Stability and synchronization of delayed fractional-order projection neural network with piecewise constant argument of mixed type.

*(English)*Zbl 1358.26012Summary: Projection equations arise in several optimization problems and possess significant applications in many areas of science and engineering. In this paper, we propose a fractional-order projection neural network to solve quadratic programming problems. We study stability and synchronization for a class of delayed projection neural networks of mixed type via impulsive control. Using concepts of fractional calculus, we investigate the existence of solution and study its global asymptotic stability. Moreover, we propose an effective impulsive control scheme to achieve synchronization for the system. We demonstrate the validity and transient behaviour of the proposed neural network with the help of suitable examples.

##### MSC:

26A33 | Fractional derivatives and integrals |

34D23 | Global stability of solutions to ordinary differential equations |

90C20 | Quadratic programming |

34D06 | Synchronization of solutions to ordinary differential equations |

##### Keywords:

fractional-order projection neural network; Mittag-Leffler stability; piecewise constant argument of generalized type; synchronization
PDF
BibTeX
XML
Cite

\textit{S. Tyagi} and \textit{S. Abbas}, Tbil. Math. J. 10, No. 1, 57--74 (2017; Zbl 1358.26012)

Full Text:
DOI

##### References:

[1] | Nocedal J, Wright S. Numerical Optimization. Springer Science & Business Media; 2006. · Zbl 1104.65059 |

[2] | Rao SS. Engineering Optimization: Theory and Practice. John Wiley & Sons; 2009. |

[3] | Bazaraa MS, Sherali HD, Shetty CM. Nonlinear programming: Theory and Algorithms. John Wiley & Sons; 2013. |

[4] | Effati S, Nazemi AR. Neural network models and its application for solving linear and quadratic programming problems. Applied Mathematics and Computation 2006; 172(1): 305-331. · Zbl 1093.65059 |

[5] | Tank DW, Hopfield JJ. Simple neural optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit. IEEE Transactions on Circuits and Systems 1986; 33(5): 533-541. |

[6] | Kennedy MP, Chua LO. Neural networks for nonlinear programming. IEEE Transactions on Circuits and Systems 1988; 35(5): 554-562. |

[7] | Hu X. Applications of the general projection neural network in solving extended linearquadratic programming problems with linear constraints. Neurocomputing 2009; 72(4): 1131-1137. |

[8] | Xia Y, Leung H, Wang J. A projection neural network and its application to constrained optimization problems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 2002; 49(4): 447-458. · Zbl 1368.92019 |

[9] | Zhang S, Constantinides AG. Lagrange programming neural networks. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 1992; 39(7): 441-452. · Zbl 0758.90067 |

[10] | Effati S, Ranjbar M. A novel recurrent nonlinear neural network for solving quadratic programming problems. Applied Mathematical Modelling 2011; 35(4): 1688-1695. · Zbl 1217.90141 |

[11] | Yang Y, Cao J, Xu X, Hu M, Gao Y. A new neural network for solving quadratic programming problems with equality and inequality constraints. Mathematics and Computers in Simulation 2014; 101: 103-112. |

[12] | Xia YS, Wang J. On the stability of globally projected dynamical systems. Journal of Optimization Theory and Applications. 2000; 106(1): 129-150. · Zbl 0971.37013 |

[13] | Hu X, Wang J. Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network. IEEE Transactions on Neural Networks 2006; 17(6): 1487-1499. |

[14] | Xia Y, Wang J. A recurrent neural network for solving linear projection equations. Neural Networks 2000; 13(3): 337-350. |

[15] | Hu X, Wang J. Design of general projection neural networks for solving monotone linear variational inequalities and linear and quadratic optimization problems. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 2007; 37(5): 1414-1421. |

[16] | Xia Y, Wang J. A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Transactions on Neural Networks 2004; 15(2): 318-328. |

[17] | Liu Q, Yang Y. Global exponential system of projection neural networks for system of generalized variational inequalities and related nonlinear minimax problems. Neurocomputing 2010; 73(10): 2069-2076. |

[18] | Xue X, Bian W. A project neural network for solving degenerate convex quadratic program. Neurocomputing 2007; 70(13): 2449-2459. |

[19] | Xue X, Bian W. A project neural network for solving degenerate quadratic minimax problem with linear constraints. Neurocomputing 2009; 72(7): 1826-1838. |

[20] | Liu Q, Cao J. Global exponential stability of discrete-time recurrent neural network for solving quadratic programming problems subject to linear constraints. Neurocomputing 2011; 74(17): 3494-3501. |

[21] | Li F. Delayed Lagrangian neural networks for solving convex programming problems. Neurocomputing 2010; 73(10): 2266-2273. |

[22] | Srivastava HM, Lin SD, Chao YT, Wang PY. Explicit solutions of a certain class of differential equations by means of fractional calculus. Russian Journal of Mathematical Physics 2007; 14(3): 357-365. |

[23] | Lin SD, Tu ST, Srivastava HM,Wang PY. Certain operators of fractional calculus and their applications to differential equations. Computers & Mathematics with Applications 2002; 44(12): 1557-1565. · Zbl 1045.34002 |

[24] | Zhang S, Yu Y, Hu W. Robust stability analysis of fractional-order Hopfield neural networks with parameter uncertainties. Mathematical Problems in Engineering 2014; Article ID 302702, 14 pages, doi: |

[25] | Wu ZB, Zou YZ. Global fractional-order projective dynamical systems. Communications in Nonlinear Science and Numerical Simulation 2014; 19(8): 2811-2819. |

[26] | Friesz TL, Bernstein D, Mehta NJ, Tobin RL, Ganjalizadeh S. Day-to-day dynamic network disequilibria and idealized traveler information systems. Operations Research 1994; 42(6): 1120-1136. · Zbl 0823.90037 |

[27] | Wu H, Shi R, Qin L, Tao F, He L. A nonlinear projection neural network for solving interval quadratic programming problems and its stability analysis. Mathematical Problems in Engineering 2010; Article ID 403749, 13 pages, doi: · Zbl 1195.90072 |

[28] | Abbas S. Existence and attractivity of k-pseudo almost automorphic sequence solution of a model of bidirectional neural networks. Acta Applicandae Mathematicae 2012; 119(1): 57-74. · Zbl 1261.39004 |

[29] | Abbas S, Yonghui X. Existence and attractivity of k-almost automorphic sequence solution of a model of cellular neural networks with delay. Acta Mathematica Scientia 2013; 33(1): 290-302. · Zbl 1289.39027 |

[30] | Abbas S. Pseudo almost periodic sequence solutions of discrete time cellular neural networks. Nonlinear Analysis, Modeling and Control 2009; 14(3): 283-301. · Zbl 1300.93021 |

[31] | Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their applications. SIAM; 1980. · Zbl 0457.35001 |

[32] | Miller RK, Michel RK. Ordinary Differential Equations. Academic Press; 1982. |

[33] | Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Elsevier Science Limited; 2006. |

[34] | Chen J, Zeng Z, Jiang P. Global Mittag-Leffler stability and synchronization of memristorbased fractional-order neural networks. Neural Networks 2014; 51: 1-8. · Zbl 1306.34006 |

[35] | Stamova I. Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays. Nonlinear Dynamics 2014; 77(4): 1251-1260. · Zbl 1331.93102 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.