# Positive solutions and exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays

- 140 Downloads

## Abstract

This paper is concerned with positive solutions and global exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays. By utilizing the comparison principle via differential inequalities, we first explore conditions on damping coefficients and self-excitation coefficients to ensure that, with nonnegative connection weights and inputs, all state trajectories of the system initiating in an admissible set of initial conditions are always nonnegative. Then, based on the method of using homeomorphisms, we derive conditions in terms of linear programming problems via M-matrices for the existence, uniqueness, and global exponential stability of a positive equilibrium of the system. Two examples with numerical simulations are given to illustrate the effectiveness of the obtained results.

## Keywords

Inertial neural networks Positive equilibrium Exponential stability Time-varying delay## Notes

### Compliance with ethical standards

### Conflicts of interest

The authors declare that no potential conflict of interest to be reported to this work.

## References

- 1.Soulié FF, Gallinari P (1998) Industrial applications of neural networks. World Scientific Publishing, SingaporeCrossRefGoogle Scholar
- 2.Venketesh P, Venkatesan R (2009) A survey on applications of neural networks and evolutionary techniques in web caching. IETE Tech Rev 26:171–180CrossRefGoogle Scholar
- 3.Mrugalski M, Luzar M, Pazera M, Witczak M, Aubrun C (2016) Neural network-based robust actuator fault diagnosis for a non-linear multi-tank system. ISA Trans 61:318–328CrossRefGoogle Scholar
- 4.Witczak P, Patan K, Witczak M, Mrugalski M (2017) A neural network approach to simultaneous state and actuator fault estimation under unknown input decoupling. Neurocomputing 250:65–75CrossRefGoogle Scholar
- 5.Kiakojoori S, Khorasani K (2016) Dynamic neural networks for gas turbine engine degradation prediction, health monitoring and prognosis. Neural Comput Appl 27:2157–2192CrossRefGoogle Scholar
- 6.Gong M, Zhao J, Liu J, Miao Q, Jiao J (2016) Change detection in synthesis aperture radar images based on deep neural networks. IEEE Trans Neural Netw Learn Syst 27:125–138MathSciNetCrossRefGoogle Scholar
- 7.Baldi P, Atiya AF (1995) How delays affect neural dynamics and learning. IEEE Trans Neural Netw 5:612–621CrossRefGoogle Scholar
- 8.Lu H (2012) Chaotic attractors in delayed neural networks. Phys Lett A 298:109–116CrossRefGoogle Scholar
- 9.Zhang H, Wang Z, Liu D (2014) A comprehensive review of stability analysis of continuous-time recurrent neural networks. IEEE Trans Neural Netw Learn Syst 25:1229–1262CrossRefGoogle Scholar
- 10.Liu B (2015) Pseudo almost periodic solutions for CNNs with continuously distributed leakage delays. Neural Process Lett 42:233–256CrossRefGoogle Scholar
- 11.Arik S (2016) Dynamical analysis of uncertain neural networks with multiple time delays. Int J Syst Sci 47:730–739MathSciNetCrossRefGoogle Scholar
- 12.Li L, Yang YQ, Lin G (2016) The stabilization of BAM neural networks with time-varying delays in the leakage terms via sampled-data control. Neural Comput Appl 27:447–457CrossRefGoogle Scholar
- 13.Liu B (2017) Global exponential convergence of non-autonomous SICNNs with multi-proportional delays. Neural Comput Appl 28:1927–1931CrossRefGoogle Scholar
- 14.Manivannan R, Samidurai R, Sriraman R (2017) An improved delay-partitioning approach to stability criteria for generalized neural networks with interval time-varying delays. Neural Comput Appl 28:3353–3369CrossRefGoogle Scholar
- 15.Hai-An LD, Hien LV, Loan TT (2017) Exponential stability of non-autonomous neural networks with heterogeneous time-varying delays and destabilizing impulses. Vietnam J Math 45:425–440MathSciNetCrossRefGoogle Scholar
- 16.Lee TH, Trinh MH, Park JH (2017) Stability analysis of neural networks with time-varying delay by constructing novel Lyapunov functionals. IEEE Trans Neural Netw Learning Syst. https://doi.org/10.1109/TNNLS.2017.2760979 CrossRefGoogle Scholar
