论文标题
使用最小二乘方法的半线性1D波方程的精确对照近似
Approximation of exact controls for semi-linear 1D wave equations using a least-squares approach
论文作者
论文摘要
假设$ g $满足生长条件$ \ vert g（s）\ vert /（\ vert /（\ vert s \ vert s \ vert s \ log^{2} $ vert，$ vert^{$ vert， s \ vert \ rightArrow \ infty $和l^\ infty_ {loc}中的$ g^\ prime \（\ mathbb {r}）$已由Zuazua在90年代获得。基于LERAY-SCHAUDER固定点参数的证明利用了线性波动方程的可观察性常数的精确估计。但是，它没有提供无效控制的明确结构。假设$ g^\ prime \ in l^\ infty_ {loc}（\ mathbb {r}）$，那个$ \ sup_ {a，b \ in \ mathbb {r {r}某些$ r \ in（0,1] $中的a-b \ vert^r <\ infty $，该$ g^\ prime $满足增长条件$ \ \ vert g^\ prime（s）\ prime（s）\ prime/\ log^{2}（\ vert s \ vert s \ vert s \ vert）在最小二乘方法的情况下，该方法的无效控制可以确保序列的初始元素，尤其是在有限的迭代之后，融合是超级线性的。
The exact distributed controllability of the semilinear wave equation $y_{tt}-y_{xx} + g(y)=f \,1_ω$, assuming that $g$ satisfies the growth condition $\vert g(s)\vert /(\vert s\vert \log^{2}(\vert s\vert))\rightarrow 0$ as $\vert s\vert \rightarrow \infty$ and that $g^\prime\in L^\infty_{loc}(\mathbb{R})$ has been obtained by Zuazua in the nineties. The proof based on a Leray-Schauder fixed point argument makes use of precise estimates of the observability constant for a linearized wave equation. It does not provide however an explicit construction of a null control. Assuming that $g^\prime\in L^\infty_{loc}(\mathbb{R})$, that $\sup_{a,b\in \mathbb{R},a\neq b} \vert g^\prime(a)-g^{\prime}(b)\vert/\vert a-b\vert^r<\infty $ for some $r\in (0,1]$ and that $g^\prime$ satisfies the growth condition $\vert g^\prime(s)\vert/\log^{2}(\vert s\vert)\rightarrow 0$ as $\vert s\vert \rightarrow \infty$, 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 guarantees the convergence whatever the initial element of the sequence may be. In particular, after a finite number of iterations, the convergence is super linear with rate $1+r$. This general method provides a constructive proof of the exact controllability for the semilinear wave equation.
