State-dependent Delay Research Articles

Stability Analysis of Impulsive Switched Nonlinear Systems With Double State-Dependent Delays

This brief addresses the problem of global asymptotic stability for impulsive switched nonlinear systems (ISNSs) with double state-dependent delays (DSDDs), where the state-dependent delays exist in both subsystem functions and impulsive functions. The innovations of our results are that: 1) Due to the ubiquitous asynchronous phenomenon between the switches and impulses in many practical systems, we investigate the asynchronous phenomenon for ISNSs with DSDDs. 2) Meanwhile, the stabilizing and destabilizing effects of state-dependent delays impulses are fully considered. 3) By utilizing multiple Lyapunov functions, the switching strategy and the impulsive strategy, based on the asynchronous phenomenon, some stability criteria on global asymptotic stability are presented for ISNSs with DSDDs. As a result, the existing results are improved and can be regarded as a special case of the stability criteria in this brief. A simulation example is provided to verify the effectiveness of the developed methods.

An efficient IMEX method for nonlinear functional differential equations with state-dependent delay

Implicit-explicit midpoint (IMEX MP) formula is proposed to solve numerically nonlinear neutral functional differential equations (NFDEs) with state-dependent delay. Since the deviating argument will be explicitly treated, the proposed method is extremely effective. Second order convergence of this method is shown theoretically and numerically for NFDEs with smooth solutions and Volterra functional differential equations (VFDEs). For general NFDEs with nonsmooth solutions, regularization problem by smoothing the initial data is introduced and analyzed. The regularization error is derived in maximum norm. The global errors of IMEX MP method are derived and reveal that it is only first order of convergence for this regularization error or for the non-smoothness of the original problems. Numerical examples are provided to support the theoretical analysis, compared with IMEX Euler method.

About Stability of Nonlinear Stochastic Differential Equations with State-Dependent Delay

A nonlinear stage-structured population model with a state-dependent delay under stochastic perturbations is investigated. Delay-independent and delay-dependent conditions of stability in probability for two equilibria of the considered system are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The model under consideration is not the aim of the work and was chosen only to demonstrate the proposed research method, which can be used for the study of other types of nonlinear systems with a state-dependent delay.

Perimeter Control With State-Dependent Delays: Optimal Control Model and Computational Method

Perimeter control is to manipulate traffic flows in different regions through adjusting traffic signals at the border of the regions. Traditionally, the complete trips in a region are assumed to be dependent on the current accumulated vehicles in this region. This assumption is invalid because the vehicle needs time to complete its trip in a region. In order to solve this shortcoming, some recent studies have introduced a time-delay dynamical system to describe the dynamical behaviour of the accumulated vehicles in a region. However, in these studies, the perimeter control problem is formulated as a tracking problem with a given optimal reference point. In reality, such an optimal reference point is unavailable in advance. This paper will fill this gap through formulating perimeter control as an optimal control problem governed by a state-dependent time delay system. The control parametrization technique and an exact penalty method are introduced to solve such a challenging optimal control problem. Model predictive control is applied to obtain close-loop solution through solving a series of online optimal control problems. Some experiments are performed to demonstrate the effectiveness of our method.

Functional Integro-Differential Equations with State-Dependent Delay and Non-Instantaneous Impulsions: Existence and Qualitative Results

This paper addresses some existence, attractivity and controllability results for semilinear integrodifferential equations having non-instantaneous impulsions on an infinite interval via resolvent operators in case of neutral and state-dependent delay problems. Our criteria were obtained by applying a Darbo’s fixed-point theorem combined with measures of noncompactness. The obtained result is illustrated by an example at the end.

Almost Periodicity in Time-Dependent and State-Dependent Delay Differential Equations

We study the response of various linear and nonlinear differential equations to almost periodic forced oscillations. A special attention is given to differential equations with state-dependent and almost periodic type delays. To the best of our knowledge, there are no results in the literature that address this problem.

Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

Abstract This paper deals with the control problems governed by fractional impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder’s fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the existing literature. Moreover, we point out some shortcomings of the existing works in the context of characterization of mild solution, phase space, and approximate controllability of fractional order impulsive systems in Banach spaces. Finally, we investigate the approximate controllability of the fractional order heat equation with non-instantaneous impulses and delay by using the developed results.

Finite fractal dimensional global attractor for abstract differential equations with state-dependent delay

We study the existence of a finite fractal dimensional global attractor for the multivalued semigroup generated by a general class of abstract differential equations with state-dependent delay. In the last section, some examples concerning partial differential equations with state-dependent delay are presented.

Exploiting delay differential equations solved by Eta functions as suitable mathematical tools for the investigation of thickness controlling in rolling mill

This work is dedicated to introducing the properties and application of Eta functions. We derive the properties of the Eta function, such as the generating function, integral representation, and the Laplace transform. Also, some properties of the Eta-based functions are introduced. To show the advantages of the Eta-based functions in the computational method, we develop a new numerical method to solve the state-dependent and time-dependent neutral delay differential equation based on the Eta-based function. We introduce the operational matrix of derivative for the Eta-base functions to develop the new numerical method. This method uses the operational matrix of derivative and collocation method to convert the delay differential equation to a system of nonlinear algebraic equations. We derive the technique’s error bound and establish the method’s accuracy by solving some examples, which are state-dependent and time-dependent delay differential equations. In the end, we study the model of the metal forming process by rolling the mill using the new numerical method to show the advantages of using the Eta-based function for solving a more practical problem.

Results on controllability and well-posedness of functional abstract second-order differential equations with state-dependent delay

The major goal of this study is to demonstrate the existence and controllability of a second-order abstract differential system with state-dependent delay. We also discuss the well-posedness of the problem at hand. Finally, some examples are provided for the illustration of the main results.

