A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 20

output operators see Curtain ( 1984 )and Curtain and Salamon ( 1986 ). Some recent
approaches can be found in Smyshlyaev and Krstic ( 2008 , 2009 )and Li et al. ( 2012 )
and references therein.

1.5.2 Nonlinear Case

Observer design for systems governed by nonlinear PDEs is very challenging. There
are very few results available. Instead of trying to cover a broad class of issues, here
we introduce a new idea that may directly connect to finite dimension observer
design, presented in Sect. 1.2 .
A mathematical description of the long-term behavior of a dynamical system
ultimately is to determine its attractor. However, the global attractor can be quite
complicated geometrically and can attracts solutions at algebraic rate. It has been
found that in many cases the global attractor can be embedded into exponen-
tially attractive finite dimensional manifolds ( Chueshov 2002 ; Chow et al. 1992 ;
Demengel and Ghidaglia 1991 ; Foias and Temam 1977 ; Garcia-archilla et al. 1999 ;
Marion 1989 ; Temam 1997 ). It turns out that inertial manifolds are an appropriate
mathematical tool which has been used in the study of the long-term behavior of
dynamical systems. These are finite-dimensional Lipschitz manifolds, which attract
all the orbits at an exponential rate. Inertial manifolds are positively invariant under
the state dynamics and thus contain the global attractors.
If a system possesses an inertial manifold, the long-time dynamics of the
system can be captured by the finite-dimensional dynamical flow on the manifold
because the inertial manifold exponentially attracts all the orbits of the system.
Hence, inertial manifolds can be used for the reduction of a PDE to a finite
dimensional ODE in which the ! -limit set of the solution of the PDE coincides

with the -limit set of a system of ODEs. In a way, the long-time dynamics of

a PDE with an inertial manifold is completely determined by the solutions of a
system of ODEs in finite dimensions, and one can use the well-established ODE
theory for the qualitative analysis in an infinite dimensional setting. Many infinite-
dimensional systems, including the well-known Navier-Stokes equations, actually
possess inertial manifolds.
Consider an abstract evolution equation of the form

!

d u

dt

(1.31)

C A u D f. u /; u .0/ D u 0 ;

in a Banach space X ,where f is assumed to be continuous from a Banach space E

into another Banach space F , with

E F X I

The injections are continuous and each space is dense in the following one.
Typically, u

.t/ is a function, for instance U.t;x/ with a space variable x and a