Observation and Prediction for the Heat Equation. IV: Patch Observability and Controllability

It is shown first that the problem of predicting the state at $t = T$ of a solution of the heat equation with homogeneous boundary conditions from observation on a patch of complementary boundary data is well-posed in the presence of an a priori bound on initial data. Next, it is shown that Russell’s star-complementarily sufficient condition for exact boundary controllability is far from necessary by demonstrating controllability from the inner face of an annular sector in $\mathbb{R}^n $. Finally, some results are obtained as to the dependence of the optimal null-control on the patch considered and on T.