k xk eA(zu)» and

^ ( ^ )

ad- 3x Taking norms ve then find

J _ k (k-l)|| 3x | fy 1 V 9 k - (k- ° "3x X ad JL 3x*' “3x Vk > 1.

ad 3_{la-j-x. Thus, A cannot have a Banach} contradicting the boundedness of

structure.

[As an aside, note that this example also illustrates the

insufficiency of Brocketts Principle, for, as we have already remarked,

2

A(Iu) is isomorphic to a Lie algebra of vector fields on 1R . But from

m a a

Th (3.2.4), x • h is not f.d.c. since the corresponding observation space is infinite dimensional and, in fact, contains !RCx]\lR].

The set up described above has analogies with the problem considered in OmotiEl] and Omori.de la Harpe [1], of classifying those Banach Lie groups acting smoothly on a finite dimensional manifold. Let us assume that $ is an f.d.c. statistic and that the corresponding estimation algebra A is Banach. We denote by it the Brockett homomorphism taking A

A

into the Lie algebra of vector fields on the state manifold, M, of \l>.

If we further suppose that tt is also continuous with respect to the

oo

usual topology on

r

ir(A)

Banach structure. As we remarked previously, without loss of generality we can take the realisation of ip on M to be minimal, and if we also assume that ir(A) is a Lie algebra of complete vector fields, then, by Theorem A of Omori [1], there is a Banach Lie subgroup G of Diff (M) which (by minimality) acts smoothly, effectively and transitively on M. This imposes immediate restrictions on ir(A) as the following

result demonstrates. THEOREM 3.2.5 (Omori [1])

Let G be a connected Banach Lie group acting smoothly* effectively and transitively on a (finite dimensional) manifold M. Then

a) if M is compact, G is finite dimensional

b) if M is non-compact, G is almost solvable, ie the Lie algebra (Jof G contains a solvable, finite codimensional, closed ideal p

(solvability in this case requires that if p q » p, and p^ is defined as

the closure of Cp . ,p .] then3 N<=> s.t. pM .. - (0>).

n— 1 n - 1 N + 1

□

Of course, in the case that the estimation algebra is finite dimensional, Thm (3.2.5b) is an immediate consequence of Levis Theorem that any finite dimensional Lie algebra is the direct sum of a solvable ideal with a semi-simple subalgebra (Jacobson [2]). The full implications of Thm 0.2.5) in the present context have yet to be explored, but Banach Lie groups have been generated by considering parameter estimation algorithms as nonlinear filtering problems (Krishnaprasad, Hazewinkel and Marcus [1], [2], C3]). However, f r o m the above remarks it seems clear that, in general, some weaker topology on the estimation algebra will be found. In some sense, this brings us full circle, since Fliess'

construction of the MacMillan degree is based in turn on the work of

Singer and Sternberg Cl] and Guillemin and Sternberg Cl], who show that any linearly compact Lie algebra possessing a fundamental subalgebra is isomorphic to a Lie algebra of formal vector fields on a finite

dimensional vector space. Without going into too much details, for which we refer to the recent text of Conn Cl], we remark that a sub­ algebra Lq of a complete topological Lie algebra L is fundamental if it

has finite codimension and the induced chain of subalgebras

forms a fundamental system of neighbourhoods of the origin and ,11 t, = {0}. Thus, the topology on L is much weaker than that induced

since L also satisfies a descending chain condition on closed ideals. Other connections can be made and this is clearly an area which could be usefully further researched.

V) We close this section, and the chapter, by remarking that the

class of systems, studied originally by Marcus and Willsky Cl], taking the form

systems it can be shown that statistics of the x process which are f.d.c. do exist - thus they form one of the few known such classes exhibiting truely nonlinear behaviour. It also turns out that the associated estimation algebras have a strong algebraic structure and possess many ideals. This structure has been fully explored in

L.

l ; CX,Y] eL . ^ VYeL)

i50 i

by a norm, however Thm (3.2.5(b)) does have a parallel (Conn Cl], Thm 1.1)

necessary condition derived in Thm (3.1.4) is trivially satisfied by the

dx » Ax dt + B dw dx^ - f(x^)dt + G(x^)dx i - 1 , 2 t (3.2.8) dy - Cx'dt + dv 87

Hazewinkel, Liu and Marcus [ 1] and related papers. (We cannot leave this example without pointing out the obvious: Linear systems are included in the class of systems defined by (3.2.8). In this case, the calculation of the estimation algebra is quite straightforward and it turns out to be both solvable and finite dimensional (Brockett [2])).

Hazewinkel, Liu and Marcus [ I] and related papers. (We cannot leave this example without pointing out the obvious: Linear systems are included in the class of systems defined by (3.2.8). In this case, the calculation of the estimation algebra is quite straightforward and it turns out to be both solvable and finite dimensional (Brockett [ ₂])).