how to regularize an ill posed problem

Hi,

This is my first contribution to this forum

I have some problems to get regression to this equation:

http://www.futura-sciences.com/cgi-b...m%20dt%20=%200

my goal is to estimate the different a,b ans c knowing u and y.

What I know are u and y vectors over time. It would not be difficult to compute the different integrals.

My fist trial was to wright the linear system:

http://www.futura-sciences.com/cgi-b...?b%20=%20A%20x

http://www.futura-sciences.com/cgi-b..._2,c_1,c_2]%27

Like that, the problem seems to be ill posed, in fact the estimation of a, b and c do not converge (the algorithm is unstable). I tried Tikhonov regularization, improved the stability for some parameters but not for all:

http://www.futura-sciences.com/cgi-b...28b-A%20x_0%29

I choosed Q and P ad-hoc.

I think the illness is due to:

- the choice of A and b: why should I wright u= and not http://www.futura-sciences.com/cgi-b...%5Cmathrm%20dt= or y= or somethink else?

- The fact that in A there are some signals that are not independent:some signals are the integrals of others.

How can I regularize this ill problem? or how to wright it as a well posed problem? How to consider the integral relationship? rigorously.

thanks