Theoretical Foundations

This area is the theoretical foundation for the lab’s applied work — the methodology that supplies the guarantees, algorithms, and limits underlying the group’s contributions to sustainable energy, nanoscience, systems biology, and machine learning.

Signature theoretical contributions include:

• Multi-objective and structured controller synthesis — methods where frequency-domain, time-domain, and noise-rejection performance measures are incorporated seamlessly, with provable guarantees. A Matlab-based multi-objective control package was released with collaborators.
• Less-conservative stability analysis — relaxations of existing Lyapunov and frequency-domain criteria for stability and stability regions, including generalized Zames-Falb multipliers and integral quadratic constraints.
• Network structure and causal discovery from time-series data — a unified thread spanning three closely related concerns: recovering topology of dynamical networks (with unobserved nodes and cycle-carrying, non-DAG structures), causal discovery from passive and partial observations (Wiener-filter-based causal recovery, connections between graphical-model notions of causation and observable statistics in networks of dynamical systems), and information-theoretically optimal sample-complexity bounds for learning dynamical DAGs. Recent work shows when frequency-domain analysis outperforms time-domain approaches for causal-structure recovery.
• Distributed consensus and optimization — the first distributed stopping criteria for consensus-type dynamics over networks; a gradient-consensus method with linear convergence; D-DistADMM for distributed optimization in directed graphs.
• Thermodynamics at the small scale — non-equilibrium and fluctuation-dominated thermodynamics of small systems, where classical equilibrium notions break down. Contributions include Landauer-bound results on the energetic cost of erasing memory, methods for realizing information erasure in finite time, quantifying errors in the Jarzynski estimator, mixed entropy power inequalities and log-concavity of equilibrium distributions, and analysis of feedback-controlled stochastic ratchets. These bounds and methods feed directly into the nanoscience and systems-biology programs, where operating regimes are inherently non-equilibrium and stochastic.

The area is supported by NSF (multiple awards on multi-objective control, structured control, less-conservative stability, and information-theoretic limits) and provides the methods used by every applied direction in the lab.