract5040268