A semilinear control problem involving homogenization
Carlos Conca, Axel Osses, Jeannine Saint Jean Paulin · DOAJ (DOAJ: Directory of Open Access Journals) · 2001
We consider a control problem involving a semilinear elliptic equation with a uniformly Lipschitz non-linearity and rapidly oscillating coefficients in a bounded domain of $mathbb{R}^N$. The control is distributed on a compact subset interior to the domain. Given an $N-1$ dimensional hypersurface at the interior of the domain not intersecting the control zone, the trace of the solution on the curve has to be controlled. We prove that there exists a limit control as the homogenization parameter converges to zero, which results as the limit of fixed points for controllability problems. We link this limit control with the corresponding homogenized problem.