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u_(k+1) = u_k +v_k +2p_k

v_(k+1) = 2u_k +v_k +2p_k

p_(k+1) = 3u_k +3v_k + p_k

where k indicates the time index. we wish to apply a four- dimensional data assimilation scheme to determine the vector x_0 at time t_0.

Suppose that we take observations of both u and p together at the two times t_0 and t_1. Determine whether we have enough information to reconstruct the vector x_0 uniquely.

= we can write the dynamical system as

x_(k+1) =F * x_(k)

[1 1 2]

F= [ 2 1 2]

[3 3 1]

Where F is matrix

For observations of u and p together the observation operator is

H = [1 0 0]

[0 0 1]

The observability matrix is

P = [H]

[HF]

HF = [1 1 2]

[3 3 1]

Hence F = [1 0 0]

[0 0 1]

[1 1 2]

[3 3 1]

we can see this is not full rank. hence we do not have sufficient information to reconstruct x_0.

(This is what I have try)