By means of an equivalent invariant form of boundary conditions, the authors get the exis- tence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems...By means of an equivalent invariant form of boundary conditions, the authors get the exis- tence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions.展开更多
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 appl...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.展开更多
The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By con...The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By constructing an example, first it is illustrated that the classical solution to this kind of Cauchy problem may blow up in a finite time, even if the system is weakly linearly degenerate. Then some lower bounds of the life-span of classical solutions are given in the case that the system is weakly linearly degenerate. These estimates imply that, when the system is weakly linearly degenerate, the classical solution exists almost globally in time. Finally, it is proved that Theorems 1.1-1.3 in [2] are still valid for this kind of initial data.展开更多
By means of the theory on the semi-global C^1 solution to the mixed initialboundary value problem （IBVP） for first order quasilinear hyperbolic systems, we establish the exact controllability for general nonautonomo...By means of the theory on the semi-global C^1 solution to the mixed initialboundary value problem （IBVP） for first order quasilinear hyperbolic systems, we establish the exact controllability for general nonautonomous first order quasilinear hyperbolic systems with general nonlinear boundary conditions.展开更多
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
The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global wea...The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.展开更多
For a class of mixed initial-boundary value problem for general quasilinear hyperbolic sys-tems with zero eigenvalues, the authors establish the local exact controllability with boundary controls acting on one end or ...For a class of mixed initial-boundary value problem for general quasilinear hyperbolic sys-tems with zero eigenvalues, the authors establish the local exact controllability with boundary controls acting on one end or on two ends and internai controls acting on a part of equations in the system.展开更多
In this paper, we study a nonlinear hyperbolic-parabolic system modeling some biological phenomena. By semigroup theory and Leray-Schauder fixed point argument, the local existence and uniqueness of the weak solutions...In this paper, we study a nonlinear hyperbolic-parabolic system modeling some biological phenomena. By semigroup theory and Leray-Schauder fixed point argument, the local existence and uniqueness of the weak solutions for this system are proved. For the spatial dimension N = 1, the global existence of the weak solution will be established by the bootstrap argument.展开更多
This paper develops delay-independent fuzzy hyperbolic guaranteed cost control for nonlinear continuous-time systems with parameter uncertainties. Fuzzy hyperbolic model (FHM) can be used to establish the model for ce...This paper develops delay-independent fuzzy hyperbolic guaranteed cost control for nonlinear continuous-time systems with parameter uncertainties. Fuzzy hyperbolic model (FHM) can be used to establish the model for certain unknown complex system. The main advantage of using FHM over Takagi-Sugeno (T-S) fuzzy model is that no premise structure identification is needed and no completeness design of premise variables space is needed. In addition, an FHM is not only a kind of valid global description but also a kind of nonlinear model in nature. A nonlinear quadratic cost function is developed as a performance measurement of the closed-loop fuzzy system based on FHM.Based on delay-independent Lyapunov functional approach, some sufficient conditions for the existence of such a fuzzy hyperbolic guaranteed cost controller via state feedback are provided. These conditions are given in terms of the feasibility of linear matrix inequalities (LMIs). A simulation example is provided to illustrate the design procedure of the proposed method.展开更多
In this paper, a new class of high resolution schemes satisfying the 'condition A' (SCA) and the 'condition B' (SCB) for hyperbolic systems of conservation laws in one and two dimensions are constructe...In this paper, a new class of high resolution schemes satisfying the 'condition A' (SCA) and the 'condition B' (SCB) for hyperbolic systems of conservation laws in one and two dimensions are constructed. Moreover, the results of the numerical experiments by using these schemes are given for the system of Euler equations in one and two dimensions.展开更多
In this paper, we prove the uniqueness of generalized solution defined by Lebesgue-Stieltjes integral for the Cauchy problem of transportation equations. Our results are based on the discussions for linear system with...In this paper, we prove the uniqueness of generalized solution defined by Lebesgue-Stieltjes integral for the Cauchy problem of transportation equations. Our results are based on the discussions for linear system with discontinuous coefficient.展开更多
The paper is devoted to the study of optimal control of a hyperbolic system,where the control enters the system through the boundary.The hyperbolic system is the wave equation on a smooth and open domain,where the bou...The paper is devoted to the study of optimal control of a hyperbolic system,where the control enters the system through the boundary.The hyperbolic system is the wave equation on a smooth and open domain,where the boundary condition involves the normal derivative at the boundary of z,the time derivative of z times a constant k,and a nonlinear term control.Here,z is the state and u is the control,satisfying some boundedness condition depending on k.The functional cost consists of the energy and the difference between the solution of the system at final time,and a desired state in L2-norm.For a closed convex set,we prove the existence of an optimal control that minimises the cost functional using a priori estimates.Then,using the differentiability of the cost functional with respect to the control,we establish the characterisation by deriving necessary conditions that an optimal control must satisfy.A numerical approach is successfully illustrated by simulations.展开更多
For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.
The global existence and uniqueness of solutions for the Cauchy problem of a non-strictly hyperbolic system are proved. Such new generalized solutions are defined by Lebesgue-Stieltjes integral. It involves the so-cal...The global existence and uniqueness of solutions for the Cauchy problem of a non-strictly hyperbolic system are proved. Such new generalized solutions are defined by Lebesgue-Stieltjes integral. It involves the so-called wave. The method can also be applied to more general systems.展开更多
In this paper, we prove the global existence of generalized solutions of Cauchy problem for transportation equations; moreover, we construct the solution by generalized potential.
By means of the continuous Glimm functional, a proof is given on the global existence of classical solutions to Cauchy problem for general first order quasilinear hyperbolic systems with small initial total variation.
