Nonautonomous Equations Research Articles

Stabilization of non-autonomous infinite-dimensional systems depending on a parameter

ABSTRACT In this paper, we investigate the practical stabilization of non-autonomous infinite-dimensional systems depending on a parameter. By the Lyapunov method, several sufficient conditions are provided to obtain the practical stabilization of two classes of non-autonomous semilinear evolution equations in Hilbert spaces. We propose some classes of memoryless state linear and nonlinear feedback controllers. Numerical examples are given to illustrate the validity of the proposed theoretical results.

Imbalance vibration suppression for asymmetric rotors via an enhanced automatic dynamic balancer

Automatic dynamic balancer (ADB) is a passive ball-bearing type of vibration control device and has been applied successfully on some rotating machineries, i.e., hand-held machine tools and optical disk drives. Recently, an enhanced automatic dynamic balancer (EADB) has been developed to extend the capability of the ADB and to achieve a perfect-balancing condition in a wider rotor speed operation range. However, the EADB cannot be applied directly to the imbalance vibration suppression for asymmetric rotors since a multi-frequency whirling limit cycle in the asymmetrically supported rotor/ADB systems prevents the assumed solution of a single-frequency whirling limit cycle for the symmetric supported rotor/ADB systems. To address the aforementioned limitation, this paper proposes an imbalance vibration suppression method for asymmetric rotors via an EADB, which can achieve rotor perfect balancing condition in a wider operation range. The non-autonomous equations of motion for an asymmetrically supported planar rotor with an EADB system is established via Lagrange's method. The perfect-balancing equilibrium of the system is solved analytically, and the multi-frequency whirling limit cycle is obtained via a multi-tone harmonic balance method whose tones are predetermined by the matrix pencil method. To destabilize the multi-frequency whirling limit cycle and to guarantee the stable perfect balancing conditions for an asymmetric rotor with an EADB, a numerical continuation method is adopted to design parameters for the EADB. The stability analysis of the perfect-balancing equilibrium and the multi-frequency whirling limit cycle are conducted by Floquet theory. The enlarged unstable whirling limit cycle solution proves the effectiveness of the proposed method.

Out-of-plane equilibrium points of extra-solar planets in the central binaries PSR B1620-26 and Kepler-16 with clusters of material points and variable masses

In this paper, the existence and stability of the out-of-plane equilibrium points of extra-solar planets using the model of the restricted three-body problem, when the central binaries are surrounded by a clusters of material points and their motion is described by the Gylden-Mestschersky problem, is investigated. The masses of the central binaries and the clusters of material points are assumed to vary with time at the same rate in accordance with the unified Mestschersky law. The governing non-autonomous equations of motion of the model is obtained and for a complete transformation to the autonomized systems to be obtainable, we assume that the mass of the clusters of materials varies at the same rate as the masses of the central binaries while the radius and the parameters which determines the density profile of the clusters vary at the same rate as the distances. Thus, we introduce a transformation which defines the mass variation of the clusters with the help of the unified Mestschersky law and the Mestschersky transformation and obtain the autonomized system with constant coefficients. Next, the coordinates of the out-of-plane points are found using perturbation method, the Newton-Raphson's method and analytical approximations in the form of power series in the cluster's mass coefficient. The existence of these points depends solely on the mass variation parameter κ, although the points are however affected by the total mass of the clusters of material points Md and the mass parameter υ. Two pairs of out-of-plane equilibrium points are found; the first pair of equilibrium points L6,7 exists for κ>1 while the second pair L8,9 exists for κ>1 and ξ<υ(κ−1), where ξ is the abscissa of the out-of-plane equilibrium points. Further, for our numerical evidence, we compute the out-of-plane equilibrium points of two extra-solar planets PSR B1620-26b and Kepler-16b, in the binary systems PSR B1620-26 and Kepler-16, respectively. Our numerical results shows that the equilibrium points for PSR B1620-26b, exist when the mass variation parameter and the mass of the clusters lie in the intervals 1<κ≤4 and 0≤Md≤0.04, respectively, while for Kepler-16b, the points exist for 1<κ≤4.1 and 0≤Md≤0.04. Also, it is seen that in the absence of the clusters more out-of-plane equilibrium points evolve and are located far away from the line joining the binaries, while in the presence of the clusters, less out-of-plane equilibrium points evolve, but the evolved points are located closest to the line joining the binaries. Finally, we analyze the stability of the equilibrium points of the autonomized and non-autonomous systems, and both were found to be unstable equilibrium points.

Uniform attractors for nonautonomous 2D MHD equations with partial dissipation

In this article, we study the long-time behavior of solutions of nonautonomous 2D magnetohydrodynamics equations with partial dissipation and time-dependent external forcing, which is weakly normal. We prove that the system possesses strong uniform attractors. Furthermore, the structure of the uniform attractors is obtained, and the fractal dimension is estimated for the kernel sections of the uniform attractors.

