Contraction Analysis of Nonlinear Distributed Systems

TL;DR

This paper extends contraction theory to analyze the local and global stability of various nonlinear distributed systems, providing explicit conditions and new insights into energy-based and entropy-producing processes.

Contribution

It introduces contraction analysis for complex distributed dynamics, including convection-diffusion-reaction and Hamilton-Jacobi systems, with explicit stability criteria and novel perspectives on energy and entropy.

Findings

Extended contraction theory to distributed nonlinear systems

Derived explicit stability conditions for Hamilton-Jacobi dynamics

Linked contraction to variational conservation laws and entropy stability

Abstract

Contraction theory is a recently developed dynamic analysis and nonlinear control system design tool based on an exact differential analysis of convergence. This paper extends contraction theory to local and global stability analysis of important classes of nonlinear distributed dynamics, such as convection-diffusion-reaction processes, Lagrangian and Hamilton-Jacobi dynamics, and optimal controllers and observers. The Hamilton-Jacobi-Bellman controller and a similar optimal nonlinear observer design are studied. Explicit stability conditions are given, which extend the well-known conditions on controllability and observability Grammians for linear time-varying systems. Stability of the Hamilton-Jacobi dynamics is assessed by evaluating the Hessian of the system state along system trajectories. In contrast to stability proofs based on energy dissipation,this principle allows to conclude on stability of energy-based systems that are excited by time-varying inputs. In this context, contraction can be regarded as describing new variational conservation laws and the stability of entropy producing processes.