# Technical Document Extraction: Control Value Over Time
## Graph Description
The image depicts a line graph illustrating the **control value (u_opt)** over **time (t)** for three distinct control strategies: **ODE**, **KOL-δ**, and **KOL-m**. The graph includes labeled axes, a legend, and numerical annotations for key data points.
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### **Axes and Labels**
- **X-axis (Time, t):**
- Range: 0 to 5 (discrete intervals).
- Units: Time (t).
- **Y-axis (Control value, u_opt):**
- Range: 0.0 to 0.7.
- Units: Control value (u_opt).
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### **Legend**
- **Colors and Corresponding Strategies:**
- **Blue (solid line):** ODE
- **Red (dashed line):** KOL-δ
- **Purple (dash-dot line):** KOL-m
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### **Key Data Points and Trends**
1. **ODE (Blue Line):**
- **Initial Value (t=0):** 0.7.
- **First Drop (t=1):** 0.3.
- **Second Drop (t=3):** 0.1.
- **Final Value (t=5):** 0.0.
- **Behavior:** Sharp, stepwise decreases at t=1 and t=3.
2. **KOL-δ (Red Dashed Line):**
- **Initial Value (t=0):** 0.2.
- **Rise (t=2):** 0.6.
- **Drop (t=4):** 0.1.
- **Final Value (t=5):** 0.0.
- **Behavior:** Stepwise increase at t=2, followed by a drop at t=4.
3. **KOL-m (Purple Dash-Dot Line):**
- **Initial Value (t=0):** 0.0.
- **Rise (t=2):** 0.7.
- **Drop (t=4):** 0.05.
- **Final Value (t=5):** 0.0.
- **Behavior:** Sharp increase at t=2, followed by a gradual decline.
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### **Parameters**
- **C_I = 1**
- **C_u = 1e-6**
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### **Observations**
- All strategies converge to **u_opt = 0.0** by t=5.
- **ODE** exhibits the most aggressive control adjustments, with the largest initial drop.
- **KOL-m** achieves the highest control value (0.7) but decays rapidly.
- **KOL-δ** balances moderate increases and decreases.
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### **Technical Notes**
- The graph uses **step-like transitions**, indicating discrete control adjustments rather than continuous changes.
- Numerical annotations on the graph confirm exact control values at critical time points.
- The legend and axis labels are explicitly defined, ensuring clarity in interpreting the strategies and metrics.
