AI 中文总结
本文研究用少于方程数的内控制实现一维拟线性双曲系统的精确能控性，区分同速与异速情形，针对异速导致的导数损失，采用代数可解性与Nash-Moser型定点定理解决。
AI 中文摘要
本文研究了由$m<n$个在空间上局部化于域内某部分的内控制所驱动的$n \ imes n$一维拟线性双曲系统的精确能控性。我们区分两种情况。第一种是系统方程具有相同速度的情形。在这种情况下，我们可以利用特征线方法，对线性系统获得简单而完整的刻画。借助一个线性检验，这也为半线性系统在轨迹附近的局部精确能控性提供了若干充分条件。然而，当方程的速度不再相同时，我们看到，若试图用减少数量的控制来驾驭拟线性系统，就会遇到导数损失的问题。为解决此问题，如同J.-M. Coron和P. Lissy先前关于Navier-Stokes控制系统的一篇文章那样，我们首先利用M. Gromov提出的代数可解性概念。然而，与那篇先前文章不同——在那里可以使用标准的定点论证来处理非线性项——我们这里采用M. Gromov提出的Nash-Moser型定点定理，以处理导数损失的问题。
英文摘要
In this paper we investigate the exact controllability of $n \times n$ first order one-dimensional quasilinear hyperbolic systems by $m<n$ internal controls that are localized in space in some part of the domain. We distinguish two situations. The first one is when the equations of the system have the same speed. In this case, we can use the method of characteristics and obtain a simple and complete characterization for linear systems. Thanks to a linear test this also provides some sufficient conditions for the local exact controllability around the trajectories of semilinear systems. However, when the speed of the equations are not anymore the same, we see that we encounter the problem of loss of derivatives if we try to control quasilinear systems with a reduced number of controls. To solve this problem, as in a prior article by J.-M. Coron and P. Lissy on a Navier-Stokes control system, we first use the notion of algebraic solvability due to M. Gromov. However, in contrast with this prior article where a standard fixed point argument could be used to treat the nonlinearities, we use here a fixed point theorem of Nash-Moser type due to M. Gromov in order to handle the problem of loss of derivatives.
