% Thrust 3: Continuous Verification
\begin{frame}{Thrust 3: Continuous controllers verified using three complementary techniques}

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        \textbf{FIGURE: Continuous Verification Visualization}\\[0.3cm]
        Top: Three mode types with phase portraits\\
        \textbf{Stabilizing} (attractor) | \textbf{Transitory} (flow) | \textbf{Expulsory} (escape)\\[0.3cm]
        Middle: State space diagram showing\\
        reachable sets at boundaries\\[0.3cm]
        Bottom: Barrier certificate illustration\\
        Safe set with vector field showing\\
        all trajectories stay inside or exit correctly
      };
    \end{tikzpicture}
  \end{center}

  %SPEAKER NOTES: See comments below
%
    \textbf{Three Types of Continuous Modes:}
    1. Stabilizing: Stay in current mode (e.g., full-power operation)
    2. Transitory: Drive toward next mode (e.g., startup heatup)
    3. Expulsory: Force to safe mode (e.g., SCRAM)

    \textbf{Three Verification Techniques:}
    1. Reachability Analysis: Compute reachable state sets, verify boundary conditions met
    2. Assume-Guarantee: Local verification, global guarantees. Each mode verified independently
    3. Barrier Certificates: Prove safe set forward invariance, guarantee transitions occur correctly

    \textbf{Key Innovation:} Design continuous controllers after synthesizing automaton
    - Automaton defines transition boundaries
    - Design each mode to satisfy its local transitions
    - Compositional verification avoids intractable global analysis

    Output: Verified continuous modes + discrete automaton = Complete hybrid controller
% (End of speaker notes)
\end{frame}