Asymptotic Behavior Of Solutions Research Articles

On the boundedness of solutions of some fuzzy dynamical control systems

<abstract> <p>The asymptotic behavior of solutions of fuzzy control systems is a component of the study of fuzzy control theory. The study of stability for T-S (Takagi-Sugeno) fuzzy systems, which process qualitative data through linguistic expressions, is the subject of this paper. Asymptotic stability is conservative in many real-world applications due to measurement noise and other disruptions. The ultimate limit, which indicates that the mistakes stay in a specific area close to the origin after a long enough amount of time, is a crucial characteristic that is frequently defined for such systems. We are interested with the problem of the state feedback controller for T-S fuzzy models with uncertainties where the global exponential ultimate boundedness of solutions is studied for certain fuzzy control systems. We use common quadratic Lyapunov function and parallel distributed compensation controller techniques to study the asymptotic behavior of the solutions of fuzzy control system in presence of perturbations. An example demonstrating the validity of the main result is discussed.</p> </abstract>

On asymptotic behavior of solutions of multivariate linear stochastic differential equations with multiplicative noise

On asymptotic behavior of solutions of multivariate linear stochastic differential equations with multiplicative noise

The existence and asymptotic behavior of solutions to 3D viscous primitive equations with Caputo fractional time derivatives

On the one hand, the primitive three-dimensional viscous equations for large-scale ocean and atmosphere dynamics are commonly used in weather and climate predictions. On the other hand, ever since the middle of the last century, it has been widely recognized that the climate variability exhibits long-time memory. In this paper, we first prove the global existence of weak solutions to the primitive equations of large-scale ocean and atmosphere dynamics with Caputo fractional time derivatives. Then we establish the existence of an absorbing set, which is positively invariant. Finally, an attractor (strictly speaking, the minimal attracting set containing all the limiting dynamics) is constructed for the time fractional primitive equations, which means that the present state of a system may have long-time influences on the states in far future. However, there was no work on the long-time behavior of the time fractional primitive equations and we fill this gap in this paper.

Asymptotic Behavior of Solutions to Difference Equations of Neutral Type

We present sufficient conditions for the existence of a solution x to an equation Δm(xn-unxn-k)=anf(xn-τ)+bn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\ au })+b_n, \\end{aligned}$$\\end{document}which is “close” to a given solution y to the linear homogeneous equation of neutral type Δm(yn-λyn-k)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta ^m(y_n-\\lambda y_{n-k})=0$$\\end{document}, where λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} is the limit of the sequence u. Closeness of solutions to above equations is understood as xn-yn=o(ωn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_n-y_n=\ extrm{o}(\\omega _n)$$\\end{document}, where ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} is a given nonincreasing sequence with positive values. Moreover, we establish under which conditions for a given solution x to Δm(xn-unxn-k)=anf(xn-τ)+bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta ^m(x_n-u_nx_{n-k})=a_nf(x_{n-\ au })+b_n$$\\end{document} and a given nonincreasing sequence with positive values ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} there exists a polynomial sequence φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} of degree less than m such that xn=φ(n)+o(ωn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_n=\\varphi (n)+\ extrm{o}(\\omega _n)$$\\end{document}. Presented conditions strongly depend on λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}.

Retracted: Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

Retracted: Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

ASYMPTOTIC REPRESENTATION OF SOLUTIONS CLOSE TO LINEAR DIFFERENTIAL EQUATIONS OF THE THIRD ORDER

In the work, using the a priori properties of the Pω(λ0) class of solutions, the conditions for the existence of one class of solutions in the binomial non-autonomous of a differential equation of the third order with a nonlinearity close in some sense to linear, in the critical case, namely when λ0 = ±∞. Asymptotic at t↑ω (ω ≤ +∞) images for such solutions and their derivatives of the first and second order in the case λ0 = ±∞. Proven for of the nonlinear equation, the lemma and the theorem are transferred to linear differential equations of the third order with asymptotically small coefficients. The transferred results do not contradict, and to some extent, complement known results regarding the asymptotic behavior of solutions of linear differential equations of the third order.

Splitting of Solutions of Weakly Nonlinear Singularly Perturbed Equations Under Regular Degeneration

We consider a weakly nonlinear singularly perturbed equation in complex domains. The problem is posed about the possibility of splitting the equation into several components. By introducing new unknown functions, a system of two equations is obtained. Next, the asymptotic behavior of solutions of the resulting equations in complex domains is studied. It has been proven that the solution to each of these equations is dominant in certain parts of the areas under consideration. The solution of one of these equations determines the boundary lines and regions, and the solution of the other system determines the regular region.