- 17.Wheeler DW, Schieve WC (1997) Stability and chaos in an inertial two neuron system. Phys D Nonlin Phenom 105:267–284CrossRefGoogle Scholar
- 18.Koch C (1984) Cable theory in neurons with active linearized membrane. Biol Cybern 50:15–33CrossRefGoogle Scholar
- 19.Babcock KL, Westervelt RM (1986) Stability and dynamics of simple electronic neural networks with added inertia. Phys D Nonlin Phenom 23:464–469CrossRefGoogle Scholar
- 20.Tu Z, Cao J, Hayat T (2016) Matrix measure based dissipativity analysis for inertial delayed uncertain neural networks. Neural Netw 75:47–55CrossRefGoogle Scholar
- 21.Wan P, Jian J (2017) Global convergence analysis of impulsive inertial neural networks with time-varying delays. Neurocomputing 245:68–76CrossRefGoogle Scholar
- 22.Tu Z, Cao J, Alsaedi A, Alsaadi F (2017) Global dissipativity of memristor-based neutral type inertial neural networks. Neural Netw 88:125–133CrossRefGoogle Scholar
- 23.Zhang G, Zeng Z, Hu J (2018) New results on global exponential dissipativity analysis of memristive inertial neural networks with distributed time-varying delays. Neural Netw 97:183–191CrossRefGoogle Scholar
- 24.Ke Y, Miao C (2013) Stability analysis of inertial Cohen–Grossberg-type neural networks with time delays. Neurocomputing 117:196–205CrossRefGoogle Scholar
- 25.Zhang Z, Quan Z (2015) Global exponential stability via inequality technique for inertial BAM neural networks with time delays. Neurocomputing 151:1316–1326CrossRefGoogle Scholar
- 26.Cui N, Jiang H, Hu C, Abdurahman A (2018) Global asymptotic and robust stability of inertial neural networks with proportional delays. Neurocomputing 272:326–333CrossRefGoogle Scholar
- 27.Tu Z, Cao J, Hayat T (2016) Global exponential stability in Lagrange sense for inertial neural networks with time-varying delays. Neurocomputing 171:524–531CrossRefGoogle Scholar
- 28.Wang J, Tian L (2017) Global Lagrange stability for inertial neural networks with mixed time-varying delays. Neurocomputing 235:140–146CrossRefGoogle Scholar
- 29.He X, Huang TW, Yu JZ, Li CD, Li CJ (2017) An inertial projection neural network for solving variational inequalities. IEEE Trans Cybern 47:809–814CrossRefGoogle Scholar
- 30.Smith H (2008) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. American Mathematical Society, ProvidenceCrossRefGoogle Scholar
- 31.Mózaryn J, Kurek JE (2010) Design of a neural network for an identification of a robot model with a positive definite inertia matrix. In: Artificial Intelligence and Soft Computing. Springer, BerlinGoogle Scholar
- 32.Ma GJ, Wu S, Cai GQ (2013) Neural networks control of the Ni-MH power battery positive mill thickness. Appl Mech Mater 411–414:1855–1858CrossRefGoogle Scholar
- 33.Lu W, Chen T (2007) \(R^n_+\)-global stability of a Cohen–Grossberg neural network system with nonnegative equilibria. Neural Netw 20:714–722Google Scholar
- 34.Liu B, Huang L (2008) Positive almost periodic solutions for recurrent neural networks. Nonlinear Anal Real World Appl 9:830–841MathSciNetCrossRefGoogle Scholar
- 35.Hien LV (2017) On global exponential stability of positive neural networks with time-varying delay. Neural Netw 87:22–2617CrossRefGoogle Scholar
- 36.He Y, Ji MD, Zhang CK, Wu M (2016) Global exponential stability of neural networks with time-varying delay based on free-matrix-based integral inequality. Neural Netw 77:80–86CrossRefGoogle Scholar
- 37.Arino O, Hbid ML, Ait Dads E (2002) Delay differential equations and applications. Springer, DordrechtGoogle Scholar
- 38.Forti M, Tesi A (1995) New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE Trans Circuits Syst-I: Fund 42:354–366MathSciNetCrossRefGoogle Scholar
- 39.Hien LV, Son DT (2015) Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Appl Math Comput 251:14–23MathSciNetzbMATHGoogle Scholar
- 40.Haykin S (1999) Neural networks: a comprehensive foundation. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar