Bopeng Rao

• Professor
• Professor (Full) at University of Strasbourg

In this paper, the synchronizable system is defined and studied for a coupled system of wave equations with the same wave speed or with different wave speeds.

We show that under Kalman's rank condition on the coupling matrices, the uniqueness of solution to a complex system of elliptic operators can be reduced to the observability of a scalar problem. Based on this result, we establish the asymptotic stability and the asymptotic synchronization for a large class of linear dissipative systems.

In this paper, we consider the exact boundary controllability and the exact boundary synchronization (by groups) for a coupled system of wave quations with coupled Robin boundary controls. Owing to the difficulty coming from the lack of regularity of the solution, we confront a bigger challenge than that in the case with Dirichlet or Neumann bounda...

We show that under Kalman’s rank condition, the observability of a scalar equation implies the uniqueness of solution to a system of elliptic operators. Using this result, we establish the asymptotic synchronization by groups for second order evolution systems.

In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when ther...

In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when ther...

In order to study the approximate boundary synchronization for system (III) with coupled Robin boundary controls, some algebraic lemmas are given in this chapter.

We consider the unique continuation for Robin problems in this chapter.

The exact boundary synchronization by groups will be considered in this chapter for system (III) with further lack of coupled Robin boundary controls.

We introduce the induced approximate boundary synchronization for system (I) with Dirichlet boundary controls and give some examples in this chapter.

The approximate boundary synchronization by p-groups is introduced and studied in this chapter for system (III) with coupled Robin boundary controls.

In the case of partial lack of boundary controls, we consider the exact boundary synchronization and the non-exact boundary synchronization in this chapter for system (II) with Neumann boundary controls.

When system (I) possesses the exact boundary synchronization by p-groups, the corresponding exactly synchronizable states by p-groups will be studied in this chapter.

In this chapter, we will study the exact boundary controllability and the non-exact boundary controllability for the coupled system (III) of wave equations with coupled Robin boundary controls

Since the exact boundary synchronization on a finite time interval is closely linked with the exact boundary null controllability, we first consider the exact boundary null controllability and the non-exact boundary null controllability for system (I) of wave equations with Dirichlet boundary controls in this chapter.

When system (I) possesses the exact boundary synchronization, the corresponding exactly synchronizable states will be studied in this chapter.

An introduction and overview of the whole book can be found in this chapter.

Based on the results of the exact boundary controllability and the non-exact boundary controllability, we study the exact boundary synchronization for system (III) with coupled Robin boundary controls.

In this chapter, we will define the approximate boundary null controllability for system (III) and the D-observability for the adjoint problem, and show that these two concepts are equivalent to each other.

The approximate boundary synchronization is defined and studied in this chapter for system (III) with coupled Robin boundary controls.

In order to consider the exact boundary controllability and the exact boundary synchronization of system (III), we first give some necessary results on problem (III) and (III0) in this chapter.

When system (II) possesses the exact boundary synchronization by p-groups, the corresponding exactly synchronizable states by p-groups will be studied in this chapter.

When system (III) is approximately synchronizable by p-groups, the corresponding approximately synchronizable states by p-groups will be considered in this chapter.

In this chapter, we will define the approximate boundary null controllability for system (II) and the D-observability for the adjoint problem, and show that these two concepts are equivalent to each other. Moreover, the corresponding Kalman’s criterion is introduced and studied.

The approximate boundary synchronization by p-groups is introduced and studied in this chapter for system (I) with Dirichlet boundary controls.

In this chapter, we will discuss the necessity of the conditions of \(C_p\)-compatibility for system (III) with coupled Robin boundary controls. This problem is closely related to the number of applied boundary controls.

The exact boundary synchronization by groups will be considered in this chapter for system (I) with further lack of Dirichlet boundary controls.

This chapter contains some algebraic preliminaries, which are useful in the whole book. In this chapter, we denote by A a matrix of order N, by D a full column-rank matrix of order \(N\times M\) with \(M\leqslant N\), and by \(C_p\) a full row-rank matrix of order \( (N-p)\times N\) with \(0<p<N\). All these matrices are of constant entries.

In this chapter, we will define the approximate boundary null controllability for system (I) and the D-observability for the adjoint problem, and show that these two concepts are equivalent to each other. Moreover, the corresponding Kalman’s criterion is introduced and studied.

When system (III) possesses the exact boundary synchronization, the corresponding exactly synchronizable states will be studied in this chapter.

