diff --git a/3-research-approach/v3.tex b/3-research-approach/v3.tex
index 3d8656d..6646411 100644
--- a/3-research-approach/v3.tex
+++ b/3-research-approach/v3.tex
@@ -314,9 +314,7 @@ the vector fields at discrete state interfaces makes reachability analysis
 computationally expensive, and analytic solutions often become intractable
 \cite{MANYUS THESIS}.
 
-These issues are circumvented by designing the hybrid system from the bottom up
-with verification in mind. The discrete transitions define each continuous
-control mode's input and output sets clearly \textit{a priori}.
+I circumvent these issues by designing the hybrid system from the bottom up with verification in mind. The discrete transitions define each continuous control mode's input and output sets clearly \textit{a priori}.
 
 Each discrete mode $q_i$ provides
 three key pieces of information for continuous controller design:
@@ -374,9 +372,7 @@ reachable within time horizon $T$:
 \text{Reach}(\mathcal{X}_{entry}, f_i, [0,T]) \cap \mathcal{X}_{exit} \neq \emptyset
 \]
 
-The discrete controller defines clear boundaries in continuous state
-space, making the verification problem for each transitory mode well-posed. The possible initial conditions, target conditions, and safety envelope are all known. The verification task then confirms that the candidate
-continuous controller achieves the objective from all possible starting points.
+The discrete controller defines clear boundaries in continuous state space. This makes the verification problem for each transitory mode well-posed. The possible initial conditions, target conditions, and safety envelope are all known before verification begins. The verification task then confirms that the candidate continuous controller achieves the objective from all possible starting points.
 
 Several tools exist for computing reachable sets of hybrid
 systems, including CORA, Flow*, SpaceEx, and JuliaReach. The choice of tool
@@ -423,12 +419,7 @@ the time derivative of $B$ is non-negative. Geometrically, this means the
 vector field points inward or tangent to the boundary, never outward. If this
 condition holds, no trajectory starting inside $\mathcal{C}$ can ever leave.
 
-Because the design of the discrete controller defines careful boundaries in
-continuous state space, the barrier is known prior to designing the continuous
-controller. This eliminates the search for an appropriate barrier and minimizes
-complication in validating stabilizing continuous control modes. The discrete
-specifications tell us what region must be invariant; the barrier certificate
-confirms that the candidate controller achieves this invariance.
+The discrete controller design defines careful boundaries in continuous state space. Therefore, the barrier is known prior to designing the continuous controller. This eliminates the search for an appropriate barrier. It minimizes complication in validating stabilizing continuous control modes. The discrete specifications tell us what region must be invariant. The barrier certificate then confirms that the candidate controller achieves this invariance.
 
 Finding barrier certificates can be formulated as a
 sum-of-squares (SOS) optimization problem for polynomial systems, or solved