Existence and Asymptotic Behavior of Solutions for Generalized Choquard Systems

Existence and Asymptotic Behavior of Solutions for Generalized Choquard Systems

Asymptotic Behavior of Solutions of the Nonstationary Schrödinger Equation with Potential that Slowly Depends on Time

Asymptotic Behavior of Solutions of the Nonstationary Schrödinger Equation with Potential that Slowly Depends on Time

The Quenching Solutions of a Singular Parabolic Equation

This article is dedicated to the study of the self-similar solutions of a nonlinear parabolic equation. More precisely, we consider the following uni-dimensional equation: (E) : ut(x, t) = (u m)_xx(x, t) − |x|^q u^−p (x, t), x ∈ R, t > 0, where m > 1, q > 1 and p > 0. Initially, we employed a fixed point theorem and an associated energy function to establish the existence of solutions. Subsequently, we derived some important results on the asymptotic behavior of solutions near the origin.

Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data

Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data

Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

Existence and asymptotic behavior of solutions for the Schrödinger–Born–Infeld system with steep potential well

On asymptotic behavior of solutions of stochastic differential equations in multidimensional space

Consider the multidimensional SDE dX(t) = a(X(t)) dt + b(X(t)) dW(t). We study the asymptotic behavior of its solution X(t) as t → ∞, namely, we study sufficient conditions of transience of its solution X(t), stabilization of its multidimensional angle X(t)/|X(t)|, and asymptotic equivalence of solutions of the given SDE and the following ODE without noise: dx(t) = a(x(t)) dt.

Asymptotic behavior of solutions and spectrum of states in the quantum scalar field theory in the Schwarzschild spacetime

Asymptotic behavior of solutions and spectrum of states in the quantum scalar field theory in the Schwarzschild spacetime

Existence and asymptotic behavior of solutions for non-linear wave equations of Kirchhoff type with viscoelasticity

"In this paper we discuss the global existence and the asymptotic behavior of solutions of an initial boundary value problem of a non-linear wave equation of Kirchhoff type. "

Mass concentration for Ergodic Choquard Mean-Field Games

We study concentration phenomena in the vanishing viscosity limit for second-order stationary Mean-Field Games systems defined in the whole space RN with Riesz-type aggregating coupling and external confining potential. In this setting, every player of the game is attracted toward congested areas and the external potential discourages agents to be far away from the origin. Assuming some restrictions on the strength of the attractive nonlocal term depending on the growth of the Hamiltonian, we study the asymptotic behavior of solutions in the vanishing viscosity limit. First, we obtain existence of classical solutions to potential-free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around minima of the potential.

A precise asymptotic description of half‐linear differential equations

AbstractWe study asymptotic behavior of solutions of nonoscillatory second‐order half‐linear differential equations. We give (in some sense optimal) conditions that guarantee generalized regular variation of all solutions, where no sign condition on the potential is assumed. For all of these solutions, we establish precise asymptotic formulas, where positive as well as negative potential is considered. We examine, as consequences, also equations with regularly varying coefficients, or with the coefficients viewed as perturbations of exponentials, or the equations under certain critical (double roots) settings. We make also asymptotic analysis of Poincaré–Perron solutions. Many of our results are new even in the linear case.

Motion planning and stabilization of nonholonomic systems using gradient flow approximations

Nonlinear control-affine systems described by ordinary differential equations with time-varying vector fields are considered in the paper. We propose a unified control design scheme with oscillating inputs for solving the trajectory tracking and stabilization problems under the bracket-generating condition. This methodology is based on the approximation of a gradient-like dynamics by trajectories of the designed closed-loop system. As an intermediate outcome, we characterize the asymptotic behavior of solutions of the considered class of nonlinear control systems with oscillating inputs under rather general assumptions on the generating potential function. These results are applied to examples of nonholonomic trajectory tracking and obstacle avoidance.

Existence and asymptotic behavior of solutions for Kirchhoff equations with general Choquard-type nonlinearities

Existence and asymptotic behavior of solutions for Kirchhoff equations with general Choquard-type nonlinearities

Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations

We study the existence and multiplicity of solutions for the Schrodinger-Bopp-Podolsky system $$\displaylines{ -\Delta u + \phi u = \omega u \quad\text{ in } \Omega \cr a^2\Delta^2\phi-\Delta \phi = u^2 \quad\text{ in } \Omega \cr u=\phi=\Delta\phi=0\quad\text{ on } \partial\Omega \cr \int_{\Omega} u^2\,dx =1 }$$ where \(\Omega\) is an open bounded and smooth domain in \(\mathbb R^{3}\), \(a>0 \) is the Bopp-Podolsky parameter. The unknowns are \(u,\phi:\Omega\to \mathbb R\) and \(\omega\in\mathbb R\). By using variational methods we show that for any \(a>0\) there are infinitely many solutions with diverging energy and divergent in norm. We show that ground states solutions converge to a ground state solution of the related classical Schrodinger-Poisson system, as \(a\to 0\). For more information see https://ejde.math.txstate.edu/Volumes/2023/66/abstr.html