Stability of neutral delay differential equations with applications in a model of human balancing

Alexander Domoshnitsky¹ ; Shai Levi¹ ; Ron Hay Kappel¹ ; Elena Litsyn¹ ; Roman Yavich¹

¹ Department of Mathematics, Ariel University, Ariel, Israel.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 21.

Résumé

In this paper the exponential stability of linear neutral second order differential equations is studied. In contrast with many other works, coefficients and delays in our equations can be variable. The neutral term makes this object essentially more complicated for the study. A new method for the study of stability of neutral equation based on an idea of the Azbelev W-transform has been proposed. An application to stabilization in a model of human balancing has been described. New stability tests in explicit form are proposed.

Détail

DOI : 10.1051/mmnp/2021008

Bibliographie

[1] N.V. Azbelev, V.P. Maksimov and L.F. Rakhmatullina, Introduction to theory of functional-differential equations. Nauka, Moscow (1991) (in Russian).

[2] N.V. Azbelev and P.M. Simonov, Stability of differential equations with aftereffect. Stability and Control: Theory, Methods and Applications, 20. Taylor and Francis, London (2003).

[3] L. Berezansky, A. Domoshnitsky On stability of the second order neutral differential equation Appl. Math. Lett 2019 90 95

[4] T.A. Burton, Stability by fixed point theory for functional differential equations. Dover Publications, Mineola, New York (2006).

[5] B. Cahlon, D. Schmidt Stability criteria for second-order delay differential equations with mixed coefficients J. Comput. Appl. Math 2004 79 102

[6] A. Domoshnitsky Nonoscillation, maximum principles and exponential stability of second order delay differential equations without damping term J. Inequal. Appl 2014 361

[7] A. Domoshnitsky Unboundedness of solutions and instability of differential equations of the second order with delayed argument Differ. Integr. Equ 2001 559 576

[8] Z. Došla, I.T. Kiguradze On boundedness and stability of solutions of second order linear differential equations with advanced arguments Adv. Math. Sci. Appl. Gakkotosho, Tokyo 1999 1 24

[9] L.N. Erbe, Q. Kong and B.G. Zhang, Oscillation theory for functional differential equations. Dekker, New York/Basel (1995).

[10] T. Erneux, Applied delay differential equations. Springer Science + Business Media (2009).

[11] V.N. Fomin, A.L. Fradkov and V.A. Yakubovich, Adaptive control of dynamical objects. Moscow: Nauka (1981).

[12] K. Gopalasamy, Stability and oscillation in delay differential equation of population dynamics. Kluwer Academic publishers, Dordrecht, Boston, London (1992).

[13] I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations. Clarendon, Oxford (1991).

[14] T. Insperger, J.G. Milton, G. Stepan Acceleration feedback improves balancing against reflex delay J. R. Soc. Interface 2013 20120763

[15] D.V. Izjumova About boundedness and stability of solutions of nonlinear functional-differential equations of the second order Proc. Georg. Acad. Sci. V 1980 285 288

[16] V. Kolmanovskii and A. Myshkis, Introduction to the theory and applications of functional differential equations. Kluwer Academic Publisher, Dordrecht/Boston/London (1999).

[17] I.D. Loram, M. Lakie Direct measurement of human ankle stiffness during quiet standing: the intrinsic mechanical stiffness is insufficient for stability J. Physiol 2002 1041 1053

[18] N. Minorski, Nonlinear oscillations. Van Nostrand, New York (1962).

[19] A.D. Myshkis, Linear differential equations with delayed argument. Moscow, Nauka, 1972, 352 p. (in Russian).

[20] M. Pinto Asymptotic solutions for second order delay differential equations Nonlin. Anal. : TMA 1999 1729 1740

[21] A.V. Shatyrko, D. Ya Khusainov Absolute interval stability of indirect regulating systems of neutral type J. Autom. Inf. Sci 2010 43 54

[22] A.H. Vette, K. Masani, K. Nakazawa, M.R. Popovic Neural-mechanical feedback control scheme generates physiological ankle torque fluctuation during quiet stance IEEE Trans. Neural. Syst. Rehabil. Eng. 2010 86 95

[23] L. Zhang, G. Stepan, T. Insperger Saturation limits the contribution of acceleration feedback to balancing against reaction delay J. R. Soc. Interface 2018 20170771

[24] L. Zhang, G. Stepan Exact stability chart of an elastic beam subjected to delayed feedback J. Sound Vib 2016 219 232