Showing papers on "Nonlinear system published in 1992"

Semigroups of Linear Operators and Applications to Partial Differential Equations

Abstract: 1 Generation and Representation.- 1.1 Uniformly Continuous Semigroups of Bounded Linear Operators.- 1.2 Strongly Continuous Semigroups of Bounded Linear Operators.- 1.3 The Hille-Yosida Theorem.- 1.4 The Lumer Phillips Theorem.- 1.5 The Characterization of the Infinitesimal Generators of C0 Semigroups.- 1.6 Groups of Bounded Operators.- 1.7 The Inversion of the Laplace Transform.- 1.8 Two Exponential Formulas.- 1.9 Pseudo Resolvents.- 1.10 The Dual Semigroup.- 2 Spectral Properties and Regularity.- 2.1 Weak Equals Strong.- 2.2 Spectral Mapping Theorems.- 2.3 Semigroups of Compact Operators.- 2.4 Differentiability.- 2.5 Analytic Semigroups.- 2.6 Fractional Powers of Closed Operators.- 3 Perturbations and Approximations.- 3.1 Perturbations by Bounded Linear Operators.- 3.2 Perturbations of Infinitesimal Generators of Analytic Semigroups.- 3.3 Perturbations of Infinitesimal Generators of Contraction Semigroups.- 3.4 The Trotter Approximation Theorem.- 3.5 A General Representation Theorem.- 3.6 Approximation by Discrete Semigroups.- 4 The Abstract Cauchy Problem.- 4.1 The Homogeneous Initial Value Problem.- 4.2 The Inhomogeneous Initial Value Problem.- 4.3 Regularity of Mild Solutions for Analytic Semigroups.- 4.4 Asymptotic Behavior of Solutions.- 4.5 Invariant and Admissible Subspaces.- 5 Evolution Equations.- 5.1 Evolution Systems.- 5.2 Stable Families of Generators.- 5.3 An Evolution System in the Hyperbolic Case.- 5.4 Regular Solutions in the Hyperbolic Case.- 5.5 The Inhomogeneous Equation in the Hyperbolic Case.- 5.6 An Evolution System for the Parabolic Initial Value Problem.- 5.7 The Inhomogeneous Equation in the Parabolic Case.- 5.8 Asymptotic Behavior of Solutions in the Parabolic Case.- 6 Some Nonlinear Evolution Equations.- 6.1 Lipschitz Perturbations of Linear Evolution Equations.- 6.2 Semilinear Equations with Compact Semigroups.- 6.3 Semilinear Equations with Analytic Semigroups.- 6.4 A Quasilinear Equation of Evolution.- 7 Applications to Partial Differential Equations-Linear Equations.- 7.1 Introduction.- 7.2 Parabolic Equations-L2 Theory.- 7.3 Parabolic Equations-Lp Theory.- 7.4 The Wave Equation.- 7.5 A Schrodinger Equation.- 7.6 A Parabolic Evolution Equation.- 8 Applications to Partial Differential Equations-Nonlinear Equations.- 8.1 A Nonlinear Schroinger Equation.- 8.2 A Nonlinear Heat Equation in R1.- 8.3 A Semilinear Evolution Equation in R3.- 8.4 A General Class of Semilinear Initial Value Problems.- 8.5 The Korteweg-de Vries Equation.- Bibliographical Notes and Remarks.

