Finite Delay Research Articles

An optimal control for non-autonomous second-order stochastic differential equations with delayed arguments

Abstract Optimal control of non-autonomous second-order stochastic differential equations with delayed arguments is indispensable for managing systems exposed to uncertainty, time-dependent dynamics, and historical influences. These equations underpin a wide range of applications, including finance, engineering, and biology, where it’s imperative to make informed decisions that mitigate risks or maximize returns while considering the inherent randomness, evolving conditions, and the impact of past states. By employing optimal control techniques, we can devise strategies that are resilient to uncertainty, adaptable to changing circumstances, and capable of accounting for the memory effects of previous events. This empowers us to optimize system performance, bolster stability, and attain desired objectives in intricate and dynamic environments. So, the goal of this article is to introduce a novel model of second-order perturbed stochastic differential equations incorporating non-local finite delay and deviated arguments in the setting of Hilbert spaces. Moreover, essential criteria are presented to examine the existence of a mild solution and evaluate the potential for approximate and optimal control of the proposed system. These results have been obtained by using evolution operators, fixed point techniques, random analytic methods, and compact semigroup theory. Further, to support the theoretical results, the optimal controllability of our model was studied by considering the Lagrange problem. Finally, the results were applied to discuss the approximate controllability of a partial differential equation. These models have the potential to advance the understanding and application of optimal control techniques for a wider range of complex systems.

Existence of solution to Hadamard–Caputo fractional differential equation with time delay

This article focuses on the study of the existence of solution to Hadamard–Caputo fractional nonlinear differential equation with time delay. For time delay, two different cases of finite delay and infinite delay are considered. Based on the weighted function solution spaces, the existence and uniqueness of solution to the differential equations are verified by Leray–Schauder selection theorem and Banach fixed point theorem respectively. Finally, examples are given to illustrate the abstract results of this article.

Existence of solutions and approximate controllability of second-order stochastic differential systems with Poisson jumps and finite delay

Existence of solutions and approximate controllability of second-order stochastic differential systems with Poisson jumps and finite delay

Existence and Approximate Controllability of Random Impulsive Neutral Functional Differential Equation with Finite Delay

This study investigates second-order neutral functional differential equations with delays, prevalent in various scientific and engineering fields. These equations, characterized by their neutral nature and delays, present unique challenges within Banach spaces. The research focuses on the existence and approximate controllability of solutions, using advanced mathematical tools like cosine family theory and the Leray-Schauder theorem to establish rigorous solution conditions. These theoretical results are empirically validated through practical examples, enhancing understanding of real-life behavior and bridging theory with practice. The study’s findings advance the understanding of delayed feedback systems, facilitating effective control strategies and practical engineering solutions, thereby contributing significantly to dynamical systems and control theory.

Forced response of transverse reacting jet in vitiated crossflow to harmonic velocity disturbances

The orthogonally oriented injector configuration typically used in axial fuel staging-capable gas turbine combustors is known to produce complicated thermoacoustic interactions between two distinct reaction zones. Despite the fundamental importance of transfer functions of transverse reacting jets in vitiated crossflow, however, their distinctive properties remain unknown, and associated low-order modeling combined with two different flame transfer functions remains entirely undemonstrated. From detailed measurements of transfer functions of lean-premixed primary and secondary flames in response to harmonic velocity disturbances, here we show that the transfer function of a transverse reacting jet in high-temperature crossflow is described as having relatively larger gain without undulating patterns, near-linear slow decay in magnitude with respect to the forcing frequency, and remarkably shorter duration response time. By leveraging a reduced-order modeling approach to account for the finite time delay between the two distinct transfer functions and by validating the calculation results against velocity and pressure measurement data, we demonstrate that self-induced instabilities in an axial fuel-staged system can be simultaneously driven by the dynamics of primary and secondary flames, and that the triggering of second-stage-induced instability is characteristically connected to higher-order acoustic modes. Such non-axisymmetric intra-combustor interactions generate a cascade of spectral peaks, including the first and third longitudinal modes and their respective higher harmonics, and additional peaks at the sum and difference of two individual frequencies. While the primary flame's dynamics are capable of selectively exciting the first mode, the secondary jet flame exhibits more complicated modal dynamics, manifested as the L3 mode-coupled jet merging-related flame surface annihilation and the L1 mode-coupled lateral movements of the transverse jet column.

