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

TL;DR

This paper introduces a new framework for estimating stabilization domains in nonlinear input-constrained discrete-time systems using value functions and neural networks, enabling accurate domain estimation and controller synthesis.

Contribution

It proposes a novel value function characterization and a physics-informed neural network approach for estimating stabilization domains in nonlinear systems.

Findings

Successfully estimates DOSs for nonlinear systems

Neural network learns value functions satisfying Bellman-type equations

Demonstrates accurate stabilization domain estimation in numerical examples

Abstract

Analyzing nonlinear systems with stabilizable controlled invariant sets (CISs) requires accurate estimation of their domains of stabilization (DOS) together with associated stabilizing controllers. Despite extensive research, estimating DOSs for general nonlinear systems remains challenging due to fundamental theoretical and computational limitations. In this paper, we propose a novel framework for estimating DOSs for controlled input-constrained discrete-time systems. The DOS is characterized via newly introduced value functions defined on metric spaces of compact sets. We establish the fundamental properties of these value functions and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework that learns the value functions by embedding the derived functional equations directly into the training process. The proposed methodology is demonstrated through two numerical examples, illustrating its ability to accurately estimate DOSs and synthesize stabilizing controllers from the learned value functions.