Modeling and numerical simulation of linear and nonlinear spacecraft attitude dynamics and gravity gradient moments: A comparative study

In this paper linear and nonlinear models of spacecraft attitude dynamics equations and gravity gradient moments are investigated. In addition, effects of gravity gradient moments on attitude dynamics of the satellite are studied. The purpose of this paper is to present a comparison between nonlinear and linear models of spacecraft attitude dynamics and gravity gradient moments in order to determine divergence of linear approximation from the nonlinear model. Simulation results indicate that designer of spacecraft attitude control subsystem should be meticulous in applying linear approximation of equations especially in low earth orbits. Consequently, finding an upper bound for small angle to keep the linear model valid and precise enough would be a vital part of using linear approximation. Results supported by numerical examples demonstrate various features of this study.     

A STRONG ERGODIC THEOREM FOR SOME NONLINEAR MATRIX MODELS FOR THE DYNAMICS OF STRUCTURED POPULATIONS

A general class of matrix difference equation models for the dynamics of discrete class structured populations in discrete time which possess a certain general type of nonlinearity introduced by Leslie for age-structured populations is considered. Arbitrary structuring is allowed in that transitions between any two classes are permitted. It is shown that normalized class distributions for such nonlinear models globally approach a “stable class distribution” and thus possess a strong ergodic property exactly like that of the classical linear theory of demography. However, unlike in the linear theory according to which the total population size grows or dies exponentially, the dynamics of total population size in these nonlinear models are shown to be governed by a nonlinear, nonautonomous scalar difference equation. This difference equation is asymptotically autonomous, and theorems which relate the dynamics of total population size to those of this limiting equation are proved. Examples in which the results are applied to some nonlinear age-structure models found in the literature are given.     

The study and applications of photochemical-dynamical gravity wave model Ⅰ--Model description

A two-dimensional, nonlinear, compressible, diabatic, nonhydrostatic photochemical- dynamical gravity wave model has been advanced. The model includes diabetic process produced by photochemistry and the effect of gravity wave on atmospheric chemical species. In the horizontal direction, the pseudospectral method is used. The finite difference approximations are used in vertical direction z and time t. The FICE method is used to solve the model. The model results on small amplitude fluctuation are very close to those of linear theory, which demonstrates the correctness of the model.     

The study and applications of photochemical-dynamical gravity wave model I

A two-dimensional, nonlinear, compressible, diabatic, nonhydrostatic photochemical-dynamical gravity wave model has been advanced. The model includes diabetic process produced by photochemistry and the effect of gravity wave on atmospheric chemical species. In the horizontal direction, the pseudospectral method is used. The finite difference approximations are used in vertical direction z and time t. The FICE method is used to solve the model. The model results on small amplitude fluctuation are very close to those of linear theory, which demonstrates the correctness of the model.

     

Hamiltonian Structure and a Variational Principle for Grounded Abyssal Flow on a Sloping Bottom in a Mid‐Latitude β‐Plane

Observations, numerical simulations, and theoretical scaling arguments suggest that in mid‐latitudes, away from the high‐latitude source regions and the equator, the meridional transport of abyssal water masses along a continental slope correspond to geostrophic flows that are gravity or density driven and topographically steered. These dynamics are examined using a nonlinear reduced‐gravity geostrophic model that describes grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. It is shown that this model possesses a noncanonical Hamiltonian formulation. General nonlinear steady solutions to the model can be obtained for arbitrary bottom topography. These solutions correspond to nonparallel shear flows that flow across the planetary vorticity gradient. If the in‐flow current along the poleward boundary is strictly equatorward, then no shock can form in the solution in the mid‐latitude domain. It is also shown that the steady solutions satisfy the first‐order necessary conditions for an extremal to a suitably constrained potential energy functional. Sufficient conditions for the definiteness of the second variation of the constrained energy functional are examined. The theory is illustrated with a nonlinear steady solution corresponding to an abyssal flow with upslope and down slope groundings in the height field.     

The similarity forms and invariant solutions of two-layer shallow-water equations

In this study symmetry group properties and general similarity forms of the two-layer shallow-water equations are discussed by Lie group theory. We represent that Lie group theory can be used as an effective approach for investigation of the self-similar solutions for the shallow-water equations with variable inflow as the generalization of dimensional analysis that was used so far for a regular approach in the literature. We also represent that the results obtained by dimensional analysis are just a special case of the results obtained by Lie group theory and it is possible to obtain the new similarity forms and the new variable inflow functions for the study of gravity currents in two-layer flow under shallow-water approximations based on Lie group theory. The symmetry groups of the system of nonlinear partial differential equations are found and the corresponding similarity and reduced forms are obtained. Some similarity solutions of the reduced equations are investigated. It is shown that reduced equations and similarity forms of the system depend on the group parameters. We show that an analytic similarity solution for the system of equations can be found for some special values of them. For other values of the group parameters, the similarity solutions of the two-layer shallow-water equations representing the gravity currents with a variable inflow are found by the numeric integration.     

