Control problems for the wave and telegrapher's equations on metric graphs

Abstract

The dissertation focuses on control problems for the wave and telegrapher's equations on metric graphs. In the first part, an algorithm is constructed to solve the exact control problems on finite intervals. The algorithm is implemented numerically to solve the exact control problems on finite intervals. Moreover, we developed numerical algorithms for the solution of control problems on metric graphs based on the recent boundary controllability results of wave equations on metric graphs. We presented numerical solutions to shape control problems on quantum graphs. Specifically, we presented the results of numerical experiments involving a three-star graph. Our second part deals with the forward and control problems for the telegrapher's equations on metric graphs. We consider the forward problem on general graphs and develop an algorithm that solves equations with variable resistance, conductance, constant inductance, and constant capacitance. An algorithm is developed to solve the voltage and current control problems on a finite interval for constant inductance and capacitance, and variable resistance and conductance. Numerical results are also presented for this case. Finally, we consider the control problems for the telegrapher's equations on metric graphs. The control problem is considered on tree graphs, i.e. graphs without cycles, with some restrictions on the coefficients. Specifically, we consider equations with constant coefficients that do not depend on the edge. We obtained the necessary and sufficient conditions of the exact controllability and indicate the minimal control time.