本文研究了具有空间和不定常系数函数的线性热方程的Dirichlet和Neumann边界控制律的控制问题。该方程模拟了一维杆中的热扩散和与空间、不定常的热生成。在没有控制的情况下，如果系数函数为正且较大，则系统是不稳定的。利用基于状态反馈的边界控制，证明了对于时间解析系数$a(x,t)$，系统在任意速率下都是指数稳定的。进一步证明了Dirichlet边值系统和Neumann边值系统的控制都可以用这种方法镇定。在这样做时，控制核被{ \it明确}计算为一系列近似，并且它们被用于仿真。通过数值模拟验证了理论分析的正确性和系统的可控性。最后讨论了今后可能的工作。这是刘[{ \it SIAM J. Cont. Optim.}第42卷，第100页。1033-1043]和Smyshlyaev和Krstic [接受{ \it Automatica}]。
英文摘要：
In this paper, we study the control of the linear heat equation with a space and time dependent coefficient function by the Dirichlet and Neumann boundary control laws. This equation models the heat diffusion and space, time dependent heat generation in a one dimensional rod. Without control, the system is unstable if the coefficient function is positive and large. With boundary control based on the state feedback, we show that for the time analytic coefficient $a(x,t)$, the exponential stability of the system at any rate can be achieved. It is further shown that both the control of the Dirichlet and Neumann boundary value systems can be stabilized using this method. In doing this, the control kernels are {\it explicitly} calculated as series of approximation and they are used in the simulations. The numerical simulation confirmed the theoretical arguments and the controllability of the system. The possible future works are discussed at the end. This is the continuation of the recent work of Liu [{\it SIAM J. Cont. Optim.} vol. 42, pp. 1033-1043] and Smyshlyaev and Krstic [accepted by {\it Automatica}].
