Observability and Control of Parabolic Equations on Networks

In this article, we show some results we obtained related to observability and control of parabolic equations on networks with loops. By using a novel Carleman inequality, we found that the observability of the entire network could be achieved under certain hypothesis about the position of the observation domain. The main difficulty we tackled, due to the existence of loops, was to avoid entering into a circular fallacy, notably in the construction of the auxiliary function for the Carleman inequality. The difficulty was overcame with a careful treatment of the boundary terms on the junctions. Finally, we used the observability to prove the null controllability of the network and to obtain the Lipschitz stability for an inverse problem consisting on retrieving a stationary potential in the parabolic equation from measurements on the observation domain.