Dynamic Spectral and Systems-Theoretic Approaches for Inference and Control of Heterogeneous Complex Networks

Networks of nonlinear systems are commonly employed to describe a diverse range of phenomena across physics, engineering, neuroscience, and biology. The undesirable behaviors of such systems, in the form of neurological disorders, power grid failures, or ecological collapses, have spurred significant interest in understanding their dynamic structures and developing effective control strategies. These systems are typically large-scale, consist of heterogeneous units, and are partially observable with limited measurement data, presenting theoretical and computational challenges for control design and connectivity inference. This thesis addresses these challenges by developing novel algorithms for pattern formation in populations of stable limit-cycle oscillators and connectivity reconstruction in complex networks from time-series data. The first part focuses on controlling nonlinear oscillator networks with heterogeneous units under aggregate population-level measurement constraints. The primary contribution is the development of a unified framework that leverages spectral approximation to transform the optimal control problem into a simple convex quadratic program with linear constraints, applicable to both open-loop and feedback control tasks within a model-based setting. We demonstrate its efficacy through numerical simulations and experimental validations on ensembles of electrochemical oscillators. The proposed framework is further extended to enable data-driven learning using aggregated measurements by characterizing the network synchronization patterns using the Fourier coefficients of the population mean. The second part of the thesis addresses the network reconstruction challenges arising from limited measurement data and partial node observations. We first present a data-efficient algorithm by formulating the network inference task as a bilinear optimization problem. An iterative algorithm with sequential initialization is proposed to solve the resulting bilinear program. We then tackle partial observability by integrating time-delay embedding with statistical learning techniques. The performance of the proposed algorithms is compared with existing methods across experimental and simulated datasets, comprising oscillatory, non-oscillatory, and chaotic dynamics. This thesis advances both control strategies for oscillator networks and inference techniques for nonlinear networks, contributing to the manipulation and understanding of complex dynamical networks.