Thesis (Ph.D.) University of Alaska Fairbanks, 2009The thesis is devoted to control and inverse problems (dynamical and spectral) for systems on graphs and on the half line. In the first part we study the boundary control problems for the wave, heat, and Schrodinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. The exact controllability in L2-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. The null controllability for the heat equation and exact controllability for the Schrodinger equation in arbitrary time interval are obtained. In the second part we consider the in-plane motion of elastic strings on a tree-like network, observed from the 'leaves.' We investigate the inverse problem of recovering not only the physical properties, i.e. the 'optical lengths' of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed. In the third part of the thesis we consider Inverse dynamical and spectral problems for the Schrodinger operator on the half line. Using the connection between dynamical (Boundary Control method) and spectral approaches (due to Krein, Gelfand-Levitan, Simon and Remling), we improved the result on the representation of so-called A---amplitude and derive the "local" version of the classical Gelfand-Levitan equations
