Nonlinear Optimal Control Problems for Parabolic Equations

Let u be the solution of the heat equation with diffusion coefficient $k(x)( - 1 < x < 1)$, initial values $h(x)$ and boundary values 0 on $x = \pm 1$. The function $k(x)$ is a control variable to be chosen from a suitable class so as to minimize $\int_{ - 1}^1 {u^2 (x,T)dx} $ for some given $T > 0$. An explicit characterization of the optimal k is given. Other related problems are considered.