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

Real and Complex Analysis

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Observation and Control for Operator Semigroups

Diffusion and Reactions in Fractals and Disordered Systems

In this paper we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties. Mathematics Subject Classication. 93B52, 93D15, 93B07.

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Stabilization of second order evolution equations by a class of unbounded feedbacks

We study the energy decay of a Bernoulli--Euler beam which is subject to a pointwise feedback force. We show that both uniform and nonuniform energy decay may occur. The uniform or nonuniform decay depends on the boundary conditions. In the case of nonuniform decay in the energy space we give explicit polynomial decay estimates valid for regular initial data. Our method consists of deducing the decay estimates from observability inequalities for the associated undamped problem via sharp trace regularity results.

Stabilization of Bernoulli--Euler Beams by Means of a Pointwise Feedback Force

In this paper, we characterize the stabilization of some delay systems. The proof of the main result uses the method introduced in Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001) where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined with a boundedness property of the transfer function of the associated open loop system.

Feedback stabilization of a class of evolution equations with delay

Some backgrounds.- Stabilization of second order evolution equations by a class of unbounded feedbacks.- Stabilization of second order evolution equations with unbounded feedback with delay.- Asymptotic behaviour of concrete dissipative systems.- Systems with delay.- Bibliography

Kaïs Ammari

Papers

Stabilization of second order evolution equations by a class of unbounded feedbacks

Stabilization of Bernoulli--Euler Beams by Means of a Pointwise Feedback Force

Feedback stabilization of a class of evolution equations with delay

Stabilization of Elastic Systems by Collocated Feedback

Feedback boundary stabilization of wave equations with interior delay