Exact Controllability for Nonautonomous First Order Quasilinear Hyperbolic Systems

By means of the theory on the semi-global C1 solution to the mixed initial-boundary 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.     

Exact Boundary Controllability for a Kind of Second-Order Quasilinear Hyperbolic Systems

Based on the theory of semi-global C ¹ solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems, the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.     

On Quasilinear Generalized Canonical Hyperbolic Systems of First-Order Partial Differential Equations

By using the method of characteristics, we prove theorems on continuous solvability and on properties of solutions of the mixed Cauchy boundary-value problem for the generalized canonical hyperbolic system of quasilinear partial differential equations of the first order in a general connected domain in (m+ 1)-dimensions.     

Galerkin Method for First-Order Hyperbolic Systems with Two Independent Variables

We investigate a mixed problem for a weakly nonlinear first-order hyperbolic system with two independent variables in bounded and unbounded domains. Assuming that the nonlinearities are monotonic, we obtain conditions for the existence and uniqueness of a generalized solution; these conditions do not depend on the behavior of a solution as x +.     

Exact Controllability of a Damped Wave Equation with Distributed Controls

We study the controllability problem of the one-dimensional damped wave equation $$\rho {\text{(}}x{\text{)}}u_{tt} - \frac{d}{{dx}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {p(x)u_x } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2d(x)\rho (x)u_t + q(x)\rho (x)u = h(x,t),{\text{ }}x \in {\text{(0,1)}}$$ This equation describes the forced motion of a nonhomogeneous string subject to a viscous damping. It is proved that the solution can be exactly controlled in finite time by means of distributed control forces h which vanish outside of any fixed nonempty subinterval of (0, 1). Moreover the optimal time of controllability is given.     

半线性热方程精确内部能控性

主要采用了构造辅助系统u′ △u=0的方法，同时也利用了强单调算子 的满射性。在条件（H)的修改下，得到在有限时间内的精确可控性，研究半线性热方程系统ey/et=△y-f(x,t,y,倒△y) h(x,t)的精确内部能控性。     

Cauchy Problem for Quasilinear Hyperbolic Systems with Higher Order Dissipative Terms

Abstract In this paper, the author studies the global existence, singularities and life span of smoothsolutions of the Cauchy probleth for a class of quasilinear hypetbolic systems with higher order dissipativeterms and gives their applications to nonlinear wave equations with higher order dissipative terms.     

Formation of Singularities for Quasilinear Hyperbolic Systems with Characteristics with Constant Multiplicity

In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity(> 1), under the assumptions that there is a genuinely nonlinear simple characteristic and the initial data possess certain decaying properties, the blow-up result is obtained for the C¹ solution to the Cauchy problem.     

Mixed Problem for Singular Quasilinear Hyperbolic Systems with a Single Spatial Variable

Mathematical Notes -     

Initial Value Problem for General Quasilinear Hyperbolic Systems with Characteristics with Constant Multiplicity

The authors consider the global existence and the blow-up phenomenon of classical solutions with small amplitude to the Cauchy problem for general quasilinear hyperbolic systems with characteristics with constant multiplicity and given some applications.     

Cauchy Problem for General Rst Order Inhomogeneous Quasilinear Hyperbolic Systems

In this paper, we consider Cauchy problem for general first order inho- mogeneous quasilinear strictly hyperbolic systems. Under the matching condition, we first give an estimate on inhomogeneous terms. By this estimate, we obtain the asymptotic behaviour for the life-span of C¹ solutions with “slowly” decaying and small initial data and prove that the formation of singularity is due to the envelope of characteristics of the same family.     

对角型拟线性双曲组的整体经典解

本文在特征值部分线性退化、部分弱线性退化时,考察一阶拟线性对角型严格双曲组的柯西问题,当初值满足适当的条件时得到了其整体经典解的存在性.     

可化约拟线性双曲组带一类非局部边界条件的精确边界能控性

先证明了一类非线性积分方程解的存在唯一性,利用此结果,建立了可化约拟线性双曲组带一类非局部边界条件的单侧精确边界能控性.     

拟线性双曲型方程(组)的精确能控性

本文为作者在中国科学院数学与系统科学研究院举办的第六届华罗庚数学讲座上的讲稿.§1　引言——从常微分方程谈起考虑如下的线性常微分方程组dXdt=AX+Bu,(1.1)其中,t为自变量(时间),X=(X1,…,XN)为状态变量,u=(u1,…,um)为控制变量,而A及B分别为N×N及N×m常数阵.(1.1)是一个有限维的动力系统.说该系统在时间区间[0,T](T>0)上具有精确能控性,是指对于在t=0时任意给定的初值X0及在t=T时任意给定的终值XT,一定能找到[0,T]上的控制函数u=u(t),使Cauchy问题dXdt=AX+Bu(t),(1.2)t=0:X=X0(1.3)的解X=X(t)精确地满足终端条件t=T:X=X…     

Global Smooth Solutions for a Class of Quasilinear Hyperbolic Systems with Disstipative Terms

In this paper the authors prove the existence and uniqueness of global smooth solutions to the Cauchy problem for quasilinear hyperbolic systems with some kinds of dissipative terms.     

Petrovsky板方程的精确内部能控性

本文研究在局部分布反馈控制下边界夹住的Petrovsky板方程.利用逐片乘子技巧和紧唯一性推理,证明了在任意指定时间内,系统都是精确内部能控的.