In this talk we describe an approach to inverse problems (the so-called Boundary Control method) which is based on deep connections between controllability and identification problems and is applicable to a wide range of linear systems.
As an example of the approach we consider control and inverse problems for differential equations on graphs. We suppose that on each edge of the graph the wave (or heat, or Schrodinger) equation is defined, and standard compatibility conditions are satisfied at the internal vertices. We prove that the system is exactly controllable if the graph is a tree and the control is applied to all (or to all but one) boundary vertices. Otherwise the system is not generally exactly controllable but may be spectrally controllable. We show how to recover a tree (its connectivity and the lengths of the edges together with coefficients of the equation) by given response operator or Weyl matrix function.
We also demonstrate effectiveness of the Boundary Control method on example of a classical problem of signal processing -- the spectral estimation problem.
The talk is based in part on joint work with A. Bulanova and P. Kurasov.
Back to Control Seminars Page.