Contraction analysis of non-linear distributed systems

Contraction theory is a comparatively recent dynamic analysis and non-linear 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 non-linear distributed dynamics, such as convection-diffusion-reaction processes, Lagrangian and Hamilton–Jacobi dynamics, and optimal controllers and observers. By contrast with stability proofs based on energy dissipation, stability and convergence can be determined for energy-based systems excited by time-varying inputs. The Hamilton–Jacobi–Bellman controller and a similar optimal non-linear observer design are studied based on explicitly computable conditions on the convexity of the cost function. These stability conditions extend the well-known conditions on controllability and observability Grammians for linear time-varying systems, without requiring the unknown transition matrix of the underlying differential dynamics.