State-dependent Delay Research Articles

Existence and Stability of Neutral Stochastic Impulsive and Delayed Integro-Differential System via Resolvent Operator

In this paper, we present the existence of a mild solution for a class of a neutral stochastic integro-differential system over a Hilbert space. Such systems are influenced by both multiplicative and fractional noise, alongside non-instantaneous impulses, with a Hurst index H in the interval (12,1). Additionally, the systems under consideration feature state-dependent delays (SDDs). To address this, we develop an approach to reformulate the neutral stochastic integro-differential system, incorporating SDDs and non-instantaneous impulses, into an equivalent fixed-point (FP) problem via an appropriate integral operator. By integrating stochastic analysis with the theory of resolvent operators, we employ Banach’s FP theorem to establish both the existence and uniqueness of the solution. Furthermore, we explore the Ulam–Hyers–Rassias stability of the system. Lastly, we provide illustrative examples to demonstrate the practical applicability of our results.

Controllability of Impulsive Damped Fractional Order Systems Involving State Dependent Delay

In this article, the concept of controllability on fractional order impulsive systems involving state dependent delay and damping behavior is analysed by utilizing Caputo fractional derivative. The main motivation is to derive the sufficient conditions for the controllability of the considered systems. Based on the Laplace transform and inverse Laplace transform, the solution of fractional-order dynamical systems are obtained. The results are established by utilizing basic ideas of fractional calculus, Mittag-Leffler function and Banach fixed point theorem. Finally, an application is provided to illustrate the derived result.

A Comprehensive View of the Solvability and Stability of a Feedback Control Problem with a State-Dependent Delay Implicit Pantograph Equation

In this paper, we examine the existence of a unique solution of a feedback control problem with an implicit state-dependent pantograph equation. Additionally, the study implements the problem’s Hyers-Ulam stability and the continuous dependence of the unique solution on the initial data and the parameters. Furthermore, we investigate this problem in the absence of feedback control. We also provide some examples to illustrate our results.

A note on the approximate controllability of damped second-order differential inclusions with state-dependent delay

In this paper, we use Bohnenblust–Karlin's fixed point theorem and the theory of strongly continuous cosine families of bounded linear operators to study the approximate controllability of damped second-order differential inclusions with state-dependent delay in Banach spaces. In particular, we obtain a new set of sufficient conditions for the approximate controllability of damped second-order differential inclusions with state-dependent delay under the assumption that the corresponding linear system is approximately controllable. An example is provided to illustrate our main results.

Existence solutions for nondensely integro-differential equations with state-dependent delay

Existence solutions for nondensely integro-differential equations with state-dependent delay

Differentiability of Semi-Flow for Impulsive Evolution Equation with State-Dependent Delay

Differentiability of Semi-Flow for Impulsive Evolution Equation with State-Dependent Delay

Approximate controllability of second-order neutral measure driven systems with state-dependent delay

This article investigates the solvability and approximate controllability of a new class of measure driven control systems with state-dependent delay, which can deal with hybrid system without any restriction Zeno behaviour. By applying fixed point technology and Lebesgue Stieltjes integral, a new set of sufficient conditions is constructed that ensures the existence of mild solution and approximate controllability of the considered system. Lastly, an example is formulated to demonstrate the effectiveness of obtained results.

Approximate controllability of damped second-order neutral differential equations with state-dependent delay

ABSTRACT This article discusses the approximate controllability of damped second-order neutral differential equations with state-dependent delay. The results are obtained using strongly continuous cosine family theories and fixed-point theories. Assuming approximate controllability of the corresponding equation, we obtain sufficient conditions for approximate controllability of the nonlinear second-order equations. An application are provided to demonstrate how the proposed result can be applied.

