Optimal complexity solution of space-time finite element systems for state-based parabolic distributed optimal control problems

In this paper we consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The approach presented here is based on the variational formulation of the parabolic evolution equation in anisotropic Sobolev spaces, considering the control in ${\left[H_{0;,0}^{1,1/2}\right(Q\left)\right]}^{⁎}$. Since the state equation defines an isomorphism from $H_{0;0,}^{1,1/2}\left(Q\right)$ onto ${\left[H_{0;,0}^{1,1}\right(Q\left)\right]}^{⁎}$, we can eliminate the control to end up with a minimization problem in $H_{0;0,}^{1,1/2}\left(Q\right)$ where the anisotropic Sobolev norm can be realized using a modified Hilbert transformation. In the unconstrained case, the minimizer is the unique solution of a singularly perturbed elliptic equation. In the case of a space-time tensor-product mesh, we can use sparse factorization techniques to construct a solver of almost linear complexity. Numerical examples also include additional state constraints, and a nonlinear state equation.