Exponential stability criteria for linear neutral systems with applications to neural networks of neutral type. (English) Zbl 1506.93074
Summary: Linear neutral vector equations
\[
\dot{x}(t)=A_0(t)\dot{x}(h_0(t))+\sum\limits_{k=1}^mA_k(t)x(h_k(t))+\int_{g(t)}^tP(t,s)x(s)ds
\]
are considered on interval \([0,\infty)\). Here \(x=(x_1,\dots,x_n)^T\), \(m\) is a positive integer, the entries of matrices \(A_l\), \(l=0,\dots,m,P\), and the delays \(h_k\), \(k=0,\dots,m,g\) are assumed to be Lebesgue measurable functions. New explicit criteria are derived on uniform exponential stability. Comparisons are made and discussed based on an overview of the existing results. An application is presented to local exponential stability of non-autonomous neural network models of neutral type.
MSC:
| 93D23 | Exponential stability |
| 93B70 | Networked control |
| 93C05 | Linear systems in control theory |
| 26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |
References:
| [1] | Arik, S., A modified Lyapunov functional with application to stability of neutral-type neural networks with time delays, J. Frankl. Inst., 356, 276-291 (2019) · Zbl 1405.93188 |
| [2] | Arik, S., New criteria for stability of neutral-type neural networks with multiple time delays, IEEE Trans. Neural Netw. Learn. Syst., 31, 1504-1513 (2020) |
| [3] | English translation: Differential Equations 13 (1977) 1331-1339 (1978) · Zbl 0396.34005 |
| [4] | Azbelev, N.; Maksimov, V.; Rakhmatullina, L., Introduction to the theory of linear functional differential equations, Advanced Series in Mathematical Science and Engineering, vol. 3 (1995), World Federation Publishers Company: World Federation Publishers Company Atlanta, GA · Zbl 0867.34051 |
| [5] | Azbelev, N. V.; Simonov, P. M., Stability of differential equations with aftereffect, Stability and Control: Theory, Methods and Applications, vol. 20 (2002), Taylor & Francis: Taylor & Francis London · Zbl 1049.34090 |
| [6] | Baker, C. T.H.; Bocharov, G. A.; Rihan, F. A., Neutral delay differential equations in the modelling of cell growth, J. Egyptian Math. Soc., 16, 133-160 (2008) · Zbl 1167.93003 |
| [7] | Balasubramaniam, P.; Rakkiyappan, R., Global exponential stability for neutral-type BAM neural networks with time-varying delays, Int. J. Comput. Math., 87, 2064-2075 (2010) · Zbl 1205.34091 |
| [8] | Art. ID 956121 · Zbl 1214.34059 |
| [9] | 124893 · Zbl 1478.34073 |
| [10] | Berezansky, L.; Braverman, E., On stability of linear neutral differential equations with variable delays, Czechoslovak Math. J., 69, 144, 863-891 (2019) · Zbl 1513.34267 |
| [11] | Berezansky, L.; Diblík, J.; Svoboda, Z.; Šmarda, Z., Simple tests for uniform exponential stability of a linear delayed vector differential equation, IEEE Trans. Autom. Control, 67, 3, 1537-1542 (2022) · Zbl 1537.93600 |
| [12] | Berezansky, L.; Diblík, J.; Svoboda, Z.; Šmarda, Z., Simple uniform exponential stability conditions for a system of linear delay differential equations, Appl. Math. Comput., 250, 605-614 (2015) · Zbl 1328.34066 |
| [13] | Berezansky, L.; Diblík, J.; Svoboda, Z.; Šmarda, Z., Uniform exponential stability of linear delayed integro-differential vector equations, J. Differ. Equ., 270, 573-595 (2021) · Zbl 1464.45021 |
| [14] | Burton, T. A., Stability by Fixed Point Theory for Functional Differential Equations (2006), Dover Publications, Inc. · Zbl 1160.34001 |
| [15] | Chukwu, E. N., Differential Models and Neutral Systems for Controlling the Wealth of Nations (2001), World Scientific · Zbl 0970.91056 |
| [16] | Chukwu, E. N., Stability and Time-Optimal Control of Hereditary Systems with Application to the Economic Dynamics of the US (2001), World Scientific · Zbl 1008.93003 |
| [17] | Coppel, W. A., Dichotomies in Stability Theory, Lecture Notes in Mathematics, vol. 629 (1978), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0376.34001 |
| [18] | Cruz, M.; Hale, J. K., Stability of functional differential equations of neutral type, J. Differ. Equ., 7, 224-355 (1970) · Zbl 0191.38901 |