Maximal Regularity for Fractional Difference Equations with Finite Delay on UMD Spaces

Maximal Regularity for Fractional Difference Equations with Finite Delay on UMD Spaces

Partial Stability in Probability of Nonlinear Stochastic Discrete-Time Systems with Delay

A system of nonlinear stochastic functional-difference equations with finite delay is considered. By assumption, this system has a “partial” trivial equilibrium (with respect to part of the state variables). The problem under study is to analyze partial stability in probability of this equilibrium: stability is considered with respect to part of the variables determining it. The problem is solved using a discrete-stochastic modification of the method of Lyapunov–Krasovskii functionals. Conditions for partial stability in probability are established. An example is provided to illustrate the features of the proposed approach and the rationale for introducing a one-parameter family of functionals.

Finite difference delay modelling for analysis of shielding effectiveness of perforated conductive enclosure

AbstractThis paper introduces finite difference delay modelling (FDDM) for computing the shielding effectiveness (SE) of a conductive enclosure with an aperture against electromagnetic pulses. FDDM offers optimal accuracy and stability for analysing complex structures. The time domain electric field integral equation (TD‐EFIE) for the conductive enclosure is derived by imposing boundary conditions on the perfect electrical conductor (PEC) surface in the Laplace domain. Time discretization is based on finite differences, and the Laplace‐to‐Z transform mapping is utilized. Fast Fourier Transform (FFT) is employed to expedite the FDDM solution process. Both frequency domain shielding effectiveness (FD‐SE) and time domain shielding effectiveness (TD‐SE) are evaluated using this approach. Finally, to validate the accuracy of the proposed method, results are compared with simulations using CST‐MWS software and the frequency domain moment method.

The Fractional Navier-Stokes Equations with delay conditions

Throught this paper, we study the cauchy problem for the conformable fractional Navier-Stokes Equations (FNSE) with finite delay external forces, contains some hereditary features, on a bounded domain. We prove that there exist a unique local mild solutions for the initial datum, by using semigroup theory, conformable fractional calculus and Banach contraction theorem.With more conditions on delay external forces we establish the globality and continuation of the mild solutions.

On stability of equations with an infinite distributed delay

For scalar equations of population dynamics with an infinite distributed delay x′(t)=r(t)[∫−∞tf(x(s))dsR(t,s)−x(t)], x(t)=φ(t), t⩽t0, where f is the delayed production function, we consider asymptotic stability of the zero and a positive equilibrium K. It is assumed that the initial distribution is an arbitrary continuous function. Introducing conditions on the memory decay, we characterize functions f such that any solution with nonnegative nontrivial initial conditions tends to a positive equilibrium. The differences between finite and infinite delays are outlined, in particular, we present an example when the weak Allee effect (meaning that f′(0)=1 together with f(x)>x , x∈(0,K) ) which has no effect in the finite delay case (all solutions are persistent) can lead to extinction in the case of an infinite delay.

Global exponential synchronization of complex networks with reaction diffusions and finite distributed delays coupling

This article investigates global exponential synchronization of complex networks with reaction diffusions and finite distributed delays coupling (RDDCCNs). Some restrictions on the coupling configuration matrix and time-varying delays are removed. By constructing comparison functions, utilizing analytic method and inequality techniques, a criterion for exponential synchronization of RDDCCNs is attained under adaptive intermittent control. A corresponding outcome without RDs is also acquired. At last, two simulations are proposed to illustrate validity of the obtained criteria.

Predicting the coupling of HPEM waveform with a power amplifier in perforated conductive enclosure using FDDM/VF

AbstractOne of the most common methods to protect electronic and telecommunication systems in a high‐power electromagnetic (HPEM) environment is using a conductive enclosure. Amplifiers are the vital parts in transmitter and receiver systems as the main components of a telecommunication system, so investigating the effects of HPEM waveform on their performance is very important. The Presence of apertures in the body of the conductive enclosure, as well as non‐linear elements such as transistors in amplifiers, adds to the complexity of examining the coupling of electromagnetic fields on the performance of amplifiers. Considering the importance of the topic, in this study, the impact of the HPEM waveform on the power amplifier inside the conductive enclosure has been investigated. By using the finite difference delay modelling (FDDM) numerical method as a stable method followed by the vector fitting (VF) method, the electromagnetic problem has become a circuit problem, which allows examining the response of the power amplifier and any circuit, including non‐linear elements against the interference field. Responses of bipolar linear and non‐linear components in the perforated conductive enclosure by the proposed method have been compared with the results obtained by computer simulation technology microwave studio (CST‐MWS) for verification.

Investigation on continuous dependence and regularity solutions of functional integrodifferential equations

In this work, to show the existence and continuous dependence of integral solution for neutral integro-differential equation with finite delay situation. This result established by using integrated resolvent operator with support of Lipschitz continuous and uniqueness part using Banach contraction mapping. Also study the existence of strict solution on reflexive and general Banach spaces. In the last section, an example is provided related to this theory.

