Strong (Weak) Exact Controllability and Strong (Weak) Exact Observability for Quasilinear Hyperbolic Systems

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.

From phenomena of synchronization to exact synchronization and approximate synchronization for hyperbolic systems

In this survey paper, the synchronization will be initially studied for infinite dimensional dynamical systems of partial differential equations instead of finite dimensional systems of ordinary differential equations,and will be connected with the control theory via boundary controls in a finite time interval. More precisely,various kinds of exact boundary synchronization and approximate boundary synchronization will be introduced and realized by means of fewer boundary controls for a coupled system of wave equations with Dirichlet boundary controls. Moreover, as necessary conditions for various kinds of approximate boundary synchronization, criteria of Kalman's type are obtained. Finally, some prospects will be given.

Exact Boundary Controllability on a Tree-Like Network of Nonlinear Planar Timoshenko Beams

This paper concerns a system of equations describing the vibrations of a planar network of nonlinear Timoshenko beams. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions and, using the methods of quasilinear hyperbolic systems, prove that for tree-like networks the natural initial-boundary value problem admits semi-global classical solutions in the sense of Li [Li, T. T., Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Ser. Appl. Math., vol 3,American Institute of Mathematical Sciences and Higher Education Press, 2010] existing in a neighborhood of the equilibrium solution. The authors then prove the local exact controllability of such networks near such equilibrium configurations in a certain specified time interval depending on the speed of propagation in the individual beams.

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

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.

Exact Boundary Synchronization by Groups for a Kind of System of Wave Equations Coupled with Velocities

This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM method for wave equations,however,by transforming the system into a first order hyperbolic system,the HUM method for 1-D first order hyperbolic systems,established by Li-Lu(2022)and Lu-Li(2022),can be applied to get the corresponding results.

Exact Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls

In this paper, 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 null controllability and synchronization by groups are all realized by suitable boundary controls.

Exact Boundary Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls

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 synchronization by 3-groups, respectively.

Exact Boundary Controllability for a Coupled System of Wave Equations with Neumann Boundary Controls(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

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 with respect to a weaker norm, which is obviously true in many cases of application. Next,in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System——The Constructive Method

In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally given in the framework of classical solutions is still very powerful and effective in the framework of weak solutions to prove the exact boundary(null) controllability and the exact boundary observability for first order hyperbolic systems.

Periodic Homogenization for Inner Boundary Conditions with Equi-valued Surfaces:the Unfolding Approach

Making use of the periodic unfolding method,the authors give an elementary proof for the periodic homogenization of the elastic torsion problem of an infinite 3dimensional rod with a multiply-connected cross section as well as for the general electroconductivity problem in the presence of many perfect conductors(arising in resistivity well-logging).Both problems fall into the general setting of equi-valued surfaces with corresponding assigned total fluxes.The unfolding method also gives a general corrector result for these problems.

On the Synchronizable System

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.

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System——the HUM Method

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work[Lu,X.and Li,T.T.,Exact boundary con-trollability of weak solutions for a kind of first order hyperbolic system—the constructive method,Chin.Ann.Math.Ser.B,42(5),2021,643-676].In this paper,in order to study these problems from the viewpoint of duality,the authors establish a complete the-ory on the HUM method and give its applications to first order hyperbolic systems.Thus,a deeper relationship between the controllability and the observability can be revealed.Moreover,at the end of the paper,a comparison will be made between these two methods.

A Note on the Indirect Controls for a Coupled System of Wave Equations

By a procedure of successive projections,the authors decompose a coupled system of wave equations into a sequence of sub-systems.Then,they can clarify the indirect controls and the total number of controls.Moreover,the authors give a uniqueness theorem of solution to the system of wave equations under Kalman’s rank condition.

Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions

This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.