New Approach to Inverse Problems for Differential Equation Networks

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The main goal of this project is to solve some important problems on identification, observation and control of differential equation networks (DENs). DENs are metric graphs with differential equations defined on the edges coupled by matching conditions at the vertices. DENs play a fundamental role in many problems of science and engineering. On the atomistic scale, physicists work with quantum dots and quantum waveguides. On the nanoscale, chemical engineers are interested in mechanical properties of nanorods. On the mesoscale, fiber networks modeled by strings and beams are crucial for tissue engineering, whereas on the macroscale grid structures involving pipes, plates and shells appear in civil and mechanical engineering. Control and inverse theories of DENs constitute an important part of the rapidly developing area of analysis on graphs. Being tremendously important for applications, these theories have not been, however, sufficiently developed. Control, observation and identification problems were studied almost exclusively for trees, i.e. graphs without cycles, although general DENs, which may contain cycles, are more important for applications. The project provides excellent training opportunities for graduate and undergraduate students.In this project, the principal investigator (PI) and his collaborators will develop a unified approach to control, observation, and identification of general DENs. This approach is based on the Boundary Control (BC) method in inverse theory, which uses deep connections between observability and identifiability of dynamical systems. The main results of the project will be: (a) uniqueness and stability conditions for solving inverse problems for DENs and effective algorithms for identifying the systems' parameters; and (b) controllability and observability conditions of the dynamical systems described by DENs and the methods for constructing the corresponding control and observation. The central idea is a proper choice of observations in the form of directional derivatives of the solutions to PDEs on graphs at some boundary and internal vertices to guarantee observability and identifiability of the corresponding systems. For general DENs, we will specify the sets of vertices where sensors and/or actuators should be situated. Another component of our approach is a new effective leaf-peeling (LP) method for solving inverse problems for differential equations on trees that has been recently developed by the PI and his collaborators. In the current project we plan to extend this method to general graphs with cycles. Based on the BC and LP methods, numerical algorithms for solving inverse and control problems for general DENs will be developed and extensive numerical experiments will be performed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.