The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations. First, the authors review the theory for ...The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations. First, the authors review the theory for linear partial differential equations, including the iteration method for the null controllability of the time-invariant heat equation and the Rellich-type multiplier method for the exact controllability of the time-invariant wave equation, and especially a unified controllability/observability theory for parabolic and hyperbolic equations based on a global Carleman estimate. Then, the authors present sharp global controllability results for both semi-linear parabolic and hyperbolic equations, based on linearization approach, sharp observability estimates for the corresponding linearized systems and the fixed point argument. Finally, the authors survey the local null controllability result for a class of quasilinear parabolic equations based on the global Carleman estimate, and the local exact controllability result for general hyperbolic equations based on a new unbounded perturbation techniaue.展开更多
The purpose of the present paper is to call for attention to the following question: Which of the initial data (nonsmall) admit global smooth solutions to the Cauchy problem for nonlinear wave equations. A few cases a...The purpose of the present paper is to call for attention to the following question: Which of the initial data (nonsmall) admit global smooth solutions to the Cauchy problem for nonlinear wave equations. A few cases and examples are sketched, showing that the general answer of this question may be quite complicated.展开更多
基金supported by the National Science Foundation of China under Grant Nos. 10831007,60821091,and 60974035the project MTM2008-03541 of the Spanish Ministry of Science and Innovation
文摘The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations. First, the authors review the theory for linear partial differential equations, including the iteration method for the null controllability of the time-invariant heat equation and the Rellich-type multiplier method for the exact controllability of the time-invariant wave equation, and especially a unified controllability/observability theory for parabolic and hyperbolic equations based on a global Carleman estimate. Then, the authors present sharp global controllability results for both semi-linear parabolic and hyperbolic equations, based on linearization approach, sharp observability estimates for the corresponding linearized systems and the fixed point argument. Finally, the authors survey the local null controllability result for a class of quasilinear parabolic equations based on the global Carleman estimate, and the local exact controllability result for general hyperbolic equations based on a new unbounded perturbation techniaue.
基金Project supported by the Chinese SpecialFunds for Major State Basic Research Project"NonlinearScience"
文摘The purpose of the present paper is to call for attention to the following question: Which of the initial data (nonsmall) admit global smooth solutions to the Cauchy problem for nonlinear wave equations. A few cases and examples are sketched, showing that the general answer of this question may be quite complicated.
