Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
where f : R p → R is compactly supported.
We have, for t>0:
and:
Hence,
f(z − tε) =f(z) −
∂f
(z − τε)=
dxi
∂f
(z) −
∂xi
f(z − ε) =f(z) −
p�
i=1
εi
∂f
(z)+
∂xi
Since f is compactly supported, we get :
p�
i=1
p�
j=1
εi
p�
i,j=1
εi
B.2 Proofs of the Technical Lemmas
� t ∂f
(z − τε)dτ,
∂xi
0
� τ ∂2f (z − θε)dθ.
∂xi∂xj
0
εiεj
|f(z) − f(z − ε) − df (z)ε| ≤ M
2
� 1 � τ ∂2f (z − θε)dθdτ.
∂xi∂xj
0
0
p�
|εiεj|,
where M = sup x | ∂2 f
∂xi∂xj (x)|.
Now we take f = ˜ bk, and we use the facts that ˜ bk depends only on x1, ..., xk, and that
θ ≥ 1:
This gives the result.
i,j=1
| ∂2˜bk (x)| ≤ θ
∂xi∂xj
k−1 | ∂2bk (∆
∂xi∂xj
−1 x)|.
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