| [19] | Dieudonné, J., Treatise On Analysis, Vol. II (1976), Academic Press, New York · Zbl 0315.46001 |
| [20] | Domoshnitsky, A.; Gitman, M.; Shklyar, R., Stability and estimate of solution to uncertain neutral delay systems, Bound. Value Probl., 2014, 55, 14 (2014) · Zbl 1309.34133 |
| [21] | With the assistance of W.G. Bade and R.G. Bartle |
| [22] | Faria, T.; Obaya, R.; Sanz, A. M., Asymptotic behaviour for a class of non-monotone delay differential systems with applications, J. Dyn. Differ. Equ., 30, 911-935 (2018) · Zbl 1414.34058 |
| [23] | Faria, T.; Röst, G., Persistence, permanence and global stability for an \(n\)-dimensional Nicholson system, J. Dyn. Differ. Equ., 26, 723-744 (2014) · Zbl 1316.34086 |
| [24] | Fridman, E., Introduction to time-delay systems, Analysis and Control, Systems & Control: Foundations & Applications (2014), Birkhäuser/Springer: Birkhäuser/Springer Cham · Zbl 1303.93005 |
| [25] | Gil’, M., Explicit stability conditions for neutral type vector functional differential equations. a survey, Surv. Math. Appl., 9, 1-54 (2014) · Zbl 1399.34001 |
| [26] | Gil’, M., Stability of neutral functional differential equations, Atlantis Studies in Differential Equations, vol. 3 (2014), Atlantis Press: Atlantis Press Paris · Zbl 1315.34002 |
| [27] | Gopalsamy, K., A simple stability criterion for a linear system of neutral integro-differential equations, Math. Proc. Cambridge Philos. Soc., 102, 149-162 (1987) · Zbl 0632.45001 |
| [28] | Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0752.34039 |
| [29] | Gu, K.; Kharitonov, V. L.; Chen, J., Stability of Time-Delay Systems, Control Engineering (2003), Birkhäuser, Inc.: Birkhäuser, Inc. Boston · Zbl 1039.34067 |
| [30] | Haddock, J. R.; Krisztin, T.; Terjéki, J.; Wu, J. H., An invariance principle of Lyapunov-Razumikhin type for neutral functional-differential equations, J. Differ. Equ., 107, 395-417 (1994) · Zbl 0796.34067 |
| [31] | Hadeler, K. P.; Bocharov, G., Where to put delays in population models, in particular in the neutral case, Can. Appl. Math. Q, 11, 159-173 (2003) · Zbl 1089.34065 |
| [32] | Hale, J. K.; Verduyn, L. S.M., Introduction to functional-differential equations, Applied Mathematical Sciences, vol. 99 (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002 |
| [33] | He, P.; Cao, D. Q., Algebraic stability criteria of linear neutral systems with multiple time delays, Appl. Math. Comput., 155, 643-653 (2004) · Zbl 1060.34044 |
| [34] | Hui, G. D.; Hu, G. D., Simple criteria for stability of neutral systems with multiple delays, Int. J. Syst. Sci., 28, 1325-1328 (1997) · Zbl 0899.93031 |
| [35] | Idels, L.; Kipnis, M., Stability criteria for a nonlinear nonautonomous system with delays, Appl. Math. Model., 33, 2293-2297 (2009) · Zbl 1185.74043 |
| [36] | Kolmanovskii, V.; Myshkis, A., Introduction to the theory and applications of functional-differential equations, Mathematics and its Applications, vol. 463 (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0917.34001 |
| [37] | Kolmanovskii, V. B.; Nosov, V. R., Stability of functional differential equations, Mathematics in Science and Engineering, vol. 180 (1986), Academic Press · Zbl 0593.34070 |
| [38] | Krisztin, T.; Wu, J., Asymptotic behaviors of solutions of scalar neutral functional-differential equations, Differential equations-theory, methods and applications (Hyderabad, 1995), Differ. Equ. Dyn. Syst., 4, 3-4, 351-366 (1996) · Zbl 0870.34075 |
| [39] | Kuang, Y., Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, vol. 191 (1993), Academic Press, Inc. · Zbl 0777.34002 |
| [40] | Lakshmanan, S.; Lim, C. P.; Prakash, M.; Nahavandi, S.; Balasubramaniam, P., Neutral-type of delayed inertial neural networks and their stability analysis using the LMI approach, Neurocomputing, 230, 243-250 (2017) |
| [41] | Lee, S. M.; Kwon, O. M.; Park, J. H., A novel delay-dependent criterion for delayed neural networks of neutral type, Phys. Lett. A, 374, 1843-1848 (2010) · Zbl 1236.92007 |
| [42] | Li, Y., On a periodic neutral delay Lotka-Volterra system, Nonlinear Anal., 39, 767-778 (2000) · Zbl 0943.34058 |
