高维拟线性抛物系统能控性的近期进展

本文综述高维拟线性抛物型方程、拟线性复Ginzburg-Landau方程以及只含一个控制变量的高维耦合拟线性抛物型方程组的能控性方面的一些近期的结果.通过使用不动点技术,采用主部具有C1系数的线性抛物型方程或方程组一些新的精细的Carleman估计.这一方法的要点是在古典解的框架下考虑能控性问题,并且当给定的数据具有一定的正则性时,线性抛物型方程或方程组在H¨older空间中来选取控制函数.利用类似的方法,还建立了拟线性抛物型方程不灵敏控制的存在性,其关键是将不灵敏问题转化为由拟线性抛物型方程和线性抛物型方程构成的耦合方程组在单个控制下一个非标准的能控性问题.

Recent progress in controllability of multidimensional quasilinear parabolic systems

We overview some recent controllability results for multidimensional quasilinear parabolic equations, quasi- linear complex Ginzburg-Landau equations, and coupled quasilinear parabolic systems with single control variable. When using the fixed point technique, we employ the main tools to investigate some new and delicate Carleman estimates for suitable linear parabolic equations/systems with C1coefficients in principal parts. The key points of the approach are to formulate the controllability problems in the frame of classical solutions and to seek the control functions in the H ?older spaces for linear parabolic equations/systems with given data having certain regularity. By means of a similar approach, the existence of insensitizing controls for quasilinear parabolic equations is also established. The key point is to transform this insensitizing problem into a nonstandard controllability problem for some nonlinear cascade system governed by a quasilinear parabolic equation and a linear parabolic equation with single control variable.