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

Abstract

In this paper the exponential stability of linear neutral second order differential equations is studied. In contrast with many other works, coeffcients 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.

Exact delay range for the stabilization of linear systems with input delays

“…Step 2: Based on Λ, derive n 1,𝑗 , 𝑗 = 1, • • • , s 1 for the eigenvalue 𝛽 1 via (21). Obtain Γ(𝛽 1 , h) according to (41).…”

“…For second‐order delay equations, a similar idea leads to new stability results in Chapters 4 and 5 of the book [13] and in [20]. Applications of this idea to the problem of human balancing can be found in [21]. [22] built a concept of the maximum principles for functional differential equations and obtained stability results.…”

Further results on finite-time stability of neutral nonlinear multi-term fractional order time-varying delay systems

“…Particularly, some authors have devoted attention to stability and control issues of the neutral TDS (NTDS) integer and fractional order [21][22][23][24][25][26][27][28][29][30]. Integer order NTDS in mechanical problems were presented in [21,22]; the stability chart of an elastic beam was obtained in [21] and the problem of ship rolling with control based on values of delayed acceleration was considered in [22].…”

“…Also, in [23] the delayed acceleration feedback control has been applied for chatter suppression in turning machines. The human self-balancing models have been modeled as integer order neutral TDS due to stabilizing time-delayed feedback control which depends on position, velocity and acceleration [24][25][26]. Moreover, the generalized Scot-Blair model has been studied [27], which can be modeled as neutral NITDS where a viscoelastic material is used as damping in vibration systems, on the assumption that the damping is proportional to a fractional rder derivative of the displacement variable with the non-integer order of derivative 0 < α < 1.…”

Parameter Estimation in Neutral Delay Differential Equations Using Genetic Algorithm With Multi-Parent Crossover

“…Several studies have been done on the numerical approximations and asymptotic properties of solutions to NDDEs [1]- [7]. NDDEs have been used to develop models in engi-neering [8]- [12], biosciences [1], [13]- [17], and economics [18], [19]. A review of the applications of NDDEs in various fields is discussed in [20].…”