Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2. φ(s, s) =Id,
3. for all τ ∈ I, φ(t,τ )φ(τ,s)=φ(t, s),
4. φ −1 (t, s) =φ(s, t),
5. ∂
∂
dtφ(t, s) =A(t)φ(t, s) and dsφ(t, s) =−φ(t, s)A(s),
6. in particular: ∂
dt φ−1 (t, s) ′
= −A ′
(s)φ −1 (t, s) ′
,
A.2 Weak-∗ Topology
7. the solution of (A.2) at time t with, x(s) =x0 is given by: x(t) =φ(t, s)x0,
8. the solution of (A.1) at time t with, x(s) =x0 is given by:
A.2 Weak-∗ Topology
� t
x(t) =φ(t, s)x0 + φ(t, v)b(v)dv.
Let E be a Banach space 2 and denote its norm by �.�E. E ∗ is the dual vector space 3 of
E, and E ∗∗ is the dual vector space of E ∗ : the bi-dual space of E. A dual space is embedded
with the dual norm defined as:
�f�E ∗ = sup
x∈E,�x�≤1
|f(x)|.
Therefore E ∗ has the topology induced by the dual norm. The object of this section is to
give the definition of two additional topologies that can be built on the space E ∗ : the weak
and the weak-∗ topologies. The topology described above, induced by the dual norm, is also
called strong topology.
For f ∈ E ∗ and x ∈ E we denote < f, x > instead of f(x). It is called the dual scalar
product.
Topology Associated to a Family of Applications
We consider a set X and a family of topological spaces (Yi)i∈I. For all i ∈ I, let ϕi denote
an application from X to Yi. We want to embed X with the coarser topology that makes all
the applications ϕi continuous. In the following this topology is denoted T.
Let i be an element of I. Let ωi be an open subset of Yi. If the application ϕi is continuous
then ϕ −1
i (ωi) is an open subset of X. Consequently T shall contain the family of subsets of
X defined by ϕ −1
i (ωi) when ωi scans all the open subsets of Yi. This remark applies to all
i ∈ I. We denote (Oλ)λ∈Λ the family composed by all the subsets of the form ϕ −1
i (ωi), where
ωi is an open subset of Yi, and i ∈ I.
2 i.e. A normed vector space, complete with respect to the topology of its norm.
3 The space of all the continuous linear forms on E.
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