双线性控制下抛物型发展方程的超指数能稳性
Superexponential stabilizability of evolution equations of parabolic type via bilinear control
摘要：
本文证明了一类抽象抛物型方程\begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\geq0 \end{equation*}的基态解的快速稳定性，其中算子$-A$是希尔伯特空间上的自伴增生算子，控制函数$p(\cdot)$是该算子.证明是基于线性化参数。我们证明了线性化系统是精确可控的，我们应用矩量法建立了一个控制$p(\cdot)$，引导解决方案在有限时间内的基态。最后，我们使用这样的控制，使非线性方程的解任意接近基态解的双指数收敛速率。给出了我们的结果在不同类型的抛物型方程中的几个应用。
英文摘要：
We prove rapid stabilizability to the ground state solution for a class of abstract parabolic equations of the form \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\geq0 \end{equation*} where the operator $-A$ is a self-adjoint accretive operator on a Hilbert space and $p(\cdot)$ is the control function. The proof is based on a linearization argument. We prove that the linearized system is exacly controllable and we apply the moment method to build a control $p(\cdot)$ that steers the solution to the ground state in finite time. Finally, we use such a control to bring the solution of the nonlinear equation arbitrarily close to the ground state solution with doubly exponential rate of convergence. We give several applications of our result to different kinds of parabolic equations.
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