论文标题
1D线性化压缩的Navier-Stokes System通过一个控制力的边界零可控性
Boundary null-controllability of 1d linearized compressible Navier-Stokes system by one control force
论文作者
论文摘要
In this article, we study the boundary null-controllability properties of the one-dimensional linearized (around $(Q_0,V_0)$ with constants $Q_0>0, V_0>0$) compressible Navier-Stokes equations in the interval $(0,1)$ when a control function is acting either on the density or velocity component at one end of the interval. We first prove that the linearized system, with a Dirichlet boundary control on the density component and homogeneous Dirichlet boundary conditions on the velocity component, is null-controllable in $H^s_{per}(0,1) \times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, where $H^s_{per}(0,1)$ denotes the周期功能的Sobolev空间。证明是基于解决混合的抛物线 - 纤维骨头问题的问题，为此，我们对相关的伴随操作员进行了光谱分析，这是这项工作的主要部分。作为推论，我们还证明该系统在$ l^2（0,1）\ times l^2（0,1）$中时大致可控制。$ t> 1 $。 另一方面，假设密度在两个边界点W.R.T.时间，当通过dirichlet条件上对速度部分应用控件时，我们只能证明该系统在严格的有限的condimension $ \ Mathcal {h} \ subset H^s_s_ {per}（per}（per}（0,1）\ times lime times l^2（0,1,1）$ $ s $ s $ s $ s $ s> 1 $ t $时，该系统可在严格的子空间中无效控制。更确切地说，在这种情况下，我们能够证明相关伴随操作员的所有特征函数对于较高的频率都可以观察到，而对于较低的频率，很难得出任何结论。我们在本文中证明的一种抛物线式Hyperbolic联合INGHAM型不平等，导致空间$ \ MATHCAL {H}^*$的可观察性不平等，随之而来的可控性结果。重要的一点是，当对照在速度部分作用时，矩方法并不能为无零控制性提供更好的空间。
In this article, we study the boundary null-controllability properties of the one-dimensional linearized (around $(Q_0,V_0)$ with constants $Q_0>0, V_0>0$) compressible Navier-Stokes equations in the interval $(0,1)$ when a control function is acting either on the density or velocity component at one end of the interval. We first prove that the linearized system, with a Dirichlet boundary control on the density component and homogeneous Dirichlet boundary conditions on the velocity component, is null-controllable in $H^s_{per}(0,1) \times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, where $H^s_{per}(0,1)$ denotes the Sobolev space of periodic functions. The proof is based on solving a mixed parabolic-hyperbolic moments problem and to do so, we perform a spectral analysis for the associated adjoint operator which is the main involved part of this work. As a corollary, we also prove that the system is approximately controllable in $L^2(0,1) \times L^2(0,1)$ when $T>1$. On the other hand, assuming that the density is equal on the two boundary points w.r.t. time, when a control is applied on the velocity part through a Dirichlet condition, we can only able to prove that the system is null-controllable in a strict subspace of finite codimension $\mathcal{H}\subset H^s_{per}(0,1) \times L^2(0,1)$ for $s>1/2$ when $T>1$. More precisely, in this case we are able to show that all the eigenfunctions of the associated adjoint operator are observable for higher frequencies whereas for the lower frequencies it is hard to conclude anything. A parabolic-hyperbolic joint Ingham-type inequality which we prove in this article, leads to an observability inequality in the space $\mathcal{H}^*$ and the controllability result follows. The significant point is that the moments method does not yield a better space for the null-controllability when a control acts on the velocity part.