The history of hyperbolic system of conservation laws can date back to the eighteenth century,after several works of the natural philosophers,most notably L.Euler.The compressible Euler system,consisting of mass,momen...The history of hyperbolic system of conservation laws can date back to the eighteenth century,after several works of the natural philosophers,most notably L.Euler.The compressible Euler system,consisting of mass,momentum and energy in the divergence form,is the prototypical model of the hyperbolic system of conservation laws,and its main feature is the shock formation in a finite time no matter how the initial values smooth.This poses huge challenges in the mathematical analysis.展开更多
基金the Special Funds for Major State Basic Research Projects of China.
文摘By means of an equivalent invariant form of boundary conditions, the authors get the exis- tence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions.
基金Project supported by the Special Funds forMajor State Basic Research Projects ofChina.
文摘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.
基金Project supported by the National Natural Science Foundation of China
文摘The author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with "slow" decay initial data. By constructing an example, first it is illustrated that the classical solution to this kind of Cauchy problem may blow up in a finite time, even if the system is weakly linearly degenerate. Then some lower bounds of the life-span of classical solutions are given in the case that the system is weakly linearly degenerate. These estimates imply that, when the system is weakly linearly degenerate, the classical solution exists almost globally in time. Finally, it is proved that Theorems 1.1-1.3 in [2] are still valid for this kind of initial data.
基金Project supported by Specialized Research Fund for the Doctoral Program of Higher Education.
文摘By means of the theory on the semi-global C^1 solution to the mixed initialboundary value problem （IBVP） for first order quasilinear hyperbolic systems, we establish the exact controllability for general nonautonomous first order quasilinear hyperbolic systems with general nonlinear boundary conditions.
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Project supported by the Special Funds for Major State Basic Research Projects of China
文摘The authors consider the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems and obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution.
基金the Special Funds for Major State Basic Research Projects of China.
文摘For a class of mixed initial-boundary value problem for general quasilinear hyperbolic sys-tems with zero eigenvalues, the authors establish the local exact controllability with boundary controls acting on one end or on two ends and internai controls acting on a part of equations in the system.
文摘In this paper, we study a nonlinear hyperbolic-parabolic system modeling some biological phenomena. By semigroup theory and Leray-Schauder fixed point argument, the local existence and uniqueness of the weak solutions for this system are proved. For the spatial dimension N = 1, the global existence of the weak solution will be established by the bootstrap argument.
文摘This paper develops delay-independent fuzzy hyperbolic guaranteed cost control for nonlinear continuous-time systems with parameter uncertainties. Fuzzy hyperbolic model (FHM) can be used to establish the model for certain unknown complex system. The main advantage of using FHM over Takagi-Sugeno (T-S) fuzzy model is that no premise structure identification is needed and no completeness design of premise variables space is needed. In addition, an FHM is not only a kind of valid global description but also a kind of nonlinear model in nature. A nonlinear quadratic cost function is developed as a performance measurement of the closed-loop fuzzy system based on FHM.Based on delay-independent Lyapunov functional approach, some sufficient conditions for the existence of such a fuzzy hyperbolic guaranteed cost controller via state feedback are provided. These conditions are given in terms of the feasibility of linear matrix inequalities (LMIs). A simulation example is provided to illustrate the design procedure of the proposed method.
文摘In this paper, a new class of high resolution schemes satisfying the 'condition A' (SCA) and the 'condition B' (SCB) for hyperbolic systems of conservation laws in one and two dimensions are constructed. Moreover, the results of the numerical experiments by using these schemes are given for the system of Euler equations in one and two dimensions.
文摘In this paper, we prove the uniqueness of generalized solution defined by Lebesgue-Stieltjes integral for the Cauchy problem of transportation equations. Our results are based on the discussions for linear system with discontinuous coefficient.
文摘The paper is devoted to the study of optimal control of a hyperbolic system,where the control enters the system through the boundary.The hyperbolic system is the wave equation on a smooth and open domain,where the boundary condition involves the normal derivative at the boundary of z,the time derivative of z times a constant k,and a nonlinear term control.Here,z is the state and u is the control,satisfying some boundedness condition depending on k.The functional cost consists of the energy and the difference between the solution of the system at final time,and a desired state in L2-norm.For a closed convex set,we prove the existence of an optimal control that minimises the cost functional using a priori estimates.Then,using the differentiability of the cost functional with respect to the control,we establish the characterisation by deriving necessary conditions that an optimal control must satisfy.A numerical approach is successfully illustrated by simulations.
基金supported by the National Natural Science Foundation of China(Nos.12122104,11831011)Shanghai Science and Technology Programs(Nos.21ZR1406000,21JC1400600,19JC1420101)。
文摘For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.
文摘The global existence and uniqueness of solutions for the Cauchy problem of a non-strictly hyperbolic system are proved. Such new generalized solutions are defined by Lebesgue-Stieltjes integral. It involves the so-called wave. The method can also be applied to more general systems.
文摘In this paper, we prove the global existence of generalized solutions of Cauchy problem for transportation equations; moreover, we construct the solution by generalized potential.
文摘By means of the continuous Glimm functional, a proof is given on the global existence of classical solutions to Cauchy problem for general first order quasilinear hyperbolic systems with small initial total variation.
文摘The history of hyperbolic system of conservation laws can date back to the eighteenth century,after several works of the natural philosophers,most notably L.Euler.The compressible Euler system,consisting of mass,momentum and energy in the divergence form,is the prototypical model of the hyperbolic system of conservation laws,and its main feature is the shock formation in a finite time no matter how the initial values smooth.This poses huge challenges in the mathematical analysis.