Sliding Modes in Control and Optimization

Abstract: I. Mathematical Tools.- 1 Scope of the Theory of Sliding Modes.- 1 Shaping the Problem.- 2 Formalization of Sliding Mode Description.- 3 Sliding Modes in Control Systems.- 2 Mathematical Description of Motions on Discontinuity Boundaries.- 1 Regularization Problem.- 2 Equivalent Control Method.- 3 Regularization of Systems Linear with Respect to Control.- 4 Physical Meaning of the Equivalent Control.- 5 Stochastic Regularization.- 3 The Uniqueness Problems.- 1 Examples of Discontinuous Systems with Ambiguous Sliding Equations.- 1.1 Systems with Scalar Control.- 1.2 Systems Nonlinear with Respect to Vector-Valued Control.- 1.3 Example of Ambiguity in a System Linear with Respect to Control ..- 2 Minimal Convex Sets.- 3 Ambiguity in Systems Linear with Respect to Control.- 4 Stability of Sliding Modes.- 1 Problem Statement, Definitions, Necessary Conditions for Stability ..- 2 An Analog of Lyapunov's Theorem to Determine the Sliding Mode Domain.- 3 Piecewise Smooth Lyapunov Functions.- 4 Quadratic Forms Method.- 5 Systems with a Vector-Valued Control Hierarchy.- 6 The Finiteness of Lyapunov Functions in Discontinuous Dynamic Systems.- 5 Singularly Perturbed Discontinuous Systems.- 1 Separation of Motions in Singularly Perturbed Systems.- 2 Problem Statement for Systems with Discontinuous control.- 3 Sliding Modes in Singularly Perturbed Discontinuous Control Systems.- II. Design.- 6 Decoupling in Systems with Discontinuous Controls.- 1 Problem Statement.- 2 Invariant Transformations.- 3 Design Procedure.- 4 Reduction of the Control System Equations to a Regular Form.- 4.1 Single-Input Systems.- 4.2 Multiple-Input Systems.- 7 Eigenvalue Allocation.- 1 Controllability of Stationary Linear Systems.- 2 Canonical Controllability Form.- 3 Eigenvalue Allocation in Linear Systems. Stabilizability.- 4 Design of Discontinuity Surfaces.- 5 Stability of Sliding Modes.- 6 Estimation of Convergence to Sliding Manifold.- 8 Systems with Scalar Control.- 1 Design of Locally Stable Sliding Modes.- 2 Conditions of Sliding Mode Stability "in the Large".- 3 Design Procedure: An Example.- 4 Systems in the Canonical Form.- 9 Dynamic Optimization.- 1 Problem Statement.- 2 Observability, Detectability.- 3 Optimal Control in Linear Systems with Quadratic Criterion.- 4 Optimal Sliding Modes.- 5 Parametric Optimization.- 6 Optimization in Time-Varying Systems.- 10 Control of Linear Plants in the Presence of Disturbances.- 1 Problem Statement.- 2 Sliding Mode Invariance Conditions.- 3 Combined Systems.- 4 Invariant Systems Without Disturbance Measurements.- 5 Eigenvalue Allocation in Invariant System with Non-measurable Disturbances.- 11 Systems with High Gains and Discontinuous Controls.- 1 Decoupled Motion Systems.- 2 Linear Time-Invariant Systems.- 3 Equivalent Control Method for the Study of Non-linear High-Gain Systems.- 4 Concluding Remarks.- 12 Control of Distributed-Parameter Plants.- 1 Systems with Mobile Control.- 2 Design Based on the Lyapunov Method.- 3 Modal Control.- 4 Design of Distributed Control of Multi-Variable Heat Processes.- 13 Control Under Uncertainty Conditions.- 1 Design of Adaptive Systems with Reference Model.- 2 Identification with Piecewise-Continuous Dynamic Models.- 3 Method of Self-Optimization.- 14 State Observation and Filtering.- 1 The Luenberger Observer.- 2 Observer with Discontinuous Parameters.- 3 Sliding Modes in Systems with Asymptotic Observers.- 4 Quasi-Optimal Adaptive Filtering.- 15 Sliding Modes in Problems of Mathematical Programming.- 1 Problem Statement.- 2 Motion Equations and Necessary Existence Conditions for Sliding Mode.- 3 Gradient Procedures for Piecewise Smooth Function.- 4 Conditions for Penalty Function Existence. Convergence of Gradient Procedure.- 5 Design of Piecewise Smooth Penalty Function.- 6 Linearly Independent Constraints.- III. Applications.- 16 Manipulator Control System.- 1 Model of Robot Arm.- 2 Problem Statement.- 3 Design of Control.- 4 Design of Control System for a Two-joint Manipulator.- 5 Manipulator Simulation.- 6 Path Control.- 7 Conclusions.- 17 Sliding Modes in Control of Electric Motors.- 1 Problem Statement.- 2 Control of d. c. Motor.- 3 Control of Induction Motor.- 4 Control of Synchronous Motor.- 18 Examples.- 1 Electric Drives for Metal-cutting Machine Tools.- 2 Vehicle Control.- 3 Process Control.- 4 Other Applications.- References.

User’s guide to viscosity solutions of second order partial differential equations

Abstract: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

Solitons, Nonlinear Evolution Equations and Inverse Scattering

Abstract: Solitons have been of considerable interest to mathematicians since their discovery by Kruskal and Zabusky. This book brings together several aspects of soliton theory currently only available in research papers. Emphasis is given to the multi-dimensional problems arising and includes inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multi-dimensions and the ∂ method. Thus, this book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Darboux transformations and solitons

Abstract: In 1882 Darboux proposed a systematic algebraic approach to the solution of the linear Sturm-Liouville problem. In this book, the authors develop Darboux's idea to solve linear and nonlinear partial differential equations arising in soliton theory: the non-stationary linear Schrodinger equation, Korteweg-de Vries and Kadomtsev-Petviashvili equations, the Davey Stewartson system, Sine-Gordon and nonlinear Schrodinger equations 1+1 and 2+1 Toda lattice equations, and many others. By using the Darboux transformation, the authors construct and examine the asymptotic behaviour of multisoliton solutions interacting with an arbitrary background. In particular, the approach is useful in systems where an analysis based on the inverse scattering transform is more difficult. The approach involves rather elementary tools of analysis and linear algebra so that it will be useful not only for experimentalists and specialists in soliton theory, but also for beginners with a grasp of these subjects.

