Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form
$$x_{n+1} = Ax_n + Bu_n,$$
where:
$$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 & 0 & -1 \\ 0 & 0 & 1 \end{pmatrix}$$
and:
$$B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ 1 & 0 \end{pmatrix}$$
is to be controlled for $x_0 = 0 ~ to ~ x_3 = [2, 1, 2]^T$ .
Show this target could have been achieved at $x_2$
Solution So far I have caculated the controlability matrix to be $$ C =\begin{pmatrix} 0&0&2&2&6&6\\ 0&1&-1&0&-3&-2\\ 1&0&1&0&1&0 \end{pmatrix}. $$ Thus the system is controlable Now putting Cv=x3 i have the 3 equations $$ 2c+2d+6e+6f=2\\ b-c-3e-2f=1\\ c+e+f=2\\ $$ which i have then put into augmented matrix row echleon form which i have found to be $$ a-d-2e-3f=1\\ b+d+f=2\\ c+d+3e+3e=1\\ $$
How do I now solve with so many unknowns? also can you please check my working so far is correct. many thanks