Boundary Feedback Stabilization of an Unstable Heat Equation
Boundary Feedback Stabilization of an Unstable Heat Equation
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DOI：
10.1137/s0363012902402414
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
Weijiu Liu
中科院分区：
文献类型：
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作者：
Weijiu Liu
本文研究不稳定热方程ut（x，t）= uxx（x，t）+a（x）u（x，t）的边界反馈镇定问题.该方程可以被视为导热杆的模型，其中不仅热被扩散（数学上由于扩散项uxx），而且不稳定热也被生成（数学上由于a u，其中a >0）。我们证明了对于任意给定的连续可微函数a和任意给定的正常数$\l$，我们可以显式地构造一个边界反馈控制律，使得具有该控制律的方程的解以$\l$的速度指数收敛到零。这是Boskovic，Krstic和Liu [IEEE Trans. Automat. Control，46（2001），pp. 2022- 2028]和Balogh和Krstic [European J. Control，8（2002），pp. 165- 176]。
In this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut(x,t) = uxx(x,t)+a(x) u(x,t). This equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused (mathematically due to the diffusive term uxx) but also the destabilizing heat is generating (mathematically due to the term a u with a >0). We show that for any given continuously differentiable function a and any given positive constant $\l$ we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of $\l$. This is a continuation of the recent work of Boskovic, Krstic, and Liu [IEEE Trans. Automat. Control, 46 (2001), pp. 2022--2028] and Balogh and Krstic [European J. Control, 8 (2002), pp. 165--176].