CONTROLLABILITY OF NONLINEAR-WAVE EQUATION IN SEVERAL SPACE VARIABLES

On a rectangular parallelopiped in R**N, N greater than equivalent to 2, the equation u//t//t equals DELTA u plus f(u, u//t) is considered, where f is a nonlinear perturbation meeting certain conditions. It is proved that the above system is locally controllable at u equals 0, u//t equals 0; i. e. , the set of states in a certain function space which can be reached from (0,0) in a finite time T less than infinity using boundary controls is an open neighborhood of (0,0) in that function space. These results generalize to the nonlinear case conclusions obtained by D. L. Russell for the linear wave equation, in which global controllability was established.