Wednesday, Sept 12, 4:00PM-5:00 PM, Conference room

September 07, 2018 by Duan Chen
Categories: Spring 2022
Professor: Prof. Sergei. Avdonin, Univ. of Alaska, Fairbanks
Title: Control and Inverse Problems for Differential Equations on Graphs
Abstract: Quantum graphs are metric graphs with differential equations defined on the edges. Recent interest in control and inverse problems for quantum graphs is motivated by applications to important problems of classical and quantum physics, chemistry, biology, and engineering.In this talk we describe some new controllability and identifiability results
for partial differential equations on compact graphs. In particular, we consider graph-like networks of inhomogeneous strings with masses attached at the interior vertices. We show that the wave transmitted through a mass is more regular than the incoming wave. Therefore, the regularity of the solution to the initial boundary value problem on an edge depends on the combinatorial distance of this edge from the source, that makes control and inverse problems for such systems more difficult.

We prove the exact controllability of the systems with the optimal number of controls and propose an algorithm recovering the unknown densities of the strings, lengths of the edges, attached masses, and the topology of the graph.

The proofs are based on the boundary control and leaf peeling methods de-
veloped in our previous papers. The boundary control method is a powerful method in inverse theory which uses deep connections between controllability and identifiability of distributed parameter systems and lends itself to straight-forward algorithmic implementations.