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 is an element of (0, 1) assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup(vertical bar r vertical bar ->infinity) vertical bar f (r)vertical bar/(vertical bar r vertical bar ln(p) vertical bar r vertical bar) <= beta for some beta small enough and p = 2 has been obtained by Zuazua (Ann Inst H Poincare Anal Non Lineaire 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(vertical bar r vertical bar ->infinity) vertical bar f '(r)vertical bar/(vertical bar r vertical bar ln(3/2) vertical bar r vertical bar) <= 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.