deletions | additions       diff --git a/Control Theory.tex b/Control Theory.tex  index 8ce4665..7ff9b29 100644  --- a/Control Theory.tex  +++ b/Control Theory.tex 
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Now I investigate rudimentary linear control theory.  Standard state-space representation of a linear system has this form:  $\dot{\v{x}}(t) $$\dot{\v{x}}(t)  = A(t) \v{x}(t) + B(t) \v{u}(t)\\  \v{y}(t) \v{u}(t)$$  $$\v{y}(t)  = C(t) \v{x}(t) + D(t) \v{u}(t)$ \v{u}(t)$$  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 above. 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 $A_1 \cdot = \frac{-i}{\hbar}[\mu,\cdot]$.  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: \[ \dot{\v{x}}(t) = \left( \begin{array}{cccc}  0 & 0 & 0 & 0 \\