In the case of partial lack of boundary controls, we consider the exact boundary synchronization and the non-exact boundary synchronization in this chapter for system (I) with Dirichlet boundary controls.

The exact boundary synchronization by p-groups will be considered in this chapter for system (II) with further lack of Neumann boundary controls.

The approximate boundary synchronization by p-groups is introduced and studied in this chapter for system (II) with Neumann boundary controls

In this chapter, we will consider the exact boundary controllability and the non-exact boundary controllability for system (II) of wave equations with Neumann boundary controls.

The approximate boundary synchronization is defined and studied in this chapter for system (II) with Neumann boundary controls.

The approximate boundary synchronization is defined and studied in this chapter for system (I) with Dirichlet boundary controls.

When system (III) possesses the exact boundary synchronization by p-groups, the corresponding exactly synchronizable states by p-groups will be studied in this chapter.

We consider the stability of a system of two strongly coupled wave equations by means of only one boundary feedback. We show that the stability of the system depends in a very complex way on all of the involved factors such as the type of coupling, the hidden regularity and the accordance of boundary conditions. We first show that the system is uni...

Within this carefully presented monograph, the authors extend the universal phenomenon of synchronization from finite-dimensional dynamical systems of ordinary differential equations (ODEs) to infinite-dimensional dynamical systems of partial differential equations (PDEs). By combining synchronization with controllability, they introduce the study...

In this Note, we consider a system of wave equations coupled by a nilpotent matrix with homogeneous Dirichlet boundary condition. We establish the uniqueness of the solution when partial Neumann observation satisfies Kalman's rank condition.

In this paper, for a coupled system of wave equations with Neumann boundary controls, the approximate boundary null controllability, the approximate boundary synchronization and the approximate boundary synchronization by groups are taken into account, respectively. Like in the case with Dirichlet boundary controls, the corresponding conditions of...

In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups. Moreover, the determination of the state of synchronization by groups is discussed with details for the synchronization and for the...

Résumé Dans cette Note, nous donnons une nouvelle définition plus naturelle de la synchronisation approchée par p-groupes (p≥1) pour un système couplé de N équations des ondes par des contrôles frontières de Dirichlet. Au moyen du critère de Kalman, nous introduisons la notion de nombre de contrôles totaux (directs et indirects). Nous montrons que,...

This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection wit...

Résumé Dans cette Note, nous obtenons des conditions nécessaires, exprimées sous la forme de critères du type de Kálmán, pour la contrôlabilité nulle approchée et la synchronisation approchée par groupes d'un système couplé d'équations des ondes avec des contrôles frontières de Dirichlet. De plus, nous établissons la suffisance de ces conditions po...

By means of a non-exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2013 John Wiley & Sons, Ltd.

In this Note, we consider the determination of the state of exact synchronization for a coupled system of wave equations. In a special case, the state of exact synchronization can be uniquely determined whatever the boundary controls would be chosen. In the general case, the state of exact synchronization depends on the boundary controls that reali...

Several kinds of exact synchronizations and the generalized exact synchronization are introduced for a coupled system of 1-D wave equations with various boundary conditions and we show that these synchronizations can be realized by means of some boundary controls.

Several kinds of exact synchronizations are introduced for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type and these synchronizations can be realized by means of some boundary controls.

In this paper, the asymptotic null controllability and various kinds of asymptotic synchronization, as some sorts of weakened controllability and synchronization, are introduced and studied for a coupled system of wave equations with Dirichlet boundary controls. Equivalent properties of observability are established and several examples of explanat...

In this Note, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact nul...

In this Note, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact nul...

This paper addresses a study of the eventual regularity of a wave equation with boundary dissipation and distributed damping. The equation under consideration is rewritten as a system of first order and analyzed by semigroup methods. By a certain asymptotic expansion theorem, we prove that the associated solution semigroup is eventually differentia...

In this Note we introduce the asymptotic controllability and the asymptotic null controllability for 1-D linear hyperbolic systems under the lack of boundary controls. We claim that they are equivalent, respectively, to the strong observability and the weak observability for the dual system. An example of 4×4 hyperbolic system with only one boundar...

In this Note we introduce the asymptotic controllability and the asymptotic null controllability for 1-D linear hyperbolic systems under the lack of boundary controls. We claim that they are equivalent, respectively, to the strong observability and the weak observability for the dual system. An example of 4 x 4 hyperbolic system with only one bound...

