Uniqueness of Solution to Systems of Elliptic Operators and Application to Asymptotic Synchronization of Linear Dissipative Systems II:Case of Multiple Feedback Dampings*

Tatsien LIBopeng RAO

1School of Mathematical Sciences,Fudan University,Shanghai 200433,China;Shanghai Key Laboratory for Contemporary Applied Mathematic;Nonlinear Mathematical Modeling and Methods Laboratory,Shanghai 200433,China.E-mail:dqli@fudan.edu.cn

2Corresponding author.Institut de Recherche Math′ematique Avanc′ee,Universit′e de Strasbourg,67084 Strasbourg,France.E-mail:bopeng.rao@math.unistra.fr

Abstract In this paper,the authors consider the asymptotic synchronization of a linear dissipative system with multiple feedback dampings.They first show that under the observability of a scalar equation,Kalman’s rank condition is sufficient for the uniqueness of solution to a complex system of elliptic equations with mixedobservations.The authors then establish a general theory on the asymptotic stability and the asymptotic synchronization for the corresponding evolutional system subjected to mixed dampings of various natures.Some classic models are presented to illustrate the field of applications of the abstract theory.

Keywords Kalman rank condition,Uniqueness,Asymptotic synchronization,Kelvin-Voigt damping

1 Introduction

Synchronization is a widespread natural phenomenon.It was first observed by Huygens[11]in 1665.The theoretical research on synchronization from the mathematical point of view dates back to Wiener in 1950s in[43](Chapter 10).The previous study focused on the systems described by ordinary differential equations.Since 2012,Li and Rao started the research on the exact boundary synchronization for a coupled system of wave equations(see[18,20–23,26]),later the approximate synchronization has been carried out for a coupled system of wave equations with various boundary controls(see[19,25,27,30]).The most part of their results was recently collected in the monograph[28].Consequently,this kind of study of synchronization becomes a part of research in control theory.The optimal control for the exact synchronization of parabolic system was recently investigated in[42].We quote[1,6]for the synchronization of distributed parameter systems on networks.

By duality,the approximate boundary controllability of a coupled system of wave equations can be transformed to the uniqueness of solution to the corresponding adjoint system.Since the adjoint system is constituted of many wave equations of the same type and observed by an incomplete system of observations,it is not a standard uniqueness of continuation,and only Kalman’s rank condition is not sufficient for the uniqueness.In order to obtain the uniqueness of solution to this complex system,our basic idea is to combine the uniform observability of a scalar equation and the algebraic structure of the coupling matrices,namely,Kalman’s rank condition.The first attempt for realizing this idea was carried out in[24–25]for a system of wave equations with Dirichlet boundary conditions by incomplete Neumann observations.Later,this idea was used in[19,30]for Neumann and Robin conditions,and further developed in[29]for an elliptic system with Neumann boundary conditions observed by incomplete Dirichlet observations.We quote[34]for a close work on the observability of heat equations by internal observations.

The goal of the present paper is to generalize the results in[29]from the special case of one sole damping to the general case of several dampings with different natures.

Let H and V be two separated Hilbert spaces such that V ?H with dense and compact imbedding.

Let L be the duality operator from V onto the dual space V′,such that

It is easy to show that(1.7)generates a semi-group of contractions with compact resolvent in the space V ×H.

The case M =1:

was studied in[29],and we showed that Kalman rank condition

is necessary for the asymptotic stability of system(1.8).Moreover,under suitable conditions on the pair of operators(L,g1),Kalman rank condition(1.9)is also sufficient for the asymptotic stability of system(1.8)(see[29,Theorem 3.4]).In[31],we carried out a complete study on the uniform synchronization of system(1.8).In particular,we justified the necessity of diverse conditions of compatibility on the matrices A and D1.Moreover,in[32]we considered a coupled system of wave equations in a rectangular domain,which does not satisfy the usual multiplier geometrical condition.

The aim of the present work is to investigate the asymptotic stability of system(1.7)under the common action of M feedback dampings D1G1U′,··· ,DMGMU′.In Proposition 2.2 below,we will show that Kalman rank condition

with the composite matrix by blocks:

is still necessary for the asymptotic stability of system(1.7).Moreover,under suitable conditions on the matrix A and on the pairs(L,gs)for 1 ≤s ≤M,we will show in Theorem 3.2 that Kalman rank condition(1.10)is still sufficient for the asymptotic stability of system(1.7).The involved dampings in system(1.7)can be of different types,for example,boundary damping,locally distributed viscous dampings,locally distributed Kelvin-Voigt damping or bending moment damping etc.Therefore,it provides a rich freedom for the choice of feedback controls in applications.This is the main advantage of the approach.

The materials in the paper are organized as follows.In§2,we first formulate the problem in the framework of semi-groups.Then by the classic method of frequency domain,we reduce the asymptotic stability to the uniqueness of solution to an over-determined elliptic system.In §3,under the assumptions that A is closed to a scalar matrix and L can be uniformly observed by the operator gsfor 1 ≤s ≤M,we establish the corresponding uniqueness theorem.We study the corresponding asymptotic synchronization in §4.In order to illustrate the abstract result,we give some examples of applications in §5.

2 Setting of Problem

In this section,we will characterize the asymptotic stability of system(1.7)by the method of frequency domain.We first make some necessary arrangement.

3 Uniqueness Theorem Under Kalman Rank Condition

namely,Φ ≡0.The proof is thus complete.

Theorem 3.1 can be read as “under suitable conditions,the observability of the scalar equation implies the stability of the whole system”.By this way,we provide a simple and efficient approach to solve a seemingly difficult problem of asymptotic stability of a complex system.

As a direct consequence of Theorems 2.1 and 3.1,we have the following important result.

Theorem 3.2Under the same assumptions as those in Theorem 3.1,system(1.7)is asymptotically stable.

4 Asymptotic Synchronization by Groups

5 Applications

5.1 Wave equations with mixed dampings

Remark 5.1In fact,the uniform estimate(5.8)is based on the uniform stability of equation(5.2)for a scalar equation(i.e.for N = 1),which was abundantly studied by different approaches in literatures.We only quote[13,16–17]and the references therein for boundary feedback.The uniform decay was first established by multipliers in[10]as ω is a neighbourhood of the boundary.The explicit decay rate was given in[41]under suitable geometric condition.Later,the result was generalized in[44]to semi-linear case.When Ω is a compact Riemann manifold without boundary and ω satisfies the geometric optic condition,the uniform stability was established by a micro-local approach in[39].Moreover,the volume of the damping region ω can be sufficiently small in[5]etc.

5.2 Kirchhoffplate equations with locally distributed Kelvin-Voigt dampings

5.3 Euler-Bernoulli beam equations with mixed dampings

In the two previous subsections,we have considered the case of mixed dampings for wave equations and of locally distributed dampings for plate equations.However,when ω is not a neighbourhood of Γ,the situation is technically complicated! As a beginning,we will consider a system of beam equations.There are many to pursue··· In particular,the discussion below can also be carried out for many other situations,such as Timoshenko beam[2,12],Bresse beam[35]etc.

Let a,b be smooth and positive functions in(0,1)such that