On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\)

In this article, we study the global behavior of the following higher-order nonautonomous rational difference equation &#x0D; \[&#x0D; y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}},\quad n=0,1,...,&#x0D; \]&#x0D; where \(\left\{\alpha_n\right\}_{n\geq0}\) is a bounded sequence of&#x0D; positive numbers, \(k,r\) are nonnegative integers such that \(r

Cauchy problem for non-autonomous fractional evolution equations

In this paper, we study the solvability of Cauchy problem for a class of non-autonomous fractional evolution equation with Caputo’s fractional derivative of order \(\alpha \in (1,2)\), which can be applied to model the time dependent coefficients fractional differential systems. We first introduce an operator family and analyze its properties, by the iterative method, we construct a solution to an operator-valued Volterra equation, which is the most critical ingredient to prove solvability of the problem. Finally, based on the solution operators we establish the existence and uniqueness of classical solutions.

Optimal regularity results for the one-dimensional prescribed curvature equation via the strong maximum principle

A refined version of the strong maximum principle is proven for a class of second order ordinary differential equations with possibly discontinuous non-monotone nonlinearities. Then, exploiting this tool, some optimal regularity results, recently established by López-Gómez and Omari in [15–17] for the bounded variation solutions of non-autonomous quasilinear equations driven by the one-dimensional curvature operator, are substantially improved by admitting general prescribed curvatures and by incorporating general boundary conditions. The novel approach developed here yields a new, deeper, interpretation of the assumptions introduced in our previous papers, simultaneously clarifying their meaning and making fully transparent their connection with the strong maximum principle.

Solutions to nonlocal evolution equations governed by non-autonomous forms and demicontinuous nonlinearities

We deal with the existence of solutions having L^2-regularity for a class of non-autonomous evolution equations. Associated with the equation, a general non-local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray–Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity.

Maximal regularity for semilinear non-autonomous evolution equations in temporally weighted spaces

We consider the problem of maximal regularity for the semilinear non-autonomous evolution equations u′(t)+A(t)u(t)=F(t,u),t-a.e.,u(0)=u0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\\, t \ ext {-a.e.}, \\, u(0)=u_0. \\end{aligned}$$\\end{document}Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space mathcal {H}. We prove the maximal regularity result in temporally weighted L^2-spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value u_0 and the inhomogeneous term F. Our results are motivated by boundary value problems.

Arbitrary settling time control with prescribed performance for high‐order nonlinear systems

AbstractThis paper studies an arbitrary convergence time tracking controller design problem for high‐order nonlinear systems. Our aim is to enhance the existing free‐will arbitrary time control (FATC), which cannot track time‐varying reference signals. To do so, a new nonautonomous equation is firstly introduced to enable an arbitrary settling time stability for one‐order systems. Then, a prescribed performance control (PPC) technology is integrated for general high order systems. By a backstepping Lyapunov analysis with iteratively contructed nonautonomous equations, it is proved that the closed‐loop system satisfies the prescribed performance and all signals of the closed‐loop system are arbitrary settling time stable (ASTS). Compared with the existing results, the tracking control problem with arbitrary convergence time is addressed by a continuous control law, which shows a better transient performance also. Simulation results confirm the effectiveness of the proposed control method.

Invariant manifolds for difference equations with generalized trichotomies

On an arbitrary Banach space, assuming that a linear nonautonomous difference equation \linebreak $x_{m+1} = A_m x_m$ admits a very general type of trichotomy, we establish conditions for the existence of global Lipschitz invariant center manifolds of the perturbed equation $x_{m+1} = A_m x_m + f_m(x_m)$. Our results not only improve results already existing in the literature, but also include new cases.

Monotone iterations method for fractional diffusion equations

In recent years, there has been a growing interest on non-localmodels because of their relevance in many practical applications. Awidely studied class of non-local models involves fractional orderoperators. They usually describe anomalous diffusion. Inparticular, these equations provide a more faithful representationof the long-memory and nonlocal dependence of diffusion in fractaland porous media, heat flow in media with memory, dynamics ofprotein in cells etc.&#x0D; For $a\in (0, 1)$, we investigate the nonautonomous fractionaldiffusion equation:&#x0D; $D^a_{*,t} u - Au = f(x, t,u),$&#x0D; where$D^a_{*,t}$ is the Caputo fractional derivative and $A$ is auniformly elliptic operator with smooth coefficients depending onspace and time. We consider these equations together with initialand quasilinear boundary conditions.&#x0D; The solvability of such problems in H\"older spaces presupposesrigid restrictions on the given initial data. These compatibilityconditions have no physical meaning and, therefore, they can beavoided, if the solution is sought in larger spaces, for instance inweighted H\"older spaces.&#x0D; We give general existence and uniqueness result andprovide some examples of applications of the main theorem. The maintool is the monotone iterations method. Preliminary we developed thelinear theory with existence and comparison results. The principleuse of the positivity lemma is the construction of a monotonesequences for our problem. Initial iteration may be taken as eitheran upper solution or a lower solution. We provide some examples ofupper and lower solution for the case of linear equations andquasilinear boundary conditions. We notice that this approach canalso be extended to other problems and systems of fractionalequations as soon as we will be able to construct appropriate upperand lower solutions.