| [43] | Li, L.-M., Stability of linear neutral delay-differential systems, Bull. Australian Math. Soc., 38, 339-344 (1988) · Zbl 0669.34074 |
| [44] | Liu, G. Q.; Zhou, S. M.; Li, Y. Z.; Cai, T. C., Stability conditions concerning neutral-type BAM neural networks with infinite distributed delay, Int. J. Comput. Math., 98, 502-516 (2021) · Zbl 1538.93032 |
| [45] | Lopes, O., Stability and forced oscillations, J. Math. Anal. Appl., 55, 686-698 (1976) · Zbl 0347.34058 |
| [46] | Manivannan, R.; Samidurai, R.; Cao, J.; Alsaedi, A.; Alsaadi, F. E., Stability analysis of interval time-varying delayed neural networks including neutral time-delay and leakage delay, Chaos, Solitons Fractals, 114, 433-445 (2018) · Zbl 1415.34113 |
| [47] | Mori, T., Criteria for asymptotic stability of linear time-delay systems, IEEE Trans. Autom. Control, 30, 158-161 (1985) · Zbl 0557.93058 |
| [48] | Ngoc, P. H.A.; Ha, Q., On exponential stability of linear non-autonomous functional differential equations of neutral type, Int. J. Control, 90, 438-446 (2017) · Zbl 1359.93401 |
| [49] | Ngoc, P. H.A.; Hieu, T., Novel criteria for exponential stability of linear neutral time-varying differential systems, IEEE Trans. Autom. Control, 61, 1590-1594 (2016) · Zbl 1359.34078 |
| [50] | Ozcan, N., Stability analysis of Cohen-Grossberg neural networks of neutral-type: multiple delays case, Neural Netw., 113, 20-27 (2019) · Zbl 1441.93232 |
| [51] | Park, J. H.; Kwon, O. M., Global stability for neural networks of neutral-type with interval time-varying delays, Chaos, Solitons Fractals, 41, 3, 1174-1181 (2009) · Zbl 1198.34158 |
| [52] | Popa, C.. A., Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays, Neurocomputing, 309, 117-133 (2018) |
| [53] | Shatyrko, A.; van Nooijen, R. R.P.; Kolechkina, A.; Khusainov, D., Stabilization of neutral-type indirect control systems to absolute stability state, Adv. Differ. Equ., 2015, 64, 18 (2015) · Zbl 1351.34092 |
| [54] | Shi, K.; Zhu, H.; Zhong, S.; Zeng, Y.; Zhang, Y., New stability analysis for neutral type neural networks with discrete and distributed delays using a multiple integral approach, J. Frankl. Inst., 352, 155-176 (2015) · Zbl 1307.93309 |
| [55] | Sun, L., Asymptotic stability for the system of neutral delay differential equations, Appl. Math. Comput., 218, 337-345 (2011) · Zbl 1232.34101 |
| [56] | Sun, Y., On simple stability criteria for nonlinear neutral systems with multiple time delays, Appl. Math. Lett., 25, 1911-1915 (2012) · Zbl 1259.34070 |
| [57] | Wan, L.; Zhou, Q., Exponential stability of neutral-type Cohen-Grossberg neural networks with multiple time-varying delays, IEEE Access, 9, 48914-48922 (2021) |
| [58] | Wan, L.; Zhou, Q., Stability analysis of neutral-type Cohen-Grossberg neural networks with multiple time-varying delays, IEEE Access, 8, 27618-27623 (2020) |
| [59] | Wan, L.; Zhou, Q.; Fu, H.; Zhang, Q., Exponential stability of Hopfield neural networks of neutral type with multiple time-varying delays, AIMS Math., 6, 8030-8043 (2021) · Zbl 1485.34186 |
| [60] | Weera, W.; Niamsup, P., Novel delay-dependent exponential stability criteria for neutral-type neural networks with non-differentiable time-varying discrete and neutral delays, Neurocomputing, 173, 886-898 (2016) |
| [61] | Won, S.; Park, J. H., A note on the stability analysis of neutral systems with multiple time-delays, Int. J. Syst. Sci., 32, 409-412 (2001) · Zbl 1018.34071 |
| [62] | Xiong, W., New exponential convergence on SICNNs with time-varying leakage delays and neutral type distributed delays, J. Appl. Math. Comput., 49, 157-179 (2015) · Zbl 1328.34072 |
| [63] | Zahreddine, Z., Matrix measure and application to stability of matrices and interval dynamical systems, Int. J. Math. Math. Sci., 2003, 2, 75-85 (2003) · Zbl 1016.15017 |
| [64] | Zhao, Z.; Jian, J.; Wang, B., Global attracting sets for neutral-type BAM neural networks with time-varying and infinite distributed delays, Nonlinear Anal., 15, 63-73 (2015) · Zbl 1370.92020 |
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