论文标题
使用最小二乘方法的半线性热方程式的无效控制近似
Approximation of null controls for semilinear heat equations using a least-squares approach
论文作者
论文摘要
假设$ g $满足生长条件$ g（s）/（\ vert s \ vert s \ vert s \ vert \ log^{3/2}（1+ \ vert s \ vert）$ right \ rightrow 0 $ rightrow 0 $ rightrow 0 $ g^\ prime \ in l^\ infty_ {loc}（\ mathbb {r}）$在2000年获得了Fernández-Cara和Zuazua的获得。基于固定点参数的证明，使用了线性化热方程的可观察力恒定的精确估计值。但是，它没有提供无效控制的明确结构。 Assuming that $g^\prime\in W^{s,\infty}(\mathbb{R})$ for one $s\in (0,1]$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In特别是在有限数量的迭代次数之后，收敛是超线性的，速率等于$ 1+s $。
The null distributed controllability of the semilinear heat equation $y_t-Δy + g(y)=f \,1_ω$, assuming that $g$ satisfies the growth condition $g(s)/(\vert s\vert \log^{3/2}(1+\vert s\vert))\rightarrow 0$ as $\vert s\vert \rightarrow \infty$ and that $g^\prime\in L^\infty_{loc}(\mathbb{R})$ has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that $g^\prime\in W^{s,\infty}(\mathbb{R})$ for one $s\in (0,1]$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to $1+s$. Numerical experiments in the one dimensional setting support our analysis.
