  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.
