# Technical Document Extraction: Control Value Analysis
## Chart Title
- **Title**: Control Value Analysis
- **Parameters**:
- \( C_I = 1 \)
- \( C_u = 1 \times 10^{-4} \)
## Axes
- **X-axis (Time)**:
- Label: `Time (t)`
- Range: \( 0 \leq t \leq 5 \)
- Ticks: \( 0, 1, 2, 3, 4, 5 \)
- **Y-axis (Control Value)**:
- Label: `Control value (u_opt)`
- Range: \( 0.0 \leq u_{opt} \leq 0.7 \)
- Ticks: \( 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 \)
## Legend
- **Control Strategies**:
- **ODE**: Solid blue line
- **KOL-δ**: Dashed red line
- **KOL-m**: Dash-dot purple line
## Key Trends and Data Points
1. **ODE (Blue Line)**:
- **Initial Value**: \( u_{opt} = 0.6 \) at \( t = 0 \).
- **Drops**: \( u_{opt} = 0.5 \) at \( t = 1 \).
- **Drops Further**: \( u_{opt} = 0.4 \) at \( t = 2 \).
- **Peaks**: \( u_{opt} = 0.6 \) at \( t = 3 \).
- **Sharp Drop**: \( u_{opt} = 0.15 \) at \( t = 4 \).
- **Stabilizes**: \( u_{opt} = 0.15 \) at \( t = 5 \).
2. **KOL-δ (Red Dashed Line)**:
- **Initial Value**: \( u_{opt} = 0.6 \) at \( t = 0 \).
- **Drops**: \( u_{opt} = 0.5 \) at \( t = 1 \).
- **Drops Further**: \( u_{opt} = 0.4 \) at \( t = 2 \).
- **Rises**: \( u_{opt} = 0.5 \) at \( t = 3 \).
- **Drops**: \( u_{opt} = 0.15 \) at \( t = 4 \).
- **Stabilizes**: \( u_{opt} = 0.15 \) at \( t = 5 \).
3. **KOL-m (Purple Dash-Dot Line)**:
- **Initial Value**: \( u_{opt} = 0.5 \) at \( t = 0 \).
- **Drops**: \( u_{opt} = 0.4 \) at \( t = 1 \).
- **Drops Further**: \( u_{opt} = 0.35 \) at \( t = 2 \).
- **Rises**: \( u_{opt} = 0.5 \) at \( t = 3 \).
- **Drops**: \( u_{opt} = 0.15 \) at \( t = 4 \).
- **Stabilizes**: \( u_{opt} = 0.15 \) at \( t = 5 \).
## Observations
- All strategies exhibit a **sharp drop** in control value at \( t = 4 \), converging to \( u_{opt} = 0.15 \).
- **ODE** and **KOL-δ** start at the highest initial value (\( 0.6 \)), while **KOL-m** begins at \( 0.5 \).
- **KOL-δ** and **KOL-m** show intermediate recovery at \( t = 3 \), whereas **ODE** maintains a higher value until \( t = 4 \).
## Cross-Referenced Legend Accuracy
- **Blue (ODE)**: Matches solid line trajectory.
- **Red (KOL-δ)**: Matches dashed line trajectory.
- **Purple (KOL-m)**: Matches dash-dot line trajectory.
