半线性一维heat equations的构造性精确控制
Constructive exact control of semilinear 1D heat equations
摘要：
Fernán-Cara和Zuazua在2000年得到了多维有界区域上半线性热方程$\partial_{t}y-Δy + g(y)=f \,1_ω$的精确分布能控性，假设$\partial_{t}y-Δy + g(y)=f \,1_ω$满足增长条件$\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$.基于非构造性不动点论证的证明利用了线性化热方程可观测常数的精确估计。在一维情形下，假设$g^\prime$在无穷远处的增长速度不超过$β>0$，且$g^\prime$在$\mathbb{R}$上是一致Hölder连续的，其指数为$p\in [0,1]$，我们设计了一个构造性的证明，给出了半线性方程的显式序列收敛到受控解，至少在有限次迭代后达到$1+p$阶.
英文摘要：
The exact distributed controllability of the semilinear heat equation $\partial_{t}y-Δy + g(y)=f \,1_ω$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that $g^\prime$ does not grow faster than $β\ln^{3/2}\vert r\vert$ at infinity for $β>0$ small enough and that $g^\prime$ is uniformly Hölder continuous on $\mathbb{R}$ with exponent $p\in [0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations.
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