Quasi-periodic Response Solution to Scalar State-Dependent Delay Differential Equation with Degenerate Equilibrium

Quasi-periodic Response Solution to Scalar State-Dependent Delay Differential Equation with Degenerate Equilibrium

Stick-slip vibration analysis of a coupling drilling-rock system with two delays and one delay-dependent coefficient

Drilling depths exceeding 10,000 m have recently been achieved, but the increasing depth poses challenges due to the flexible drill-string, making it more susceptible to stick-slip vibration. This vibration can lead to accidents such as excessive bit wear and tool falling into the well. Previous studies on modeling polycrystalline diamond compact (PDC) bits have focused on symmetrically distributed complete blade bits, which are not commonly used. To address this gap, this study introduces PDC bits that closely resemble the actual ones, with a structural configuration of both complete blades and part-blades symmetrically distributed. We also consider the impact of mud and rock chips on the drill bit by introducing a delay-dependent coefficient of response damping change. This study develops a stick-slip vibration model for a coupling drilling-rock system with two state-dependent delays tn1 and tn2, and one delay-dependent coefficient. Using the crossing curve method and numerical studies, we obtain different types of crossing curves reflecting the stability switching characteristics of the system in the dimensionless delay τ2τ1 space. During stick-slip vibration, τ1 mutates from 1.69 s to 2.51 s, and τ2 mutates from 1.3 s to 2.16 s, both within the stable region of the crossing curve. The critical steady state and the stable state typically cross the stable and unstable regions of the crossing curve in their respective times. Increasing the torsional damping within the range of 0.04 to 0.08 can accelerate system stabilization or delay the arrival of the critical steady state. Moreover, increasing the drive rotation speed within the range of 4 to 6 eliminates stick-slip vibration and accelerates the system towards a steady state. In conclusion, this study predicts the drill-string dynamics of a symmetrically distributed PDC bit with complete blades and part-blades. The insights gained can help minimize stick-slip vibration and provide recommendations for drilling operation parameters.

Hopf-bifurcation analysis of a stage-structured population model of cell differentiation

A cell differentiation model with maturity structure in progenitor cells is described by a system of nonlinear ODEs coupled to a PDE. The work mainly discusses Hopf-bifurcation for the model when the maturation rate of progenitor cells only depends on the number of mature cells, and is a continuation of the previous work (Doumic et al., 2011). Using the methods of characteristics and a change in variable, the system can be transformed into one of state-dependent delay differential equations and then into constant delay differential equations. A qualitative analysis of the solutions is performed, which includes well-posedness, non-negativity, uniform boundedness and linearized stability. Furthermore, we conduct Hopf-bifurcation analysis by taking the maturation level of progenitor as the bifurcation parameter, which means that the levels of stem and mature cells may vary with persistent oscillations. A numerical analysis stresses the role of some important parameters on the stability and Hopf-bifurcation.

Exponential stability of non-instantaneous impulsive second-order fractional neutral stochastic differential equations with state-dependent delay

The aim of this manuscript mainly concerned with a new class of exponential stability for non-instantaneous impulsive second-order fractional neutral stochastic differential equations (NIIFNSDEs) with state-dependent delay driven by Poisson jump in a separable Hilbert spaces. Firstly, a more appropriate concept of mild solution is introduced. Secondly, sufficient conditions are derived for the existence of mild solution by means stochastic analysis, fractional calculus approach and Mönch fixed point theorem with appropriate hypotheses on non-linear continuous functions combined with solution operator. Finally, stability results are derived based on the pth moment exponential stable with the help of new impulsive integral inequality techniques. In addition, an example is provided to validate the theoretical results. This manuscript is the unique combination of the new theoretical investigation which is simulated numerically. Our work extends the works of Arthi et al. (2014), Das et al. (2016), Huang et al. (2018), Pandey et al. (2014), Wang et al. (2018), Yan and Jia (2016).

