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

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.

[1]  Ian David Loram,et al.  Direct measurement of human ankle stiffness during quiet standing: the intrinsic mechanical stiffness is insufficient for stability , 2002, The Journal of physiology.

[2]  I. Kovacic.  Nonlinear Oscillations , 2020 .

[3]  A. Domoshnitsky.  Unboundedness of solutions and instability of differential equations of the second order with delayed argument , 2001, Differential and Integral Equations.

[4]  Qingkai Kong,et al.  Oscillation Theory for Functional Di erential Equations , 1994 .

[5]  Kolmanovskii,et al.  Introduction to the Theory and Applications of Functional Differential Equations , 1999 .

[6]  Alexander Domoshnitsky,et al.  On stability of the second order neutral differential equation , 2019, Appl. Math. Lett..

[7]  George Seifert,et al.  Oscillation Theory of Delay Differential Equations (I. Györi and G. Ladas); Oscillation Theory for Neutral Differential Equations with Delay (D. D. Bainov and D. P. Mishev) , 1993, SIAM Rev..

[8]  Gábor Stépán,et al.  Acceleration feedback improves balancing against reflex delay , 2013, Journal of The Royal Society Interface.

[9]  Baruch Cahlon,et al.  Stability criteria for certain second-order delay differential equations with mixed coefficients , 2004 .

[10]  M. Pinto.  Asymptotic solutions for second order delay differential equations , 1997 .

[11]  Alexander Domoshnitsky.  Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term , 2014 .

[12]  D. Khusainov,et al.  Absolute Interval Stability of Indirect Regulating Systems of Neutral Type , 2010 .

[13]  Z. Došlá,et al.  On boundedness and stability of solutions of second order linear differential equaitons with advanced arguments , 1999 .

[14]  M.R. Popovic,et al.  Neural-Mechanical Feedback Control Scheme Generates Physiological Ankle Torque Fluctuation During Quiet Stance , 2010, IEEE Transactions on Neural Systems and Rehabilitation Engineering.

[15]  Li Zhang,et al.  Saturation limits the contribution of acceleration feedback to balancing against reaction delay , 2018, Journal of The Royal Society Interface.

[16]  N. Azbelev,et al.  Stability of Differential Equations with Aftereffect , 2002 .

[17]  T. Erneux.  Applied Delay Differential Equations , 2009 .

[18]  Gábor Stépán,et al.  Exact stability chart of an elastic beam subjected to delayed feedback , 2016 .

[19]  N. Shkil’.  Construction of formal particular solutions of a system of linear differential equations with delayed argument , 1974 .

[20]  T. A. Burton,et al.  Stability by Fixed Point Theory for Functional Differential Equations , 2006 .