Set-Based Value Function Characterization and Neural Approximation of Stabilization Domains for Input-Constrained Discrete-Time Systems

A team of researchers including Mohamed Serry, S. Sivaranjani, and Jun Liu has published a paper introducing a novel framework for a fundamental challenge in control theory: estimating the Domains of Stabilization (DOS) for nonlinear systems with input constraints. The DOS defines the set of initial states from which a system can be driven to a target while respecting control limits. The researchers' key innovation is characterizing this DOS through newly defined value functions that operate on entire sets of states, not just single points. They establish the mathematical properties of these functions and derive the corresponding Bellman-type (or Zubov-type) functional equations that govern them.

Building on this theoretical foundation, the team developed a computational method using Physics-Informed Neural Networks (PINNs). Instead of learning from vast amounts of simulated data, the neural network is trained by directly embedding the derived functional equations into its loss function. This approach ensures the learned value function inherently satisfies the underlying system physics and stability criteria. The trained network can then accurately approximate the DOS and, crucially, synthesize a stabilizing controller for any state within that domain. The framework's effectiveness was validated through two numerical examples, showcasing its potential to solve complex stabilization problems for systems where traditional analytical or computational methods fall short.