Evaluating PDC bit-rock interaction models to investigate torsional vibrations in geothermal drilling

Polycrystalline diamond compact (PDC) bits are superior for drilling geothermal wells because of their superior drilling performance compared to conventional roller cone bits. However, the shear action of PDC bits generates detrimental vibrations during drilling. The main objective of this study was to establish a methodology to analyze and predict the stick-slip severity in hard rocks for geothermal wells. Two non-linear coupled axial-torsional bit-rock interaction (BRI) models are presented: one is based on a velocity-decaying friction model (VDF), and the other is based on a state-dependent delay friction model (SDDF). The capabilities of the two models were evaluated to assess the axial and torsional dynamic stabilities of drill stems in deep geothermal wells. The comparative analysis, along with the results from both models, were validated using geothermal well downhole data. Five distinct zones were selected for analysis, and the stick-slip severity value (SSV) was calculated using these two models (VDF and SDDF). The results from these two models for the five different zones were compared with the field data. The results indicated that VDF demonstrated superior quality when compared with field values, as the results of VDF were within the interquartile range of the observed SSV in each zone. A sensitivity analysis employing spider plots was performed for both models, considering parameters related to rock, bit, operational, and frictional aspects. In terms of the operational parameters, the weight-on-bit (WOB) and revolutions per minute (RPM) exerted the most significant influence on the SSV for both models. For the VDF model, the sensitivity analysis indicated that the frictional parameter, uniaxial compressive strength (UCS), and number of cutters (NOC) had the most pronounced impact on the SSV. In the case of the SDDF, the Intrinsic specific energy (ISE), bit diameter, and number of blades (NOB) are the key factors that predominantly affect the SSV.

On Explicit Abstract Neutral Differential Equations with State-Dependent Delay

On Explicit Abstract Neutral Differential Equations with State-Dependent Delay

Three-Body 3D-Kepler Electromagnetic Problem—Existence of Periodic Solutions

The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld’s hypothesis for the He atom.

On neutral integrodifferential equations with state-dependent delay in Banach spaces

On neutral integrodifferential equations with state-dependent delay in Banach spaces

Abstract random differential equations with state-dependent delay using measures of noncompactness

This paper is devoted to the existence of random mild solutions for a general class of second-order abstract random differential equations with state-dependent delay. The technique used is a generalization of the classical Darbo fixed point theorem for Fréchet spaces associated with the concept of measures of noncompactness. An application related to partial random differential equations with state-dependent delay is presented.

Stability analysis of stochastic differential systems with state‐dependent delay subject to average‐delay impulses

In this paper, we study the stochastic weak local exponential stability of stochastic differential systems with state‐dependent delay (SDD) subject to average‐delay impulses by using Lyapunov–Krasovskii functional and stochastic analytical skill. Unlike time‐dependent delay discussed in the previous literature, we introduce SDD in impulsive stochastic differential systems, where SDD is a stochastic variable associated with system state, consequently leading to an indeterminate delay boundary. Our findings reveal that when destabilizing delayed impulses satisfying specific conditions are generated, the stability of stochastic differential systems with destabilizing delayed impulses and stable continuous stochastic dynamics can still be maintained. Additionally, if stable delayed impulses satisfy certain conditions, stability of the systems under consideration can also be achieved, irrespective of the stable status on the continuous stochastic dynamics. Finally, two examples are given to demonstrate the validity of theoretical results.

Existence of a periodic solution for a state-dependent delay differential equation with a singular perturbation

The idea of this article came from the works dones by Khung and Smith on the one hand in [2] for and Arino and Al in [1] for on the other hand, to proves the existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard. I proposed the following question: Do the result remain valid if 1 ? by the same method with using the homotopy well defined link between the application of Khung and Smith and the application of Arino and and indeed, I came to prove existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard, therefore the result obtained in this work is generalization of two special cases. A State-dependent Delay Differential equation play important role in recent years to modeling of several phenomenon, in dynamics of populations, in biology and even in the social life for example: When we take our driving test, we learn that the reaction time of our nervous system when driving is of the order of a few seconds, and that we must take care to maintain a sufficient distance between two cars following each other, from this example, we notice that the delay can be useful. Although this work has added a new addition to the scientific field which is a generalization and linking between two special cases, it remains somewhat limited because it takes special case of Arino and Al. Therefore in the futur, we are thinking about a new work that takes into account the general case of Arino and .

Controllability of Mild Solutions for Second-Order Neutral Evolution Equations with State-Dependent Delay

Controllability of Mild Solutions for Second-Order Neutral Evolution Equations with State-Dependent Delay