Inverse Problem For Partial Differential Networks

Inverse problem of partial di?erential equations is a theoretically chal-lenging topic and has various application backgrounds. It has been intensivelydiscussed in the past ten years. Compared with the identification of the pa-rameter of network systems, there are only a few literatures studying theidentification of the structure of the networks. As far as we know, the physicalstructure of tree-shaped networks can be identified by its spectra. However,it is di?cult to provide a reasonable way to identify the structure of complexnetworks, such as the ones containing parallel edges, loops or circuits. Thereason is that the identification of the structure of complex networks is non-linear and that these systems do not possess exact observability and exactcontrollability. Avdonin[143] regarded this problem as an open question. Inthis thesis, we investigate the structure reconstruction of complex networkscontaining circuits, parallel edges and loops by introducing resolvent method.The main results are as follows:1. We present the model of 1-d complex wave networks with continuousplacements. We first define continuous function on graph, and then describethe dynamical behavior of the networks with wave equations, boundary con-ditions and interior connection conditions and dynamical conditions.2. By the analysis of the corresponding characteristic equations of threebasic networks, namely serially connected wave network, network on a circuit,and star-shaped network, we obtain the number of the edges of the network,and meanwhile give the characteristic equation of general networks.3. We introduce the resolvent method and present the expression of theGreen function matrix of wave networks by construction. We prove that theGreen function matrix and the boundary conditions of any complex networkcan be obtained by appropriately coupling the basic Green function matricesand their corresponding boundary conditions. Then we provide the relation between Green function matrix and edge-edge incident matrix.4. We analyze the algebraic and geometric properties of graphs. We firstgain some physical properties of the graph, such as the number of the vertices,the degree of each vertex, etc, by edge-edge incident matrix. Then we presentthe general procedure for the reconstruction of the shape of networks. Inparticular, given only one group of boundary values, we draw pictures to showthe structure of some complex networks by PAJEK.5. We consider the relation between non-local boundary value problemand the dynamical balance of the networks, which is a generalization of theboundary problem of di?erential operators.