zbMATH — the first resource for mathematics

Exact boundary controllability for the semilinear wave equation. (English) Zbl 0731.93011
Nonlinear partial differential equations and their applications, Lect. Coll. de France Semin., Paris/Fr. 1987-88, Vol. X, Pitman Res. Notes Math. Ser. 220, 357-391 (1991).
Summary: [For the entire collection see Zbl 0728.00013.]
The exact boundary controllability for the semilinear wave equation with Dirichlet boundary conditions is studied. The exact controllability is proven when the nonlinearity is globally Lipschitz continuous. The controllability time is the one of the linear wave equation. The method of proof combines HUM (Hilbert uniqueness method) and a fixed point argument. On the other hand, by using a penalization method, we prove the existence of a set of optimal controls that satisfy an optimality system. This optimality system consists of two coupled semilinear wave equations.

93B05 Controllability