Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
A.3 Uniform Continuity of the Resolvant
– Let x and y be two elements of E, J(x + y) is the associated element of E ∗∗ .
For all f ∈ E ∗ : =< x+ y, f >=< x, f > + < y, f >. Thus
J(x + y) ≡ Jx + Jy.
– If Jx is an identically null application, then < f, x >= 0∀f ∈ E ∗ , which means
that x = 0. The kernel is composed of the null element.
It may happen that J is not a surjective application: E is therefore identified to a subspace
of E ∗∗ .
We now consider a fixed x ∈ E and the application ϕx : E ∗ → R defined as f ↦→ ϕx(f) =<
f, x >. Since x can be any elements of E, we obtain a family of applications: (ϕx)x∈E.
Definition 71
The weak-* topology is the coarser topology that renders all the applications (ϕx)x∈E
continuous.
The weak-∗ topology makes continuous the family of applications F1 =(ϕx)x∈E and the
weak topology makes continuous the family of applications F2 =(ϕξ)ξ∈E∗∗. Since E can
be seen as a subset of E ∗∗ , the family F1 can be seen as a subfamily of F2. Therefore the
topology constructed using the family F1 is coarser than the one obtained from the family
F2.
More details concerning those topologies can be found in [33] (in Moliere’s mother tongue)
or [107] (in Shakespear’s mother tongue).
A.3 Uniform Continuity of the Resolvant
In the proof of some lemmas7 we use the fact that “the weak-* topology on a bounded
set implies uniform continuity of the resolvent”. We explain how it works in the present
subsection. We consider a control affine system of the form:
⎧
⎪⎨
˙x
(Σ)
⎪⎩
x(0)
=
=
�i=p
f(x)+ gi(x)ui,
i=1
x0.
Let PΣ : Dom (PΣ) ⊂ L ∞ ([0; T ] , R nu ) → C 0 ([0; T ] ,X), be the “input to state” mapping
ofΣ , i.e., the mapping that, to any u(.), associates the corresponding state trajectory
(t ∈ [0; T ] ↦→ x(t)).
We endow L ∞ ([0; T ] , R nu ) with the weak-* topology, and C 0 ([0; T ] ,X) with the topology
of uniform convergence. The following lemma is proven in [57].
Lemma 72
PΣ has open domain and is continuous on bounded sets (w.r.t. the topologies above).
The proof of this lemma is reproduced below (refer to [57] for the complete statement).
In order to ease the understanding of this proof we first recall a few facts concerning the
properties of the weak-* topology.
7 Lemmas 33 and 38, and their equivalent in Chapter 5.
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