Impulsive Functional-Abstract Neutral Functional Differential Nonlocal Cauchy Problem with State-dependent Delay

Impulsive Functional-Abstract Neutral Functional Differential Nonlocal Cauchy Problem with State-dependent Delay

Non-Instantaneous Impulsive Fractional Integro-Differential Equations with State-Dependent Delay

Non-Instantaneous Impulsive Fractional Integro-Differential Equations with State-Dependent Delay

Some results on the approximate controllability of impulsive stochastic integro-differential equations with nonlocal conditions and state-dependent delay

This paper presents approximate controllability results for impulsive stochastic integro-differential systems with state-dependent delay in a Hilbert space. The use of the resolvent operator in the sense of Grimmer, as well as stochastic analysis techniques, yields a new set of results. Finally, an example is given to show how the theory that has been worked out can be put into practice.

Uniqueness of solutions and linearized stability for impulsive differential equations with state-dependent delay

We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for an impulsive differential equation with state-dependent delay to exhibit non-uniqueness of solutions. On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result on linearized stability. As a specific application, we consider a scalar equation on the positive half-line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and impulse effect functional satisfying affine bounds.

Effect of state-dependent time delay on dynamics of trimming of thin-walled structures

Trimming of cantilever thin-walled structures is commonly seen in aerospace industry, including trimming of blades. Trimming with helix angle tools can cause the vibration of the thin-walled workpiece along the tool-axis, which may disturb the time delay between cutting by the current and previous teeth. The time delay dependent on the vibration state makes the stability analysis of trimming process challenging. This paper is the first attempt to uncover the effect of state-dependent time delay of the trimming process caused by workpiece vibration on chatter stability. Modeling of the cutter-workpiece interactions, state-dependent time delay and the dynamic chip generation mechanism are presented. A time-domain numerical algorithm with an improved stability metrics is constructed to analyze the trimming stability behaviors. We found that the two dominant states can occur, namely, period-n instabilities with time-varying time delay and stability with constant time delay. A focused experimental study was carried out to calibrate this new finding. This study reveals the way the workpiece vibration affects the time delay and stability in the trimming process.

Persistence and smooth dependence on parameters of periodic orbits in functional differential equations close to an ODE or an evolutionary PDE

We consider functional differential equations (FDEs) which are perturbations of smooth ordinary differential equations (ODEs). The FDE can involve multiple state-dependent delays, distributed delays, or implicitly defined delays (forward or backward). We show that, under some mild assumptions on the perturbation, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, when the perturbation depends on some parameters, we get smooth dependence of the periodic orbit and its frequency on the parameters with high regularity.The method can also be applied to treat equations with small delays appearing in electrodynamics and FDEs which are perturbations of some evolutionary partial differential equations (PDEs).The proof consists in solving functional equations satisfied by the parameterization of the periodic orbit and the frequency using a fixed-point approach. We do not need to consider the smoothness of the evolution or even the phase space of the FDEs.

Validated integration of differential equations with state-dependent delay

We present an implicit method of steps for differential equations with state-dependent delays and validated numerics to rigorously enclose solutions of initial-value problems. Our approach uses a combination of contraction mapping arguments based on a Newton–Kantorovich type theorem and piecewise polynomial interpolation. Completing multiple steps of integration is challenging, and we resolve it by smooth interpolation of the previous solution, resulting in an interval-valued polynomial initial condition for the subsequent step. A set of examples is provided.

Total Controllability of Non-Autonomous Measure Evolution Systems with Non-Instantaneous Impulses and State-Dependent Delay

This paper is concerned with the existence of mild solutions and total controllability for a class of non-autonomous measure evolution systems with non-instantaneous impulses and state-dependent delay. By using the theory of evolution family and Krasnoselskii’s fixed point theorem, the existence of mild solutions and total controllability for the considered systems is obtained. Finally, we give two applications to support the validity of the study.

Finite-time stability of state-dependent delayed systems and application to coupled neural networks

Finite-time stability and stabilization problems of state-dependent delayed systems are studied in this paper. Different from discrete delays and time-dependent delays which can be well estimated over time, the information of state-dependent delays is usually hard to be estimated, especially when states are unknown or unmeasurable. To guarantee the stability of state-dependent delayed systems in the framework of finite time, a Razumikhin-type inequality is used, following which estimations on the settling time and the region of attraction are proposed. Moreover, the relationship between the variation speed of state-dependent delays and the size of the region of attraction is proposed. Then as an application of the theoretical result, finite-time stabilization is studied for a set of nonlinear coupled neural networks involving state-dependent transmission delay, where the design of memoryless finite-time controllers is addressed. Two numerical examples are given to show the effectiveness of the proposed results.

Predictor-feedback stabilization of two-input nonlinear systems with distinct and state-dependent input delays

This paper deals with the predictor-feedback stabilization problem of two-input nonlinear systems with distinct and state-dependent input delays. Since each individual channel may induce a different delay, a sequential approach is employed for the construction of two predictors such that the corresponding input delays can be compensated. Concretely, the predictor over the prediction horizon which corresponds to the smaller input delay is first constructed, and the predictor over the remaining time horizon that corresponds to the larger input delay is in turn constructed, by utilizing the first predictor signal as a basis. The feedback controllers are given based on the predictors. Due to the inherent constraint on the rates of the state-dependent input delays, regional stability results are achieved. Specifically, if the system is forward-complete, for the closed-loop nonlinear system, an estimation of the region of attraction is provided, and for the linear system, the exponential stability is guaranteed. The methodology is applied to the bilateral boundary control problem of hyperbolic partial differential equations (PDEs) with a moving boundary, for illustration of the effectiveness of the methodology.