The exact controllability of the semilinear wave equation \(y_{tt}-y_{xx}+ f(y)=0\), \(x\in (0,1)\) assuming that f is locally Lipschitz continuous and satisfies the growth condition \(\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /(\vert r\vert \ln ^{p}\vert r\vert )\leqslant \beta \) for some \(\beta \) small enough and \(p=2\) has been obtained by Zuazua (Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129, 1993). The proof based on a non-constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized wave equation. Under the above asymptotic assumption with \(p=3/2\), by introducing a different fixed point application, we present a simpler proof of the exact boundary controllability which is not based on the cost of observability of the wave equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition \(\limsup _{\vert r\vert \rightarrow \infty } \vert f^\prime (r)\vert /\ln ^{3/2}\vert r\vert \leqslant \beta \) for some \(\beta \) small enough, we show that the above fixed point application is contracting yielding a constructive method to approximate the controls for the semilinear equation. Numerical experiments illustrate the results. The results can be extended to the multi-dimensional case and for nonlinearities involving the gradient of the solution.

半线性波动方程\(y_{tt}-y_{xx}+ f(y)=0\) , \(x\in (0,1)\)的精确可控性假设f是局部 Lipschitz 连续的并且满足生长条件\(\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /(\vert r\vert \ln ^{p}\vert r\vert )\leqslant \beta \)对于一些足够小的\(\beta \)并且\(p=2\)已由 Zuazua 获得（Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129, 1993）。基于非构造定点参数的证明利用了线性波动方程的可观测常数的精确估计。在上述渐近假设下，\(p=3/2\)，通过引入不同的定点应用，我们提出了一个更简单的精确边界可控性证明，它不是基于波动方程相对于势的可观测性成本。那么，假设f是局部 Lipschitz 连续的并且满足增长条件\(\limsup _{\vert r\vert \rightarrow \infty } \vert f^\prime (r)\vert /\ln ^{3/2} \vert r\vert \leqslant \beta \)对于一些足够小的\(\beta \)，我们表明上述定点应用正在收缩，从而产生一种构造方法来近似半线性方程的控制。数值实验说明了结果。结果可以扩展到多维情况以及涉及解梯度的非线性。