Homogenization and two-scale convergence

Abstract: Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.

Gaussian networks for direct adaptive control

Abstract: A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture uses a network of Gaussian radial basis functions to adaptively compensate for the plant nonlinearities. Under mild assumptions about the degree of smoothness exhibit by the nonlinear functions, the algorithm is proven to be globally stable, with tracking errors converging to a neighborhood of zero. A constructive procedure is detailed, which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy. A stable weight adjustment mechanism is determined using Lyapunov theory. The network construction and performance of the resulting controller are illustrated through simulations with example systems. >

A simple observer for nonlinear systems applications to bioreactors

Abstract: An observer for nonlinear systems is constructed under rather general technical assumptions (the fact that some functions are globally Lipschitz). This observer works either for autonomous systems or for nonlinear systems that are observable for any input. A tentative application to biological systems is described. >

L/sub 2/-gain analysis of nonlinear systems and nonlinear state-feedback H/sub infinity / control

Abstract: Previously obtained results on L2-gain analysis of smooth nonlinear systems are unified and extended using an approach based on Hamilton-Jacobi equations and inequalities, and their relation to invariant manifolds of an associated Hamiltonian vector field. On the basis of these results a nonlinear analog is obtained of the simplest part of a state-space approach to linear H/sub infinity / control, namely the state feedback H/sub infinity / optimal control problem. Furthermore, the relation with H/sub infinity / control of the linearized system is dealt with. >

On undamped heat waves in an elastic solid

Abstract: This paper is concerned with thermoelastic material behavior whose constitutive response functions possess thermal features that are more general than in the usual classical thermoelasticity. After a general development of the constitutive equations in the context of both nonlinear and linear theories, attention is focused on the latter. In particular, the one-dimensional version of the equation for the determination of temperature in the linearized theory provides an easy comparative basis of its predictive capability: In one special case where the Fourier conductivity is dominant, the temperature equation reduces to the classical Fourier law of heat conduction, which does not permit the possibility of undamped thermal waves; however,'in another special case in which the effect of conductivity is negligible, the equation has undamped thermal wave solutions without energy dissipation.

Multiple dipole modeling and localization from spatio-temporal MEG data

Abstract: The authors present general descriptive models for spatiotemporal MEG (magnetoencephalogram) data and show the separability of the linear moment parameters and nonlinear location parameters in the MEG problem. A forward model with current dipoles in a spherically symmetric conductor is used as an example: however, other more advanced MEG models, as well as many EEG (electroencephalogram) models, can also be formulated in a similar linear algebra framework. A subspace methodology and computational approach to solving the conventional least-squares problem is presented. A new scanning approach, equivalent to the statistical MUSIC method, is also developed. This subspace method scans three-dimensional space with a one-dipole model, making it computationally feasible to scan the complete head volume. >

Characterizing nonlinearities in business cycles using smooth transition autoregressive models

Abstract: During the past few years investigators have found evidence indicating that various time-series representing business cycles, such as production and unemployment, may be nonlinear. In this paper it is assumed that if the time-series is nonlinear, then it can be adequately described by a smooth transition autoregressive (STAR) model. The paper describes the application of these models to quarterly logarithmic production indices for 13 countries and ‘Europe’. Tests reject linearity for most of these series, and estimated STAR models indicate that the nonlinearity is needed mainly to describe the responses of production to large negative shocks such as oil price shocks.

