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 boundarycontrols acting on one end or o...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 boundarycontrols acting on one end or on two ends and internal controls acting on a part of equationsin the system.展开更多
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 authorsreview the theory for lin...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 authorsreview the theory for linear partial differential equations,including the iteration method for the nullcontrollability of the time-invariant heat equation and the Rellich-type multiplier method for the exactcontrollability of the time-invariant wave equation,and especially a unified controllability/observabilitytheory for parabolic and hyperbolic equations based on a global Carleman estimate.Then,the authorspresent sharp global controllability results for both semi-linear parabolic and hyperbolic equations,based on linearization approach,sharp observability estimates for the corresponding linearized systemsand the fixed point argument.Finally,the authors survey the local null controllability resultfor a class of quasilinear parabolic equations based on the global Carleman estimate,and the localexact controllability result for general hyperbolic equations based on a new unbounded perturbationtechnique.展开更多
This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings.The authors derive the equations and appropriate nodal conditions,determine equilibrium ...This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings.The authors derive the equations and appropriate nodal conditions,determine equilibrium solutions,and,by using the methods of quasilinear hyperbolic systems,prove that for tree networks the natural initial,boundary value problem has classical solutions existing in neighborhoods of the "stretched" equilibrium solutions.Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved.Finally,it is proved that,given two different equilibrium states satisfying certain conditions,it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a suffciently large time interval.展开更多
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 boundarycontrols acting on one end or on two ends and internal controls acting on a part of equationsin the system.
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 authorsreview the theory for linear partial differential equations,including the iteration method for the nullcontrollability of the time-invariant heat equation and the Rellich-type multiplier method for the exactcontrollability of the time-invariant wave equation,and especially a unified controllability/observabilitytheory for parabolic and hyperbolic equations based on a global Carleman estimate.Then,the authorspresent sharp global controllability results for both semi-linear parabolic and hyperbolic equations,based on linearization approach,sharp observability estimates for the corresponding linearized systemsand the fixed point argument.Finally,the authors survey the local null controllability resultfor a class of quasilinear parabolic equations based on the global Carleman estimate,and the localexact controllability result for general hyperbolic equations based on a new unbounded perturbationtechnique.
基金supported by the DFG SPP1253:Optimization with PDE-Constaintsthe DFG-CE315 Cluster of Excellence:Engineering of Advanced Materials
文摘This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings.The authors derive the equations and appropriate nodal conditions,determine equilibrium solutions,and,by using the methods of quasilinear hyperbolic systems,prove that for tree networks the natural initial,boundary value problem has classical solutions existing in neighborhoods of the "stretched" equilibrium solutions.Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved.Finally,it is proved that,given two different equilibrium states satisfying certain conditions,it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a suffciently large time interval.
