Optimal control of nonsmooth system governed by quasi-linear elliptic equations
The authors have discussed the following optimal control problem: \[ \min _{u\in L^2(\Omega)}G(y)+\Phi (u), \] where \(y\) is the solution of \[ -\sum _{i=1}^n(a_i(y_{x_i}))_{x_i}=u \quad \text{in }\Omega, \qquad y=0\quad \text{on }\partial \Omega . \] Here \(G\) is a lower semicontinuous convex function and \(\Phi \) is a convex function. Through the smoothness and the approximation of the initial problem the necessary condition has been obtained. This can be considered as the Euler-Lagrange condition in the quasi-linear case.
scientific article; zbMATH DE number 1046343