Adaptative high-gain extended Kalman filter and applications
Adaptative high-gain extended Kalman filter and applications
Adaptative high-gain extended Kalman filter and applications
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tel-00559107, version 1 - 24 Jan 2011<br />
A.3 Uniform Continuity of the Resolvant<br />
Here G(x(s)) denotes a column vector field whose elements are the gi’s of Σ, <strong>and</strong> u denote<br />
the line vector composed of elements ui.<br />
Proof.<br />
First of all, let us write:<br />
x(t) = x0 +<br />
� �<br />
t<br />
f(x(s)) +<br />
Then<br />
�xn(t) − x(t)� ≤<br />
≤<br />
≤<br />
0<br />
i=nu �<br />
gi(x(s))ui(s)<br />
� t<br />
i=1<br />
= x0 + [f(x(s)) + u(s)G(x(s))] ds.<br />
0<br />
��<br />
�<br />
� t<br />
�<br />
�<br />
� [f(xn(s)) + un(s)G(xn(s)) − f(x(s)) − u(s)G(x(s))] ds�<br />
�<br />
��0<br />
� t �<br />
�<br />
� f(xn(s)) + un(s)G(xn(s)) + un(s)G(x (s))<br />
0<br />
−un(s)G(x (s)) − f(x(s)) − u(s)G(x(s)) � ds � �<br />
��<br />
� t �<br />
�<br />
� f(xn(s)) + un(s)G(xn(s)) − f(x(s)) − un(s)G(x (s))<br />
0<br />
� �<br />
�<br />
ds�<br />
�<br />
��<br />
� t �<br />
+ �<br />
� (un(s) − u(s))G(x (s)) � �<br />
�<br />
ds�<br />
� = A + B.<br />
The terms A <strong>and</strong> B are such that:<br />
B ≤<br />
��<br />
�<br />
� θ<br />
�<br />
sup �<br />
� (un(s) − u(s))G(x(s))ds�<br />
�<br />
θ∈[0;T ] 0<br />
� t<br />
� t<br />
A ≤ �f(xn(s)) − f(x(s))� ds + �(G(xn) − G(x)� |un(s)|ds.<br />
0<br />
0<br />
Because un converges ∗−weakly, the sequence (�un�∞ : n ∈ N) is bounded. The Lipschitz<br />
� t<br />
properties of f <strong>and</strong> G give A ≤ m �xn(s) − x(s)� ds.<br />
Therefore<br />
�xn(t) − x(t)� ≤ sup<br />
θ∈[0;T ]<br />
The result is given by Gronwall’s lemma.<br />
0<br />
0<br />
��<br />
�<br />
� θ<br />
�<br />
� t<br />
�<br />
� (un(s) − u(s))G(x(s))ds�<br />
� + m �xn(s) − x(s)� ds.<br />
0<br />
For all θ ∈ [0,T], for all δ > 0, let us consider a subdivision {tj} of [0,T], such that<br />
θ ∈ [ti,ti+1] <strong>and</strong> tj+1 − tj < δ for all j. We have,<br />
��<br />
�<br />
� θ<br />
�<br />
�<br />
� (un(s) − u(s))G(x(s))ds�<br />
� ≤<br />
��<br />
�<br />
� ti<br />
�<br />
�<br />
� (un(s) − u(s))G(x(s))ds�<br />
� +<br />
��<br />
�<br />
� θ<br />
�<br />
�<br />
� (un(s) − u(s))G(x(s))ds�<br />
� .<br />
0<br />
0<br />
127<br />
0<br />
�<br />
ti<br />
ds
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