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 re...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.展开更多
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...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.展开更多
This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n=2 and cubic nonlinearity(see the resul...This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n=2 and cubic nonlinearity(see the results of T.T.Li and Y.M.Chen in 1992).For this purpose,the authors consider the following Cauchy problem:{where□=δt^2-∑i=1nδx^^2 is the wave operator,g(x)(≡/)0 is a smooth non-negative function with compact support,andε〉0 is a small parameter.It is shown that the solution blows up in a finite time,and the lifespan T(ε)of solutions has an upper bound T(ε)≤exp(Aε-2)with a positive constant A independent ofε,which belongs to the same kind of the lower bound of the lifespan.展开更多
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 conditi...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.展开更多
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...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.展开更多
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 couple...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.展开更多
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 ...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.展开更多
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 ...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.展开更多
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.
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 a...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.展开更多
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 met...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.展开更多
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 ...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.展开更多
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,th...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.展开更多
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 bou...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.展开更多
基金supported by the Basic Research Program of China(No. 2007CB814800)
文摘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.
基金supported by National Basic Research Program of China(Grant No.2013CB834100)National Natural Science Foundation of China(Grant No.111211101)
文摘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.
基金Project supported by the National Natural Science Foundation of China(No.10728101)the Basic Research Program of China(No.2007CB814800)+1 种基金the Doctoral Program Foundation of the Ministry of Education of Chinathe"111"Project(No.B08018)and SGST(No.09DZ2272900)
文摘This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n=2 and cubic nonlinearity(see the results of T.T.Li and Y.M.Chen in 1992).For this purpose,the authors consider the following Cauchy problem:{where□=δt^2-∑i=1nδx^^2 is the wave operator,g(x)(≡/)0 is a smooth non-negative function with compact support,andε〉0 is a small parameter.It is shown that the solution blows up in a finite time,and the lifespan T(ε)of solutions has an upper bound T(ε)≤exp(Aε-2)with a positive constant A independent ofε,which belongs to the same kind of the lower bound of the lifespan.
基金supported by the National Natural Science Foundation of China(No.11831011)。
文摘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.
基金supported by the National Basic Research Program of China(No.2103CB834100)the National Science Foundation of China(No.11121101)+1 种基金the National Natural Sciences Foundation of China(No.11101273)the DFG-Cluster of Excellence:Engineering of Advanced Materials
文摘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.
文摘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.
基金supported by the National Basic Research Program of China(No.2013CB834100)the National Natural Science Foundation of China(No.11121101)
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.11831011,11901082)the Natural Science Foundation of Jiangsu Province(No.BK20190323)the Fundamental Research Funds for the Central Universities of China。
文摘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.
基金Project supported by the National Natural Science Foundation of China(Nos.11831011,11725102).
文摘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.
基金Supported by the National Natural Science Foundation of China (No. 11121101)the National Basic Research Program of China (No. 2013CB834100)
文摘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.
基金supported by the National Natural Science Foundation of China (Nos. 11831011,11901082)the Natural Science Foundation of Jiangsu Province (No. BK20190323)the Fundamental Research Funds for the Central Universities of China
文摘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.
基金This work was supported by the National Natural Science Foundation of China(Nos.11831011,11901082)the Natural Science Foundation of Jiangsu Province(No.BK20190323)the Fundamental Research Funds for the Central Universities of China.
文摘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.
基金supported by the National Natural Science Foundation of China(No.11831011)
文摘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.
基金supported by the National Natural Science Foundation of China(No.11831011).
文摘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.
