关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

In 1982, the Belgian Pilots' Guild raised the question of what effect exposure to radar radiation--for example, that encountered in passing a pilot launch's radar--might have on the human body. Recapitulating investigations of this question, this article states that in 25 years of experience with radar, there have been no known incidents of pilots being affected by radar waves. In the future, however, involvement by some pilots with Vessel Traffic Service shore-based radar could affect pilots somewhat differently from limited exposure to pilot launch radar. Pilots who find themselves in new working conditions close to an emitting source should exercise care all times.

About heat waves

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

1 Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex Conjugates Exponential Form Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Regions in the Complex Plane 2 Analytic Functions Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle 3 Elementary Functions The Exponential Function The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 4 Integrals Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Some Examples Examples with Branch Cuts Upper Bounds for Moduli of Contour Integrals Antiderivatives Proof of the Theorem Cauchy-Goursat Theorem Proof of the Theorem Simply Connected Domains Multiply Connected Domains Cauchy Integral Formula An Extension of the Cauchy Integral Formula Some Consequences of the Extension Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle 5 Series Convergence of Sequences Convergence of Series Taylor Series Proof of Taylor's Theorem Examples Laurent Series Proof of Laurent's Theorem Examples Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representations Multiplication and Division of Power Series 6 Residues and Poles Isolated Singular Points Residues Cauchy's Residue Theorem Residue at Infinity The Three Types of Isolated Singular Points Residues at Poles Examples Zeros of Analytic Functions Zeros and Poles Behavior of Functions Near Isolated Singular Points 7 Applications of Residues Evaluation of Improper Integrals Example Improper Integrals from Fourier Analysis Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouche's Theorem Inverse Laplace Transforms Examples 8 Mapping by Elementary Functions Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane The Transformation w = sin z Mappings by z2 and Branches of z1/2 Square Roots of Polynomials Riemann Surfaces Surfaces for Related Functions 9 Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates Transformations of Harmonic Functions Transformations of Boundary Conditions 10 Applications of Conformal Mapping Steady Temperatures Steady Temperatures in a Half Plane A Related Problem Temperatures in a Quadrant Electrostatic Potential Potential in a Cylindrical Space Two-Dimensional Fluid Flow The Stream Function Flows Around a Corner and Around a Cylinder 11 The Schwarz-Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz-Christoffel Transformation Triangles and Rectangles Degenerate Polygons Fluid Flow in a Channel Through a Slit Flow in a Channel with an Offset Electrostatic Potential about an Edge of a Conducting Plate 12 Integral Formulas of the Poisson Type Poisson Integral Formula Dirichlet Problem for a Disk Related Boundary Value Problems Schwarz Integral Formula Dirichlet Problem for a Half Plane Neumann Problems Appendixes Bibliography Table of Transformations of Regions Index

Complex variables and applications

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/232625/1/j.automatica.2017.04.017.pdf

Boundary cooperative control by flexible Timoshenko arms

In this paper we study the Riesz basis property of serially connected Timoshenko beams with joint and boundary feedback controls. Suppose that the left end of the whole beam is clamped and the right end is free. At intermediate nodes, the displacement and rotational angle of beams are continuous but the shearing force and bending moment could be discontinuous. The collocated velocity feedback of the beams at intermediate nodes and the right end are used to stabilize the system. We prove that the operator determined by the closed loop system has compact resolvent and generates a C 0 semigroup in an appropriate Hilbert space. We also show that there is a sequence of the generalized eigenvectors of the operator that forms a Riesz basis with parentheses. Hence the spectrum determined growth condition holds. Therefore if the imaginary axis is not an asymptote of the spectrum, then the closed loop system is exponentially stable. Finally, we give a conclusion remark to explain that our result can be applied not ...

Riesz basis property of serially connected Timoshenko beams

In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.

/pdf/exponential-stability-of-timoshenko-beam-system-with-delay-1cp3bw8l1f.pdf

Exponential stability of timoshenko beam system with delay terms in boundary feedbacks

The spectrum and asymptotic behavior of the non-uniform porous-thermo-elasticity of Lord-Shulman type is considered in this
 paper.
It is shown that the corresponding system operator generates a $C_0$
semigroup of contractions
 in an appropriate Hilbert space setting.
 By a detailed spectral analysis, the asymptotic
 expressions of the spectrum of the system is gotten.
 Based on the spectral property, the Riesz basis property of the (generalized) eigenvectors
 is proved, which implies that the system satisfies the spectrum-determined-growth condition. Then the
 exponential stability of this system is deduced from the distribution of the spectrum.

Exponential decay in non-uniform porous-thermo-elasticity model of lord-shulman type

In this paper, we consider the output feedback stabilization problem of a Euler–Bernoulli beam with timedelay in boundary control. A similar question was studied in Shang and Xu (2012, Syst. Control Lett., 61, 1069–1078), wherein the full state of the system was used in the design of the dynamic feedback law. In the present paper, we consider mainly an output-based stabilization problem for the same model. Using the tricks of the Luenberger observer and partial state predictor, we deduce a kind of dynamic feedback control law that exponentially stabilizes the delayed system.

Output-based stabilization of Euler–Bernoulli beam with time-delay in boundary input

The dynamical behavior of a kind of elastic hybrid system is considered. This system consists of a nonhomogeneous Timoshenko beam linked at its boundary to a rigid body. The non-collocated input terms are required in the boundary feedback controls. It is proved that this closed loop system is well-posed. By a complete eigenfrequency analysis, it is shown that this infinite-dimensional hybrid system satisfies spectrum-determined growth condition. The stability of this system is discussed and some simulations are given to show that this kind of hybrid system can be exponentially stable under certain condition.

Zhong-Jie Han

Papers

Riesz basis property of serially connected Timoshenko beams

Exponential stability of timoshenko beam system with delay terms in boundary feedbacks

Exponential decay in non-uniform porous-thermo-elasticity model of lord-shulman type

Output-based stabilization of Euler–Bernoulli beam with time-delay in boundary input

Dynamical behavior of a hybrid system of nonhomogeneous timoshenko beam with partial non-collocated inputs