deletions | additions       diff --git a/Control Theory.tex b/Control Theory.tex  index 54e61cb..8ce4665 100644  --- a/Control Theory.tex  +++ b/Control Theory.tex 
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Standard state-space representation of a linear systemgenerally  has this form: $\dot{\v{x}}(t) = A(t) \v{x}(t) + B(t) \v{u}(t)\\  \v{y}(t) = C(t) \v{x}(t) + D(t) \v{u}(t)$  If the system is LTI (linear and time-invariant), then matrices $A,B,C,D$ will be time-independent. Our hope is to formulate the nonlinear spectroscopy problem in a form suitable for control theory analysis.  Our observation $P_3(t)$ can be mapped to $\v{y}(t)$ as shown in equation (10). above.  A question I have not answered: 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 $A_0, A_1$ such that $A(t) = A_0 + E(t) \cdot A_1$. By linearity of the commutator, applying $A_0$ (in operator form) can behave as $A_0 \cdot = \frac{-i}{\hbar}[H_0,\cdot]$, and applying $A_1$ (in operator form) can behave as $A_1 \cdot = \frac{-i}{\hbar}[\mu,\cdot]$. Now, all matrices are time-independent, but there remains coupling between $\v{x}(t)$ and $E(t)$. In particular, we have the evolution equation: