deletions | additions       diff --git a/Control Theory.tex b/Control Theory.tex  index 21f7cca..36018bd 100644  --- a/Control Theory.tex  +++ b/Control Theory.tex 
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Our observation $P_3(t)$ can be interpreted as $\v{y}(t)$ as shown above. In particular, $C(t) = C = \begin{matrix}(0 & 0 & 0 & F)\end{matrix}$ and $D(t) = \v{0}$. Can the time dependence within $H(t)$ be modeled as input $\v{u}(t)$ to the system? In general, $H(t) = H_0 + E(t) \cdot \mu$.   Let us define superoperators $A_0, A_1$ $M_0, M_1$  such that $A(t) $M(t)  = A_0 + E(t) \cdot A_1$. M_1$.  By linearity of the commutator, applying $A_0$ $M_0$  (in operator form) can behave as $A_0 $M_0  \cdot = \frac{-i}{\hbar}[H_0,\cdot]$, and applying $A_1$ $M_1$  (in operator form) can behave as $A_1 $M_1  \cdot = \frac{-i}{\hbar}[\mu,\cdot]$. Now, all matrices are time-independent, but there remains coupling between $\v{x}(t)$ and $E(t)$. In particular, we have the evolution equation:  \[ \dot{\v{x}}(t) = \left( \begin{array}{cccc}  0 & 0 & 0 & 0 \\  A_0 M_0  & 0 & 0 & 0 \\ 0 & A_0 M_0  & 0 & 0 \\ 0 & 0 & A_0 M_0  & 0 \\ \end{array} \right) \v{x}(t) + \left( \begin{array}{cccc}  0 & 0 & 0 & 0 \\  A_1 M_1  & 0 & 0 & 0 \\ 0 & A_1 M_1  & 0 & 0 \\ 0 & 0 & A_1 M_1  & 0 \\ \end{array} \right) \v{x}(t) E(t) \]   Bilinear system theory may be well-equipped to handle this coupling.