A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 18

In other words, C T has

a closed range. Although the exact observability is consistent with the one for finite
dimensional systems, there is no general observability test for infinite dimensional
systems. The observability is equivalent to the following inequality
Z T

.C;A/ is exactly observable on Œ0;T if Ker . C T / is f 0 g and

Z T

k . C T x/.s/ k Y ds D k Ce A.T s/ x k Y ds k x k 2 X

(1.26)

0 0

where >0 is a constant which may depend on T . However, in many cases the

inverse of C T may not be bounded. Thus this leads to the following definition of

weak observability:

Definition 1.3. System ( 1.24 ) is approximately observable on Œ0;T (for some

T>0 )if

ker . C T / Df 0 g :

In other words, C T is injective. This

definition has a drawback. Some observable systems can be ill-posed, i.e. the inverse
mapping from the output variable to the estimated state is extremely sensitive to
noise. In this case, studying partial observability makes more sense. In fact, in Kang
( 2011 ) it was proved that Definition 1.1 can be applied to PDEs to quantitatively
measure the observability of a finite dimensional subspace of the state variables.
A Luenberger observer for ( 1.24 ) is an abstract system in the form of

.A;C/ is approximately observable on Œ0;T if

x.t/ D Ax C L. y.t/ y.t//
y.t/ D C x.t/

(1.27)

where is a linear operator. Unlike the finite dimensional case, even

if ( 1.24 ) is exactly observable on some interval

L W Y ! X

, we may not have a convergent
observer ( 1.27 )(see Curtain and Zwart 1995 and references therein). If we define
the error

Œ0;T

e.t/ D x.t/ x.t/ ,then e.t/ approaches zero exponentially as t increases

provided that .A;C/ is exponentially detectable, which means that there exists a

linear operator L W Y ! X such that A C LC generates an exponentially stable

e .A C LC/t .

C 0 -semigroup

When C is a compact operator, a necessary condition for .A;C/ to be detectable

is that the unstable part of the spectrum of consists only of eigenvalues ( Curtain

and Zwart 1995 ). In infinite dimensions, it is impossible to achieve arbitrary
eigenvalue assignment, but some interesting results on partial assignment can be
found in Clarke and Williamson ( 1981 ), Curtain and Zwart ( 1995 ), Russell ( 1968 ),
Sun ( 1981 ), and Rebarber ( 1999 ). In the following, we present a relatively complete
result when

A

C has a finite rank, i.e. Rang .C/ is finite dimensional, a typical case in

engineering problems.
For a given real number ˛ , the spectrum of A can be decomposed into two parts

in the complex plane

˛ D .A/ \f 2 C j Re./ ˛ g
˛ D .A/ \f 2 C j Re./ < ˛ g :

(1.28)