In this paper, the authors define the strong (weak) exact boundary controllability and the strong (weak) exact boundary observability for first order quasilinear hyperbolic systems, and study their properties and the relationship between them. KeywordsStrong (weak) exact boundary controllability-Strong (weak) exact boundary observability-First ord...

In this paper we establish the theory on semiglobal classical solution to first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, and based on this, the corresponding exact boundary controllability and observability are obtained by a constructive method. Moreover, with the linearized Saint-Venant system and the 1-D l...

In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. If the longitudinal motion and heat transfer are neglected, this model reduces to the well-known thermoelastic Timoshenko beam equations. The system consists of three wave equations and two h...

We study the exact controllability of a system of two weakly coupled one-dimensional wave equations with the control acted on only one equation. Using the nonharmonic analysis, we establish the weak observability inequalities, which depend on the ratio of the wave propagation speeds. The obtained results are optimal.

The known theory on the one-side exact boundary observabil- ity for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions on the non-observation side (see (1)-(2), (11)). In this paper we illustrate, with an inspiring example, that the one-side exact...

The known theory on the one-side exact boundary controllability and the one-side exact boundary observability for first-order quasilinear hyperbolic systems requires that the unknown variables should be suitably coupled in the boundary conditions at the non-control or non-observation side. In this Note we illustrate, with an inspiring example, that...

The known theory on the one-side exact boundary observability for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions on the non-observation side (see [1]-[2], [11]). In this paper we illustrate, with an inspiring example, that the one-side exact bo...

In this paper, we study the stability of a system of wave equations which are weakly coupled and partially damped. Using a frequency domain approach based on the growth of the resolvent on the imaginary axis, we establish the polynomial energy decay rate for smooth initial data. We show that the behavior of the system is sensitive to the arithmetic...

We consider the wave equations with local viscoelastic damping distributed around the boundary of a bounded open set W Ì \mathbbRN .\Omega \subset \mathbb{R}^{N} . We show that the energy of the wave equations goes uniformly and exponentially to zero for all initial data of finite energy.

We study the stability of weakly coupled and partially damped systems by means of Riesz basis approach in higher dimension spaces. We propose a weaker distributed damping that compensates the behaviour of the eigenvalues of the system, therefore gives the optimal polynomial energy decay rate for smooth initial data.

In this paper, we study the decay rate of solutions to strongly stable, but not exponentially stable linear evolution equations. It is known that the resolvent operator of such an equation must be unbounded on the imaginary axis. Our main result is an estimate of the decay rate when the unboundedness is of polynomial order. We then apply our main t...

Using a direct approach, we establish the polynomial energy decay rate for smooth solutions of the equation of Kirchhoff plate. Consequently, we obtain the strong stability in the absence of compactness of the resolvent of the infinitesimal operator.

We consider the stability of wave equations with local viscoelastic damping distributed around the boundary of domain. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy. To cite this article: K. Liu, B. Rao, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Using the theory of semi-global piecewise C1 solution, we establish the exact boundary controllability of unsteady flows in a tree-like network of open canals with general topology. To cite this article: T.T. Li, B.P. Rao, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

In this paper we establish the exact boundary controllability of unsteady flows in a tree-like network of open canals with general topology.

Nous étudions la stabilisation d'un système de deux équations linéaires, dont une seule équation est amortie par un contrôle feedback. Nous montrons qu'un contrôle convenablement choisi peut compenser les parties réelles des valeures propres du système, et donc fournir le meilleur taux de décroissance polynomiale de l'énergie du système pour des do...

In this paper we present some results on the local exact boundary controllability for general one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions and give corresponding applications to nonlinear vibrating string equations.

Using a result on the existence and uniqueness of the semiglobal C1 solution to the mixed initial-boundary value problem for first order quasi-linear hyperbolic systems with general nonlinear boundary conditions, we establish the exact boundary controllability for quasi-linear hyperbolic systems if the C1 norm of initial and final states is small e...

The encapsulation and interaction of xylanase I (Xyl I) from Thermomonospora sp. in thermally evaporated fatty amine films by a simple beaker-based immersion technique under enzyme-friendly conditions has been described. The approach is based on the diffusion of the enzyme from aqueous solution, driven primarily by attractive electrostatic interact...

