Exact boundary controllability of 1D semilinear wave equations through a constructive approach

The exact controllability of the semilinear wave equation y tt - y xx + f ( y ) = 0 , x ∈ ( 0 , 1 ) assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup | r | → ∞ | f ( r ) | / ( | r | ln p | r | ) ⩽ β for some β 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 lim sup | r | → ∞ | f ′ ( r ) | / ln 3 / 2 | r | ⩽ β for some β 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.