Spectral Bounds for Tricomi Problems and Application to Semilinear Existence and Existence with Uniqueness Results

For the linear Tricomi problem, it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum. Exploiting the linear solvability theory and spectral information, standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems. In particular, using the Leray-Schauder principle, existence of generalized solutions with sublinear nonlinearities is established. For sublinear or asymptotically linear nonlinearities that satisfy a Lipschitz condition, the contraction mapping principle is employed to give results on existence with uniqueness. The Lipschitz constant depends on lower bounds for the spectrum of the linear problem. For certain superlinear problems, maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence.     

Some thoughts on a nonlinear Schrödinger equation motivated by information theory

The arguments leading to a nonlinear generalization of the Schrödinger equation in the context of the maximum uncertainty principle are reviewed. The exact and perturbative properties of that equation depend on a free regulating/interpolating parameter η, which can be fixed using energetics as is shown here. A linear theory with an external potential that reproduces some unusual exact solutions of the nonlinear equation is also discussed, together with possible symmetry enhancements in the nonlinear theory.     

One of the simplest means of accounting for nonlinearity in viscoelastic media

The author examines a simple extension, to the nonlinear case, of memory-type theory based on the Boltzmann-Volterra superposition principle. It is shown that given certain assumptions the quasi-linear theory of viscoelasticity reduces to introduction into the equations of linear memory theory of a single stress- or strain-intensity function. This function is determined from creep or relaxation tests. A successive-approximation method is presented for solving problems of nonlinear viscoelasticity with the aid of the equations introduced. It is shown that in the case of simple loading the equations of the theory of small elastic-plastic deformations are an analog of the equations considered.Mekhanika Polimerov, Vol. 3, No. 2, pp. 207–212, 1967     

On the Computation of Nonlinear Planetary Waves

A numerical method is developed to solve a class of nonlinear, nonlocal eigenvalue problems defined in an infinite strip, and is applied to compute solitary planetary waves in a sheared zonal current on the beta-plane. This method, an iterative procedure derived from the natural variational structure of these problems, is implemented in the physical case when the ambient parallel flow has a linear or a quadratic velocity profile. The results of the numerical experiments establish rigorous limits on the range of validity of the formal asymptotic theory of weakly nonlinear long waves, and also reveal some new phenomena involving strongly nonlinear waves. The iterative procedure is analyzed in a general setting, and is shown to be globally convergent without restriction on the wave amplitude.     

非线性Lipschitz连续算子的定量性质(Ⅲ)──glb-Lipschitz数

本文引进非线性Ｌｉｐｓｃｈｉｔｚ算子Ｔ的ｇｌｂ－Ｌｉｐｓｃｈｉｔｚ数ｌ（Ｔ）,并证明：ｌ（Ｔ）定量刻画非线性Ｌｉｐｓｃｈｉｔｚ连续算子全体所构成的赋半范算子空间中可逆算子Ｔ保持可逆的最大扰动半径,因而具有特别重要意义。所获结果被应用来建立“非线性扰动引理”、非线性算子条件数、推广线性算子逼近理论和建立与矩阵理论中Ｇｅｒｓｃｈｇｏｒｉｎ圆盘定理对应的非线性Ｌｉｐｓｃｈｉｔｚ连续算子谱集的包含域。     

Optimality conditions and duality in continuous programming II. The linear problem revisited

This is Part II of a two-part paper; the purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of infinite-dimensional programming, and (b) to obtain fruitful applications in continuous time programming. In Part II the continuous time version of Farkas' theorem developed in Part I serves as the foundation for the duality theory for a broad class of linear continuous time programming problems distinct from those previously examined. In particular, we establish duality under analytic conditions, e.g., whether the given functions are measurable or continuous, that are weaker, and algebraic conditions that are more general, than those previously imposed. The new class of problems arising from these conditions allows for several important resource allocation problems previously excluded from consideration. In addition, an assumption needed to prove the Kuhn-Tucker theorem for the nonlinear problem of Part I is shown in the linear case to be completely analogous to the well-known Slater condition utilized in finite-dimensional programming theory. An example is given that exhibits the essential role of the constraint qualification in linear continuous time programming, a result at variance with the theory in finite dimensions but consistent with other results concerning linear programs in infinite-dimensional spaces.     

Intelligent control of time-varying dynamical systems using CMAC artificial neural network

In this paper, some successful case studies are presented on the intelligent control of time-varying linear/nonlinear dynamical systems using the Cerebellar Model Arithmetic Computer (CMAC) artificial neural network, and a recently developed unified linear system theory and a novel control technique called Extended-Mean Assignment Control (EMAC). Thanks to CMAC's simple and effective training algorithm and fast learning convergence, satisfactory and encouraging simulation results of angle-of-attack control of an aerobreak re-entry spacecraft and attitude control of an earth orbiting space vehicle subject to gravity gradient torque are obtained and presented in this paper. Suggestions for further investigations on the control of time-varying dynamical systems using CMAC and EMAC are proposed.     

