Induced generalized exact boundary synchronizations for a coupled system of wave equations

Taking a coupled system of wave equations with Dirichlet boundary controls as an example,by splitting and merging some synchronization groups of the state variables cor-responding to a given generalized synchronization matrix,this paper introduces two kinds of induced generalized exact boundary synchronizations to better determine its generalized exactly synchronizable states.

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 Internal Controllability and Synchronization for a Coupled System of Wave Equations

In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established.Based on this result,the exact internal synchronization and the exact internal synchronization by p-groups are successfully considered.

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 iNeumann 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.

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

This paper deals with the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls in the framework of weak solutions.A necessary and sufficient condition for the generalized exact boundary synchronization is obtained,and some results for its generalized exactly synchronizable states are given.

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.

Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls

In this paper,we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls,and make a deep discussion on it.We analyze the relationship between the partial approximate boundary synchronization and the partial exact boundary synchronization,and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman’s criterion.In addition,with the help of partial synchronization decomposition,a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given.

Determination of generalized exact boundary synchronization matrix for a coupled system of wave equations

For a coupled system of wave equations with Dirichlet boundary controls,this paper deals with the possible choice of its generalized synchronization matrices so that the admissible generalized exact boundary synchronizations for this system are obtained.

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.

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.