On the nonlocal hybrid $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer inverse problem with delay and anticipation

<p>This paper focused on establishing results regarding the existence of solutions for a class of nonlocal terminal value problems involving hybrid implicit nonlinear fractional differential equations with the $ ({\mathsf{k}}, {\rm{\mathsf{φ}}}) $-Hilfer fractional derivative, which includes both finite delay and anticipation arguments. Our analysis was based on the Banach fixed point technique, and the Schauder and Krasnoselskii fixed point theorems. Moreover, illustrative examples were considered to support our new results.</p>

Existence and Regularity of solutions in alpha norm for some second order partial neutral functional differential equations in Banach Spaces

The purpose of this paper is to study the existence and regularity of solutions in the $\alpha$-norm for some second order partial neutral functional differential equations with finite delay in Banach spaces. To do this, we use the cosine family theory and Schauder's fixed point Theorem to establish the local existence of solutions and we give some sufficients conditions to enssure the regularity of solutions. Finally, we give an example to illustrate the theoretical results.

Existence and regularity of pullback attractors for a 3D non-autonomous Navier–Stokes–Voigt model with finite delay

In this manuscript previous results [Nonlinearity 25(2012), 905–930] are extended to a non-autonomous 3D Navier–Stokes–Voigt model in which a forcing term contains memory effects. Under suitable assumptions on the function driving the delay time, the existence and uniqueness of weak solution are proved. Existence and relationships among pullback attractors in several phase-spaces are analyzed for two possible choices of the attracted universes, namely, the standard one of fixed bounded sets, and another one given by a tempered condition. Some regularity results for these attractors are also established. Compactness and attraction norms are strengthened. Since the model does not have a regularizing effect, obtaining asymptotic compactness for the associated process is a more involved task. Our proofs rely on a sharp use of the energy equality, an energy method, bootstrapping arguments and by using bi-space attractors results.

Efficient Solution of Transient Electromagnetic Scattering Using ACA‐SVD‐Enhanced FDDM Method

A finite difference delay modeling (FDDM) method accelerated by adaptive cross approximation‐singular value decomposition (ACA‐SVD) is developed to solve time‐domain combined field integral equations for transient electromagnetic scattering. In the proposed method, the variable s in the Laplace domain is expressed as a difference function about z in the z‐transform domain to achieve the temporal discretization, thus improving the stability of the solution. And the method is purely algebraic and does not depend on the Green’s function. It takes advantage of the rank‐deficient nature of the impedance submatrix blocks in the FDDM to reduce the memory requirement and the computational cost. The rank‐deficient submatrix blocks can reach the maximum compression level through the ACA‐SVD. Numerical results about the electromagnetic scattering from perfect electric conducting objects are given to verify the validity and efficiency of the proposed method.

Analysis of solution of fractional summation-difference equation of finite delay in cone metric space

Analysis of solution of fractional summation-difference equation of finite delay in cone metric space

On the stability of a nonlinear nonautonomous scalar equation with variable delay

The stability problem of a scalar functional differential equation is a classical one. It has been most fully studied for linear equations. Modern research on modeling biological, infectious and other processes leads to the need to determine the qualitative properties of the solutions for more general equations. In this paper we study the stability and the global limit behavior of solutions to a nonlinear one-dimensional (scalar) equation with variable delay with unbounded and bounded right-hand sides. In particular, our research is reduced to a problem on the stability of a non-stationary solution of a nonlinear scalar Lotka-Volterra-type equation, on the stabilization and control of a non-stationary process described by such an equation. The problem posed is considered depending on the delay behavior: is it a bounded differentiable function or a continuous and bounded one. The study is based on the application of the Lyapunov-Krasovsky functionals method as well as the corresponding theorems on the stability of non-autonomous functional differential equations of retarded type with finite delay. Sufficient conditions are derived for uniform asymptotic stability of the zero solution, including global stability, for every continuous initial function. Using the theorem proven by one of the co-authors on the limiting behavior of solutions to a non-autonomous functional differential equation based on the Lyapunov functional with a semidefinite derivative, the properties of the solutions’ attraction to the set of equilibrium states of the equation under study are obtained. In addition, illustrative examples are provided.

On deformable fractional impulsive implicit boundary value problems with delay

This paper deals with some existence and uniqueness results for a class of deformable fractional differential equations. These problems encompassed nonlinear implicit fractional differential equations involving boundary conditions and various types of delays, including finite, infinite, and state-dependent delays. Our approach to proving the existence and uniqueness of solutions relied on the application of the Banach contraction principle and Schauder’s fixed-point theorem. In the last section, we provide different examples to illustrate our obtained results.