非线性 Lipschitz 连续算子的定量性质(III)------glb-Lipschitz数

本文引进非线性Lipschitz算子T的glb-Lipschitz数l(T),并证明l(T)定量刻画非线性Lipschitz连续算子全体所构成的赋半范算子空间中可逆算子T保持可逆的最大扰动半径,因而具有特别重要意义.所获结果被应用来建立``非线性扰动引理'、非线性算子条件数、推广线性算子逼近理论和建立与矩阵理论中Gerschgorin圆盘定理对应的非线性Lipschitz连续算子谱集的包含域.     

Unconditional Nonlinear Stability in Temperature-Dependent Viscosity Flow in a Porous Medium

The equations of flow in porous media attributable to Forchheimer are considered. In particular, the problem of thermal convection in such a medium is addressed when the viscosity varies with temperature. It is shown that nonlinear stability may be achieved naturally for all initial data by working with L ³ or L ⁴ norms. It is also shown that L ² theory is not sufficient for such unconditional stability. Previous work has established nonlinear stability for vanishingly small initial data thresholds, but we believe this is the first analysis that addresses the important physical problem of unconditional stability. It is shown how to extend the nonlinear analysis for a viscosity linear in temperature to the cases when the viscosity may be quadratic or when penetrative convection is allowed in the layer.     

On the Equation of a Rotating Film

We study positive periodic solutions to a nonautonomous nonlinear third-order ordinary differential equation of the theory of motion of a viscous incompressible fluid with free boundary. This equation describes the steady motions of a thin layer of a fluid film on the surface of a rotating horizontal cylinder in the gravity field. The linear operator on the left-hand side of the equation has a three-dimensional kernel. Moreover, the equation contains two nonnegative parameters proportional to the gravity acceleration and surface tension. Depending on these parameters the problem in question may have either two solutions or no solutions at all. We establish some qualitative properties of solutions to the problem: in particular, their asymptotic behavior at the extremal values of the parameters.     

On the Transition from Two-Dimensional to Three-Dimensional Water Waves

There are essentially two types of three-dimensional water waves: waves that bifurcate from the state of rest (these waves are commonly called short-crested waves or forced waves), and waves that bifurcate from a two-dimensional wave of finite amplitude (these waves are sometimes called spontaneous waves). This paper deals with spontaneously generated three-dimensional waves. To understand this phenomenon better from a mathematical point of view, it is helpful to work on model equations rather than on the full equations. Such an attempt was made formally by Martin in 1982 on the nonlinear Schrödinger equation, but it is shown here that it is hard to justify his results mathematically because of the hyperbolicity of the nonlinear Schrödinger equation for gravity waves. On the other hand, in some parameter regimes, the nonlinear Schrödinger equation becomes elliptic. In that case, the appearance of spontaneous three-dimensional waves can be shown rigorously by using a dynamical systems approach. The results are extended to the Benney–Roskes–Davey–Stewartson equations when they are both elliptic. Various types of three-dimensional waves bifurcating from a two-dimensional periodic wave are obtained.     

有限变形弹性杆中三种非线性弥散波

在一维弹性细杆拉压、扭转和弯曲波的经典线性理论基础上,分别计入有限变形和弥散效应,借助Hamilton变分原理,由统一的方法导出了3种非线性弥散波的演化方程．对3种演化方程进行了定性分析．结果表明,这些方程在相平面上存在同宿轨道或异宿轨道,分别相应于孤波解或冲击波解．根据齐次平衡原理,用Jacobi椭圆函数展开对这些演化方程进行了求解,在一定的条件下它们均可能存在孤立波解或冲击波解,这与方程的定性分析完全一致．     

Global nonlinear stability of convection in a heat generating fluid filled channel with a moving boundary

The energy method is employed to investigate the stability of a steady convective flow in a heat generating fluid arising due to the combined effect of buoyancy, shear and pressure gradient. By introducing a suitable generalized energy functional and using energy inequalities sufficient conditions for the existence of such a flow are found. An analysis through the variational principles is then made to find sharper limits for nonlinear stability. Comparisons are made with linear results in the literature and it is shown that the linear theory fails to capture the physics of the onset of secondary flow.     

Variational Characterizations of Weak Solutions to the Dirichlet Problem for Mixed‐Type Equations

For linear and nonlinear second‐order partial differential equations of mixed elliptic‐hyperbolic type, we prove that weak solutions to the Dirichlet problem are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functional restricted to suitable infinite‐dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights associated to the linear operator. This spectral theory has been recently developed by the authors, which in turn exploits weak well‐posedness results obtained by Morawetz and the authors. © 2015 Wiley Periodicals, Inc.