scholarly journals Relative controllability of multiagent systems with pairwise different delays in states

2022 ◽  
Vol 27 ◽  
pp. 1-19
Author(s):  
Yuanchao Si ◽  
JinRong Wang
Keyword(s):  
Differential Equation ◽  
Multiagent Systems ◽  
Delay System ◽  
Multiple Delays ◽  
Information Flows ◽  
Relative Controllability ◽  
Delay Differential ◽  
Time Invariant ◽  
Interaction Topology ◽  
Generic Time

In this manuscript, relative controllability of leader–follower multiagent systems with pairwise different delays in states and fixed interaction topology is considered. The interaction topology of the group of agents is modeled by a directed graph. The agents with unidirectional information flows are selected as leaders, and the others are followers. Dynamics of each follower obeys a generic time-invariant delay differential equation, and the delays of agents, which satisfy a specified condition, are different one another because of the degeneration or burn-in of sensors. With a neighbor-based protocol steering, the dynamics of followers become a compact form with multiple delays. Solution of the multidelayed system without pairwise matrices permutation is obtained by improving the method in the references, and relative controllability is established via Gramian criterion. Further rank criterion of a single delay system is dealt with. Simulation illustrates the theoretical deduction.

scholarly journals Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiong Meng ◽  
Zhen Jin ◽  
Guirong Liu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Multiple Delays ◽  
All Solutions ◽  
Sufficient And Necessary Conditions ◽  
Delay Differential ◽  
T Tau ◽  
Sufficient And Necessary Condition

AbstractThis paper studies the linear fractional-order delay differential equation $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ D − α C x ( t ) − p x ( t − τ ) = 0 , where $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ 0 < α = odd integer odd integer < 1 , $p, \tau >0$ p , τ > 0 , ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s . We obtain the conclusion that $$ p^{1/\alpha } \tau >\alpha /e $$ p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.

scholarly journals Maximum Likelihood Inference for Univariate Delay Differential Equation Models with Multiple Delays

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Ahmed A. Mahmoud ◽  
Sarat C. Dass ◽  
Mohana S. Muthuvalu ◽  
Vijanth S. Asirvadam
Keyword(s):  
Differential Equation ◽  
Maximum Likelihood ◽  
Delay Differential Equation ◽  
Information Matrix ◽  
Likelihood Inference ◽  
Multiple Delays ◽  
Unknown Parameters ◽  
Differential Equation Models ◽  
Delay Differential ◽  
Initial Starting

This article presents statistical inference methodology based on maximum likelihoods for delay differential equation models in the univariate setting. Maximum likelihood inference is obtained for single and multiple unknown delay parameters as well as other parameters of interest that govern the trajectories of the delay differential equation models. The maximum likelihood estimator is obtained based on adaptive grid and Newton-Raphson algorithms. Our methodology estimates correctly the delay parameters as well as other unknown parameters (such as the initial starting values) of the dynamical system based on simulation data. We also develop methodology to compute the information matrix and confidence intervals for all unknown parameters based on the likelihood inferential framework. We present three illustrative examples related to biological systems. The computations have been carried out with help of mathematical software: MATLAB® 8.0 R2014b.

scholarly journals Uncharted Stable Peninsula for Multivariable Milling Tools by High-Order Homotopy Perturbation Method

2020 ◽  
Vol 10 (21) ◽  
pp. 7869 ◽  
Author(s):  
Jose de la Luz Sosa ◽  
Daniel Olvera-Trejo ◽  
Gorka Urbikain ◽  
Oscar Martinez-Romero ◽  
Alex Elías-Zúñiga ◽  
...  
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Delay Differential Equation ◽  
Homotopy Perturbation Method ◽  
Multiple Delays ◽  
Third Order ◽  
Method Performance ◽  
Homotopy Perturbation ◽  
The Stability ◽  
Delay Differential

In this work, a new method for solving a delay differential equation (DDE) with multiple delays is presented by using second- and third-order polynomials to approximate the delayed terms using the enhanced homotopy perturbation method (EMHPM). To study the proposed method performance in terms of convergency and computational cost in comparison with the first-order EMHPM, semi-discretization and full-discretization methods, a delay differential equation that model the cutting milling operation process was used. To further assess the accuracy of the proposed method, a milling process with a multivariable cutter is examined in order to find the stability boundaries. Then, theoretical predictions are computed from the corresponding DDE finding uncharted stable zones at high axial depths of cut. Time-domain simulations based on continuous wavelet transform (CWT) scalograms, power spectral density (PSD) charts and Poincaré maps (PM) were employed to validate the stability lobes found by using the third-order EMHPM for the multivariable tool.

scholarly journals On stability and boundedness of solutions of a certain fourth-order delay differential equation

1989 ◽  
Vol 12 (3) ◽  
pp. 589-602 ◽  
Author(s):  
Emmanuel O. Okoronkwo
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Order Differential Equation ◽  
Type Theorem ◽  
Sufficient Conditions ◽  
Delay System ◽  
Boundedness Of Solutions ◽  
Uniform Asymptotic Stability ◽  
Delay Differential

Using a Razumikhin - type theorem, we deduce sufficient conditions that guarantee the uniform asymptotic stability and boundedness of solutions of a scalar real fourth-order delay differential equation. The Lyapunov function constructed for an ordinary fourth-order differential equation is seen to work for the delay system.

scholarly journals Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: An averaging approach

2016 ◽  
Vol 16 (04) ◽  
pp. 1650013 ◽  
Author(s):  
N. Lingala ◽  
N. Sri Namachchivaya
Keyword(s):  
Differential Equation ◽  
Characteristic Equation ◽  
Complex Plane ◽  
Martingale Problem ◽  
Delay System ◽  
Additive White Noise ◽  
Linear Delay ◽  
The Law ◽  
Delay Differential ◽  
Countably Infinite

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

scholarly journals Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 71 ◽  
Author(s):  
Olga Tsekhan
Keyword(s):  
Sufficient Conditions ◽  
Singularly Perturbed ◽  
Degenerate System ◽  
Multiple Delays ◽  
Complete Controllability ◽  
Singularly Perturbed System ◽  
System With Delay ◽  
Perturbed System ◽  
Time Invariant ◽  
Controllability Conditions

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

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