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If $S_u$ is the set of all solutions to the continuous time problem x(dot) = Ax + Bu and $\phi_h :(R^n)^R to (R^n)^N$ be an operator which maps every state x into the sequence (x(0), x(h),....,(x(ih),...) where h greater than 0.

Let input u be constant on [ih,(i+1)h) and define $u_h$ := $\phi_hu$.

Show that there exist $A_h$ and $B_h$ such that

$x_h$ belongs to $\phi_hS_u$ if and only if $x_h$ solves x(k+1) = $A_h$x(k) +$B_hu_h$

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