Disturbance attenuation and H/sub infinity /-control via measurement feedback in nonlinear systems

Abstract: A solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system. It is shown that the concept of disturbance attenuation, in the sense of truncated L/sub 2/ norms, can be given an interpretation in terms of the response to periodic inputs in the sense of RMS amplitude, even in the nonlinear setup. In the case of state feedback, a family of controllers is also provided. The proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem. >

Control and stabilization of nonholonomic dynamic systems

Abstract: A class of inherently nonlinear control problems has been identified, the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system. Models are presented for mechanical systems with nonholonomic constraints represented both by differential-algebraic equations and by reduced state equations. Control issues for this class of systems are studied and a number of fundamental results are derived. Although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback, a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested. >

The joy of feedback: nonlinear and adaptive

Abstract: It is argued that, for a cautious design, a nonlinear analysis is needed to reveal when and why linear tools fail. Emerging nonlinear tools that can be used to overcome the limitations of nonlinear designs are discussed. It is shown that some of these tools can be made adaptive and applied to nonlinear systems with unknown parameters. The emergence of robust designs for nonlinear 'interval' plants is pointed out. >

Output feedback stabilization of fully linearizable systems

Abstract: An observer-based controller is designed to stabilize a fully linearizable nonlinear system. The system is assumed to be left-invertible and minimum-phase. The controller is robust to uncertainties in modelling the nonlinearities of the system. The design of the controller and the stability analysis employs the techniques of singular perturbations. A new ‘Tikhonov-like’ theorem is presented and used to analyse the system when the control is globally bounded.

A steepest descent method for oscillatory Riemann-Hilbert problems

Abstract: In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg de Vries (MKdV) equation, $$y_t-6y^2y_x+y_{xxx}=0,\qquad -\infty

A viscosity solutions approach to shape-from-shading

Abstract: The problem of recovering a Lambertian surface from a single two-dimensional image may be written as a first-order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions. A new approach based on Hamilton–Jacobi–Bellman equations and viscosity solutions theories enables one to study non-uniqueness phenomenon and thus to characterize the surface among the various solutions.A consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle, and numerical results are presented.

Nonlinear control via approximate input-output linearization: the ball and beam example

Abstract: Approximate input-output linearization of nonlinear systems which fail to have a well defined relative degree is studied. For such systems, a method for constructing approximate systems that are input-output linearizable is provided. The analysis presented is motivated through its application to a common undergraduate control laboratory experiment-the ball and beam-where it is shown to be more effective for trajectory tracking than the standard Jacobian linearization. >

Associated coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation

Abstract: This paper presents a complete formulation of a model of coupled associative thermoplasticity at finite strains, addresses in detail the numerical analysis aspects involved in its finite element implementation, and assesses the performance of the proposed mechanical and finite element models in a comprehensive set of numerical simulations. On the thermomechanical side, novel aspects of the proposed model of thermoplasticity are (1) the explicit characterization of the plastic (configurational) entropy as an independent internal variable, (2) a thermomechanical extension of the principle of maximum dissipation consistent with the multiplicative decomposition of the deformation gradient, and (3) the exploitation of this extended principle in the formulation of an associative flow which characterizes the evolution of the plastic entropy in terms of the change of the flow criterion with respect to temperature. On the numerical analysis side, salient features of the proposed approach are (4) a new global product formula algorithm constructed via an operator split of the nonlinear initial value problem, which leads to a two-step solution procedure, (5) a unified class of local return mapping algorithms which preserves exactly the incompressibility constraint on the plastic flow and reduces to the classical radial return method for isothermal J 2 - flow theory, and (6) the formulation of a mixed finite element method in terms of the elastic entropy and the temperature field which circumvents well-known difficulties associated with the incompressibility constraint on the plastic flow. The exact linearization of both the product formula algorithm and an alternative simulataneous solution scheme for the coupled thermomechanical problem is given in two appendices.

The quantum-state diffusion model applied to open systems

Abstract: A model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation, derived from a density operator equation such as the Bloch equation, and consistent with it. The advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described. The method is applied to the nonlinear absorber, cascades of quantum transitions, second-harmonic generation and a measurement reduction process. The model provides graphic illustrations of these processes, with statistical fluctuations that mimic those of experiments. The stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics. The model is compared with the quantum-jump model of Dalibard (1992), Carmichael and others, which originated among experimenters looking for intuitive pictures and rules of computation.

A Monte Carlo Approach to Nonnormal and Nonlinear State-Space Modeling

Abstract: A solution to multivariate state-space modeling, forecasting, and smoothing is discussed. We allow for the possibilities of nonnormal errors and nonlinear functionals in the state equation, the observational equation, or both. An adaptive Monte Carlo integration technique known as the Gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework. The methodology is a general strategy for obtaining marginal posterior densities of coefficients in the model or of any of the unknown elements of the state space. Missing data problems (including the k-step ahead prediction problem) also are easily incorporated into this framework. We illustrate the broad applicability of our approach with two examples: a problem involving nonnormal error distributions in a linear model setting and a one-step ahead prediction problem in a situation where both the state and observational equations are nonlinear and involve unknown parameters.