论文标题
具有矩阵电势的1D波方程的动态反问题中的数据表征
Data characterization in dynamical inverse problem for the 1d wave equation with matrix potential
论文作者
论文摘要
正在考虑的动态系统是\ begin {align*}＆ u_ {tt} -u_ {xx}+vu = 0，\ qquad x> 0，\，\，\，\，\，\，\ \ \ \ | _ { u | _ {x = 0} = f，\，\，t \ geqslant 0，\ end {align*}其中$ v = v（x）$是matrix-valued函数（{\ it portival}）; $ f = f（t）$是时间的$ \ mathbb r^n $值函数（{\ it边界控制}）; $ u = u^f（x，t）$是{\ it轨迹}（$ x $和$ t $的$ \ mathbb r^n $ valued函数）。系统的输入/输出映射是{\ it响应运算符} $ r：f \ mapsto u^f_x（0，\ cdot），\，\，\，\，t \ geqslant0 $。 {\ it逆问题}是从给定的$ r $确定$ v $。表征其数据是为了提供确保其解决性的$ r $的必要条件。 解决此问题的过程长期以来已经众所周知，并且已经宣布了特征（Avdonin和Belishev，1996）。但是，没有提供证明，而且事实证明必须纠正配方。我们的论文填补了这一空白。
The dynamical system under consideration is \begin{align*} & u_{tt}-u_{xx}+Vu=0,\qquad x>0,\,\,\,t>0;\\ & u|_{t=0}=u_t|_{t=0}=0,\,\,x\geqslant 0;\quad u|_{x=0}=f,\,\,t\geqslant 0, \end{align*} where $V=V(x)$ is a matrix-valued function ({\it potential}); $f=f(t)$ is an $\mathbb R^N$-valued function of time ({\it boundary control}); $u=u^f(x,t)$ is a {\it trajectory} (an $\mathbb R^N$-valued function of $x$ and $t$). The input/output map of the system is a {\it response operator} $R:f\mapsto u^f_x(0,\cdot),\,\,\,t\geqslant0$. The {\it inverse problem} is to determine $V$ from given $R$. To characterize its data is to provide the necessary and sufficient conditions on $R$ that ensure its solvability. The procedure that solves this problem has long been known and the characterization has been announced (Avdonin and Belishev, 1996). However, the proof was not provided and, moreover, it turned out that the formulation must be corrected. Our paper fills this gap.
