We consider a linear system composed of $N+1$ one dimensional heat equations connected by point-mass-like interface conditions. Assume an $L^2$ Dirichlet boundary control at one end, and Dirichlet boundary condition on the other end. Given any $L^2$-type initial temperature distribution, we show that the system is null controllable in arbitrarily small time. The proof uses known results for exact controllability for the associated wave equation. An argument using the Fourier Method reduces the control problem for both the heat equation and the wave equation to certain moment problems. Controllability is then proved by relating minimality properties of the family of exponential functions associated to the wave with the family associated to the heat equation. Based on the controllability result we solve the dynamical inverse problem, i.e. recover unknown parameters of the system from the Dirichlet-to-Neumann map given at a boundary point.

我们考虑由 $ñ+\mathrm{1个}$一维热方程由点状质量的界面条件连接。假设${\mathrm{大号}}^2$一端是Dirichlet边界控制，另一端是Dirichlet边界条件。给予任何${\mathrm{大号}}^2$式的初始温度分布，我们表明系统在任意短的时间内都是空可控的。该证明使用已知结果来精确控制相关波动方程。使用傅立叶方法的参数将热方程和波动方程的控制问题都简化为某些矩问题。然后通过将与波相关的指数函数族的极小性质与与热方程相关的族相关联来证明可控性。根据可控性结果，我们解决了动力学反问题，即从边界点处的Dirichlet到Neumann映射中恢复系统的未知参数。