论文标题
通过最小二乘方法对半线性1D波方程的建设性精确控制
Constructive exact control of semilinear 1D wave equations by a least-squares approach
论文作者
论文摘要
It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $\partial_{tt}y-\partial_{xx}y + g(y)=f 1_ω$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)\cap L^2(0,1)$ with controls $f\in l^2（（0,1）\ times（0，t））$，对于任何$ t> 0 $和任何非空的开放子集$（0,1）$的$ g \ in \ nathcal {c}^1（\ r）$ in \ mathcal in \ mathcal in \ mathcal in \ $β\ vert x \ vert \ vert \ ln^^2} $β> 0 $小。但是，基于Leray-Schauder固定点定理的证明不是建设性的。在本文中，我们为半线性1D波方程的确切可控性设计了一个建设性的证明和算法。 Assuming that $g^\prime$ does not grow faster than $β\ln^{2}\vert x\vert$ at infinity for some $β>0$ small enough and that $g^\prime$ is uniformly Hölder continuous on $\R$ with exponent $s\in[0,1]$, we design a least-squares algorithm yielding an explicit sequence converging to a半线性方程的受控解决方案，至少在有限的迭代次数之后，订单$ 1+s $。
It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $\partial_{tt}y-\partial_{xx}y + g(y)=f 1_ω$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)\cap L^2(0,1)$ with controls $f\in L^2((0,1)\times(0,T))$, for any $T>0$ and any nonempty open subset $ω$ of $(0,1)$, assuming that $g\in \mathcal{C}^1(\R)$ does not grow faster than $β\vert x\vert \ln^{2}\vert x\vert$ at infinity for some $β>0$ small enough. The proof, based on the Leray-Schauder fixed point theorem, is however not constructive. In this article, we design a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations. Assuming that $g^\prime$ does not grow faster than $β\ln^{2}\vert x\vert$ at infinity for some $β>0$ small enough and that $g^\prime$ is uniformly Hölder continuous on $\R$ with exponent $s\in[0,1]$, we design a least-squares algorithm yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+s$ after a finite number of iterations.