Upper semicontinuity of optimal attractors for viscoelastic equations lacking strong damping

ABSTRACT This paper devotes to investigating the asymptotic behavior of viscoelastic equation with fading memory lacking strong damping term where . Its key feature is the lack of strong mechanical damping term replaced by weaker dissipative term (i.e. memory term). At present, the existing results only considered the existence and regularity of global attractors, but do not obtain the existence of the strong attractors, because the compactness is difficult to get in highly regular space. Accordingly, in current work, we will prove the existence of strongly global attractors with only weak dissipation when the nonlinearity f satisfies critical growth by using the dominated convergence theorem and the asymptotically contractive function method. Moreover, the upper semicontinuity of the strongly global attractors is also obtained when the perturbed parameter , which deepens some conclusions that exist in Conti et al. [Asymptotics of viscoelastic materials with nonlinear density and memory effects. J Differ Equ. 2018;264:4235–4259] and Qin et al. [Uniform attractors for a nonautonomous viscoelastic equation with a past history. Nonlinear Theory Methods Appl. 2014;101:1–15].

Nonlinearizable solutions in an eigenvalue problem for Maxwell's equations with nonhomogeneous nonlinear permittivity in a layer

AbstractThe paper focuses on a problem arising in nonlinear guided wave optics. The problem for Maxwell's equations with nonhomogeneous nonlinear constitutive law is reduced to an eigenvalue problem for a system of nonlinear ordinary differential equations with local boundary conditions. We suggest a novel approach to study eigenvalue problems for nonlinear nonautonomous equations based on studying an integral characteristic equation (ICE) of the problem. Using asymptotical analysis of the ICE, we prove existence of infinitely many eigenvalues and find their distribution. It is important that the corresponding linear problem has only a finite number of eigenvalues.

Asymptotic Autonomy of Attractors for Stochastic Fractional Nonclassical Diffusion Equations Driven by a Wong–Zakai Approximation Process on ℝn

In this paper, we consider the backward asymptotically autonomous dynamical behavior for fractional non-autonomous nonclassical diffusion equations driven by a Wong–Zakai approximations process in Hs(Rn) with s∈(0,1). We first prove the existence and backward time-dependent uniform compactness of tempered pullback random attractors when the growth rate of nonlinearities have a subcritical range. We then show that, under the Wong–Zakai approximations process, the components of the random attractors of a non-autonomous dynamical system in time can converge to those of the random attractor of the limiting autonomous dynamical system in Hs(Rn).

Well-posedness and stability results for some nonautonomous abstract linear hyperbolic equations with memory

Well-posedness and stability results for some nonautonomous abstract linear hyperbolic equations with memory

Non-autonomous maximal regularity for fractional evolution equations

We consider the problem of maximal regularity for the semilinear non-autonomous fractional equations $$\begin{aligned} B^\alpha u(t)+A(t)u(t)=F(t,u),\, t \text {-a.e}. \end{aligned}$$ Here $$B^\alpha $$ denotes the Riemann–Liouville fractional derivative of order $$\alpha \in (0,1)$$ w.r.t. time and the time- dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space $${\mathcal {H}}.$$ We prove maximal $$L^p$$ -regularity results and other regularity properties for the solution of the above equation under minimal regularity assumptions on the forms and the inhomogeneous term F.

On the admissibility of observation operators for evolution families

This paper is concerned with unbounded observation operators for non-autonomous evolution equations. Fix \(\tau > 0\) and let \(\left( A(t)\right) _{t \in [0,\tau ]} \subset \mathcal {L}(D,X)\), where D and X are two Banach spaces such that D is continuously and densely embedded into X. We assume that the operator A(t) has maximal regularity for all \(t \in [0,\tau ]\) and that \(A(\cdot ) : [0,\tau ] \rightarrow \mathcal {L}(D,X)\) satisfies a regularity condition (viz. relative p-Dini for some \(p \in (1,\infty )\)). At first sight, we show that there exists an evolution family on X associated to the problem Then we prove that an observation operator is admissible for \(A(\cdot )\) if and only if it is admissible for each A(t) for all \(t \in [0,\tau )\).

One-component and two-component Peregrine bump and integrated breather solutions for a partially nonlocal nonlinearity with a parabolic potential

Objective:We establish a projecting transformation from nonautonomous equation into autonomous one, and seek out explicit one-component and two-component solutions of a (2+1)-dimensional coupled nonautonomous nonlinear Schrödinger model for a partially nonlocal nonlinearity with a parabolic potential. Methods:We combine these solutions of autonomous one into the projecting transformation and use the Darboux method to find solutions. Result:We take into account a (2+1)-dimensional coupled nonautonomous nonlinear Schrödinger model for a partially nonlocal nonlinearity with a parabolic potential, and establish a projecting transformation from nonautonomous equation into autonomous one. Moreover, we seek out explicit one-component and two-component solutions, e.g. explicit Peregrine bump solution and explicit integrated breather solutions.

A Non-autonomous Damped Wave Equation with a Nonlinear Memory Term

A Non-autonomous Damped Wave Equation with a Nonlinear Memory Term