By means of global Carleman-type estimate, we study the stabilization problem of the wave equations with potential and indefinite damping. The energy decay rate of the system is given explicitly. Also, we obtain an upper bound estimate on the negative damping to guarantee the energy of the system decays exponentially.

For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an application, the authors show the local exact boundary controllability for a kind of nonlinear vibrating string problem.

In this Note, we first get the existence and uniqueness of semi-global solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions, and then we establish the corresponding exact boundary controllability, provided that the norm of the initial and final states is small enough.

In this paper we consider an abstract linear system with perturbation of the form $$ \frac{dy}{dt}= Ay + \varepsilon By $$ on a Hilbert space ${\cal H}$, where A is skew-adjoint, B is bounded, and $\varepsilon$ is a positive parameter. Motivated by a work of Freitas and Zuazua on the one-dimensional wave equation with indefinite viscous damping [P....

A highly thermostable xylanase (Xyl I) produced by Thermomonospora sp. was purified to homogeneity and was classified as a family 10 xylanase based on its molecular weight (38,000 Da) and isoelectric point (4.1). K2d analysis showed that the secondary structure of Xyl I was made up of 38% alpha-helix and 10% beta-sheet. The optimal temperature for...

We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by one partial differential equation and two ordinary differential equations. Using the HUM method, we prove that the hybrid system is exactly controllable in an arbit...

By means of a result on the semi-global C 1 solution, we establish the exact boundary controllability for the reducible quasilinear hyperbolic system if the C 1 norm of initial data and final state is small enough.

We consider the one-dimensional wave equation with an indefinite sign damping and a zero order potential term. Using a shooting method, we establish the asymptotic expansion of eigenvalues and eigenvectors of the damped wave equation for a large class of coefficients. In addition, if the damping coefficient is "more positive than negative," we prov...

By means of a result on the semi-global C1 solution, we establish the exact boundary controllability for the reducible quasilinear hyperbolic system if the C1 norm of initial data and final state is small enough.

In this paper we consider an abstract linear system with perturbation of the form (dy/dt) = Ay + εBy on a Hilbert space H, where A is skew-adjoint, B is bounded, and ε is a positive parameter. Motivated by a result of Freitas and Zuazua on the one-dimensional wave equation with indefinite viscous damping [JDE, 1996], we obtain sufficient conditions...

Using a direct approach, we prove the asymptotic stability of Kirchhoff plates in the absence of compactness of the resolvent. We also establish the polynomial energy decay rate for the smooth solutions.

Using a direct approach, we prove the asymptotic stability of Kirchhoff plates in the absence of compactness of the resolvent. We also establish the polynomial energy decay rate for the smooth solutions.

We consider a clamped Rayleigh beam subject to a positive viscous damping y tt -γ 2 y xxtt +y xxxx -2(ay tx ) x =0,0<x<1,t>0, y(0,t)=y x (0,t)=y(1,t)=y x (1,t)=0,t>0, y(x,0)=y 0 (x),y t (x,0)=y 1 (x)· Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the underlying system. We next identify...

We consider a hybrid system composed of a plate equation and two ordinary differential equations. We prove that the system is strongly but not uniformly stable. By a new approach, we show that the smooth solution has a rational decay rate. Finally we establish the uniform energy decay rate for a simplified hybrid system.

We consider the Rayleigh beam equation subject to a viscous damping. Using a constructive approximation, we first give the asymptotic form of eigenvalues and eigenvectors of the underlying system. We next identify the optimal energy decay rate as the supremum of the real part of the spectrum of the infinitesimal generator of the associated semigrou...

We consider the Rayleigh beam equation subject to a viscous damping. Using a constructive approximation, we first give the asymptotic form of eigenvalues and eigenvectors of the underlying system. We next identify the optimal energy decay rate as the supremum of the real part of the spectrum of the infinitesimal generator of the associated semigrou...

We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by one partial differential equation and two ordinary differential equations. Using the HUM method, we prove that the hybrid system is exactly controllable in an arbit...

We prove that the Rayleigh beam equation can be uniformly exponentially stabilized by only one control moment. We also prove the strong asymptotic stabilization and the lack of uniform exponential stabilization in the case of only one control force.

We consider the Dirichlet problem for the Marguerre-von Kármán equations of a nonlinear shallow shell model. We first show that the solutions of the shallow shell model converge to the solution of the membrane model as the intensity of the traction converges to infinity. Next, in the case without forcing terms, we prove that the Marguerre-von Kármá...